11 Team 4 - Lab 2

Lab 2 MARSS models

Author

Madison Heller-Shipley, Dylan Hubl

Data

We are examining the sockeye population within the Middle Columbia River System. Within this system there are four major population groups. The Cascades, John Day, Walla Walla, and Yakima. The John Day group has the longest running dataset with records reaching back to 1959. The other major population groups generally start their datasets in the 1980’s. A noteable exception is the Umatilla River within the Walla Walla group which also has data beginning in the 1960s. In general all of the salmon running times occur in the summer in the Middle Columbia River System.

Code

library(tidyr)
library(ggplot2)
library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.2     ✔ readr     2.1.4
✔ forcats   1.0.0     ✔ stringr   1.5.0
✔ lubridate 1.9.2     ✔ tibble    3.2.1
✔ purrr     1.0.1     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

Code

library(dplyr)
library(forecast)

Registered S3 method overwritten by 'quantmod':
  method            from
  as.zoo.data.frame zoo

Code

library(MARSS)
library(corrplot)

corrplot 0.92 loaded

Code

library(knitr)
load(here::here("Lab-2", "Data_Images", "columbia-river.rda"))

We are only interested in the rivers in the Middle Columbia River Unit

Code

#plot the unique Rivers in Middle Columbia
dat <- columbia.river
esuname <- unique(dat$esu_dps)
years<-length(unique(dat$spawningyear))
plotesu <- function(esuname){
  df <- dat %>% subset(esu_dps %in% esuname)
  ggplot(df, aes(x=spawningyear, y=log(value), color=majorpopgroup)) + 
    geom_point(size=1, na.rm = TRUE) + 
    theme(strip.text.x = element_text(size = 8)) +
    theme(axis.text.x = element_text(size = 8, angle = 90)) +
    facet_wrap(~esapopname) +
    ggtitle(paste0(esuname, collapse="\n"))
}
#plot the unique Rivers in Middle Columbia
plotesu(esuname[1])

Next, the data are arranged so the columns are the years and rows are unique rivers

Code

esuname <- esuname[1]
dat <- columbia.river %>% 
  subset(esu_dps == esuname) %>% # get only this ESU
  mutate(log.spawner = log(value)) %>% # create a column called log.spawner
  dplyr::select(esapopname, spawningyear, log.spawner) %>% # get just the columns that I need
  pivot_wider(names_from = "esapopname", values_from = "log.spawner") %>% 
  column_to_rownames(var = "spawningyear") %>% # make the years rownames
  as.matrix() %>% # turn into a matrix with year down the rows
  t() # make time across the columns
# MARSS complains if I don't do this
dat[is.na(dat)] <- NA
any(is.null(dat))

[1] FALSE

Code

any(is.infinite(dat))

[1] TRUE

Code

dat[is.infinite(dat)] <- NA

Let’s take a look at the Middle Columbia River area and formulate some hypotheses:

Code

here::here("Lab-2", "Team-4", "Middle Columbia River sockeye.png") |>
  knitr::include_graphics()

General Questions

Each group has the same general tasks, but you will adapt them as you work on the data.

Create estimates of spawner abundance for all missing years and provide estimates of the decline from the historical abundance.
Evaluate support for the major population groups. Are the populations in the groups more correlated than outside the groups?
Evaluate the evidence of cycling in the data.

Data Notes

Make some assumptions about underlying population structure. This can help you fill in missing data areas.

Adult run timing (when they’re coming into fresh water, look at run timing–any correlation?)

John Day Data set spans the entire time period, and we will look at the appropriatness of drawing inference from these data to fill in other missing values.

Methods

Address the following in your methods

Describe your assumptions about the x and how the data time series are related to x.
- How are the x and y (data) related? 1 x for 1 y or will you assume 1 x for all y or 1 x for each major population group? How will you choose?
- What will you assume about the U for the x’s?
- What will you assume about the Q matrix?
Write out your assumptions as different models in matrix form, fit each and then compare these with AIC or AICc.
Do your estimates differ depending on the assumptions you make about the structure of the data, i.e. you assumptions about the x’s, Q, and U.

Hypotheses

There were four main hypotheses explored in this modeling exercise.

Hypothesis 1: All underlying states are the same and one underlying population.
Hypothesis 2: There are four underlying states, each associated with one of the main distinct population centers (DPC), the Cascades, John Day, Walla Walla, and Yakima tributaries.
Hypothesis 3: There are two underlying states, one representing the northern area (Walla Walla and Yakima) and on representing the southern area (John Day and Cascades).
Hypothesis 4: There are two underlying states, Yakama and the rest of the areas. Salmon swim eastward to a bend in the river where salmon can choose to go north to the Yakama DPC, or south to other DPCs.

For Hypothesis 1, only one model was tested that assumed the Q matrix was diagonal and equal. We only tested this as a baseline for simplicity and time sake, as it is the model we had the least amount of confidence in (and was primarily used for conceptualization and initial MARSS model testing). For Hypotheses 2-4 four sub-hypotheses based on the Q matrix were tested.

Hypotheses:

X.1 = Diagonal and Equal
X.2 = Diagonal and Unequal
X.3 = Equal variance and covariance
X.4 = Unconstrained

This allowed us to get a better idea of the impacts of changing the amount of correlation in the process errors for each of these systems.

Other Assumptions

You can assume that R="diagonal and equal" and A="scaling". Assume that “historical” means the earliest years available for your group.

States

Your abundance estimate is the “x” or “state” estimates.

Pick best Hypothesis

We will compare AICs, all models should be comparable.

Evidence of cycling

We will see which hypothesis performs the best, and then explore cycling assumptions with a simple cycling model, and some variant on periodicity with our best performing model to see if we can improve fits and AICc.

Tips

Assumptions

tsSmooth(fit)

where fit is from fit <- MARSS()

plotting

Estimate of the mean of the spawner counts based on your x model.

autoplot(fit, plot.type="fitted.ytT")

diagnostics

autoplot(fit, plot.type="residuals")

Results

Hypothesis 1

Hypothesis 1 assumes that there is a single hidden state (X) for each stream (n=15) in the time series. The Q matrix for the variance of process errors is “diagonal and equal” meaning each state (x) model has the same variance but they are not correlated to each other.

\[ \text{Hypothesis One}: \begin{bmatrix} y_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6\\ y_7\\ y_8\\ y_9\\ y_{10}\\ y_{11}\\ y_{12}\\ y_{13}\\ y_{14}\\ y_{15}\\ \end{bmatrix}_t= \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ \end{bmatrix}* \begin{bmatrix} x_1\\ \end{bmatrix}_t+ \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6\\ a_7\\ a_8\\ a_9\\ a_{10}\\ a_{11}\\ a_{12}\\ a_{13}\\ a_{14}\\ a_{15}\\ \end{bmatrix}+ \begin{bmatrix} w_1\\ w_2\\ w_3\\ w_4\\ w_5\\ w_6\\ w_7\\ w_8\\ w_9\\ w_{10}\\ w_{11}\\ w_{12}\\ w_{13}\\ w_{14}\\ w_{15}\\ \end{bmatrix}_t \] \[ \text{Where }w \sim MVN \begin{pmatrix} \text{0,}\begin{bmatrix} R \end{bmatrix} \end{pmatrix} \]

Code

mod.list1 <- list(
  U = "unequal", #each of the rivers are estimated separately (different U)
  R = "diagonal and equal", #Process errors are all assumed to be the same 
  Q = "diagonal and equal" #Observation error 
)

m1 <- MARSS(dat, model=mod.list1, method="BFGS")

Success! Converged in 56 iterations.
Function MARSSkfas used for likelihood calculation.

MARSS fit is
Estimation method: BFGS 
Estimation converged in 56 iterations. 
Log-likelihood: -603.7266 
AIC: 1271.453   AICc: 1274.985   
 
                                                                                              Estimate
R.diag                                                                                         0.18209
U.X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer                   -0.05594
U.X.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                   -0.01546
U.X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                          -0.02689
U.X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer  -0.01958
U.X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer              -0.01838
U.X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                 -0.01218
U.X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                  -0.03634
U.X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                  -0.00526
U.X.Steelhead (Middle Columbia River DPS) Touchet River - summer                              -0.03313
U.X.Steelhead (Middle Columbia River DPS) Umatilla River - summer                              0.00777
U.X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          -0.03239
U.X.Steelhead (Middle Columbia River DPS) Naches River - summer                                0.03833
U.X.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                 0.02195
U.X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                             0.02892
U.X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer                 0.05429
Q.diag                                                                                         0.10296
x0.X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer                   8.74764
x0.X.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                   6.59490
x0.X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                          7.21538
x0.X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer  8.27871
x0.X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer              7.06973
x0.X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                 7.83286
x0.X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                  8.59051
x0.X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                  6.73248
x0.X.Steelhead (Middle Columbia River DPS) Touchet River - summer                              7.34999
x0.X.Steelhead (Middle Columbia River DPS) Umatilla River - summer                             7.16269
x0.X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          8.00358
x0.X.Steelhead (Middle Columbia River DPS) Naches River - summer                               5.16366
x0.X.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                5.81032
x0.X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                            4.27677
x0.X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer                3.12784
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

The model converged! Let’s take a look at the plots:

Code

autoplot(m1)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model doesn’t perform very well in areas that lack data, and, related, some of the QQ plots don’t hold assumptions of normality. This makes sense, given that stream missing data have nothing to inform them. In the states plots, the areas with missing data are characterized by confidence intervals that balloon out. Let’s look at the abundance estimates for this model.

Code

print(fit1_smooth<-tsSmooth(m1))

                                                                                     .rownames
1                    X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
2                    X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
3                    X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
4                    X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
5                    X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
6                    X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
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10                   X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
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14                   X.Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer
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128                  X.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer
129                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
130                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
131                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
132                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
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183                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
184                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
185                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
186                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
187                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
188                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
189                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
190                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
191                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
192                         X.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter
193 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
194 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
195 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
196 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
197 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
198 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
199 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
200 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
201 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
202 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
203 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
204 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
205 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
206 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
207 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
208 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
209 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
210 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
211 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
212 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
213 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
214 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
215 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
216 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
217 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
218 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
219 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
220 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
221 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
222 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
223 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
224 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
225 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
226 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
227 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
228 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
229 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
230 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
231 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
232 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
233 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
234 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
235 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
236 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
237 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
238 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
239 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
240 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
241 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
242 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
243 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
244 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
245 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
246 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
247 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
248 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
249 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
250 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
251 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
252 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
253 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
254 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
255 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
256 X.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer
257             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
258             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
259             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
260             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
261             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
262             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
263             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
264             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
265             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
266             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
267             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
268             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
269             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
270             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
271             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
272             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
273             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
274             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
275             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
276             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
277             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
278             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
279             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
280             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
281             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
282             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
283             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
284             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
285             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
286             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
287             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
288             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
289             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
290             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
291             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
292             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
293             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
294             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
295             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
296             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
297             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
298             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
299             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
300             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
301             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
302             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
303             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
304             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
305             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
306             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
307             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
308             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
309             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
310             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
311             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
312             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
313             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
314             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
315             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
316             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
317             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
318             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
319             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
320             X.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer
321                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
322                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
323                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
324                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
325                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
326                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
327                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
328                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
329                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
330                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
331                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
332                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
333                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
334                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
335                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
336                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
337                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
338                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
339                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
340                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
341                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
342                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
343                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
344                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
345                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
346                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
347                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
348                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
349                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
350                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
351                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
352                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
353                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
354                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
355                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
356                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
357                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
358                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
359                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
360                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
361                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
362                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
363                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
364                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
365                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
366                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
367                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
368                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
369                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
370                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
371                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
372                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
373                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
374                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
375                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
376                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
377                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
378                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
379                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
380                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
381                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
382                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
383                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
384                X.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer
385                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
386                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
387                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
388                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
389                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
390                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
391                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
392                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
393                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
394                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
395                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
396                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
397                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
398                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
399                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
400                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
401                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
402                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
403                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
404                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
405                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
406                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
407                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
408                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
409                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
410                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
411                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
412                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
413                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
414                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
415                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
416                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
417                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
418                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
419                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
420                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
421                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
422                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
423                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
424                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
425                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
426                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
427                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
428                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
429                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
430                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
431                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
432                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
433                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
434                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
435                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
436                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
437                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
438                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
439                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
440                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
441                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
442                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
443                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
444                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
445                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
446                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
447                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
448                 X.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer
449                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
450                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
451                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
452                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
453                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
454                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
455                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
456                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
457                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
458                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
459                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
460                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
461                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
462                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
463                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
464                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
465                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
466                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
467                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
468                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
469                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
470                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
471                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
472                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
473                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
474                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
475                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
476                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
477                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
478                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
479                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
480                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
481                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
482                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
483                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
484                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
485                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
486                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
487                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
488                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
489                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
490                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
491                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
492                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
493                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
494                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
495                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
496                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
497                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
498                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
499                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
500                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
501                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
502                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
503                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
504                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
505                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
506                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
507                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
508                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
509                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
510                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
511                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
512                 X.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer
513                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
514                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
515                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
516                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
517                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
518                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
519                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
520                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
521                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
522                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
523                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
524                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
525                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
526                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
527                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
528                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
529                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
530                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
531                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
532                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
533                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
534                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
535                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
536                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
537                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
538                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
539                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
540                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
541                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
542                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
543                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
544                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
545                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
546                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
547                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
548                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
549                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
550                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
551                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
552                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
553                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
554                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
555                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
556                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
557                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
558                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
559                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
560                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
561                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
562                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
563                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
564                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
565                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
566                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
567                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
568                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
569                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
570                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
571                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
572                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
573                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
574                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
575                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
576                             X.Steelhead (Middle Columbia River DPS) Touchet River - summer
577                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
578                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
579                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
580                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
581                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
582                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
583                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
584                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
585                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
586                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
587                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
588                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
589                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
590                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
591                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
592                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
593                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
594                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
595                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
596                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
597                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
598                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
599                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
600                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
601                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
602                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
603                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
604                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
605                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
606                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
607                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
608                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
609                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
610                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
611                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
612                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
613                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
614                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
615                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
616                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
617                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
618                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
619                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
620                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
621                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
622                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
623                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
624                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
625                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
626                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
627                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
628                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
629                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
630                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
631                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
632                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
633                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
634                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
635                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
636                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
637                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
638                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
639                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
640                            X.Steelhead (Middle Columbia River DPS) Umatilla River - summer
641                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
642                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
643                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
644                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
645                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
646                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
647                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
648                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
649                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
650                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
651                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
652                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
653                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
654                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
655                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
656                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
657                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
658                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
659                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
660                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
661                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
662                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
663                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
664                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
665                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
666                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
667                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
668                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
669                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
670                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
671                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
672                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
673                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
674                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
675                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
676                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
677                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
678                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
679                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
680                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
681                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
682                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
683                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
684                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
685                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
686                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
687                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
688                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
689                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
690                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
691                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
692                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
693                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
694                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
695                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
696                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
697                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
698                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
699                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
700                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
701                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
702                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
703                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
704                         X.Steelhead (Middle Columbia River DPS) Walla Walla River - summer
705                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
706                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
707                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
708                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
709                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
710                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
711                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
712                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
713                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
714                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
715                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
716                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
717                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
718                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
719                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
720                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
721                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
722                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
723                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
724                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
725                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
726                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
727                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
728                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
729                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
730                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
731                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
732                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
733                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
734                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
735                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
736                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
737                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
738                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
739                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
740                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
741                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
742                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
743                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
744                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
745                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
746                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
747                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
748                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
749                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
750                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
751                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
752                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
753                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
754                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
755                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
756                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
757                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
758                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
759                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
760                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
761                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
762                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
763                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
764                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
765                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
766                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
767                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
768                              X.Steelhead (Middle Columbia River DPS) Naches River - summer
769                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
770                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
771                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
772                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
773                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
774                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
775                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
776                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
777                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
778                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
779                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
780                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
781                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
782                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
783                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
784                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
785                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
786                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
787                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
788                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
789                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
790                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
791                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
792                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
793                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
794                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
795                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
796                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
797                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
798                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
799                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
800                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
801                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
802                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
803                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
804                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
805                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
806                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
807                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
808                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
809                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
810                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
811                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
812                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
813                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
814                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
815                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
816                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
817                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
818                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
819                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
820                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
821                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
822                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
823                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
824                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
825                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
826                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
827                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
828                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
829                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
830                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
831                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
832                               X.Steelhead (Middle Columbia River DPS) Satus Creek - summer
833                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
834                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
835                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
836                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
837                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
838                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
839                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
840                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
841                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
842                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
843                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
844                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
845                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
846                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
847                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
848                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
849                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
850                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
851                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
852                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
853                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
854                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
855                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
856                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
857                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
858                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
859                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
860                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
861                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
862                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
863                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
864                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
865                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
866                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
867                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
868                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
869                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
870                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
871                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
872                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
873                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
874                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
875                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
876                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
877                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
878                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
879                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
880                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
881                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
882                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
883                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
884                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
885                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
886                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
887                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
888                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
889                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
890                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
891                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
892                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
893                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
894                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
895                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
896                           X.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer
897               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
898               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
899               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
900               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
901               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
902               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
903               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
904               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
905               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
906               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
907               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
908               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
909               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
910               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
911               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
912               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
913               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
914               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
915               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
916               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
917               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
918               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
919               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
920               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
921               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
922               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
923               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
924               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
925               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
926               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
927               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
928               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
929               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
930               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
931               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
932               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
933               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
934               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
935               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
936               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
937               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
938               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
939               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
940               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
941               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
942               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
943               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
944               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
945               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
946               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
947               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
948               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
949               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
950               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
951               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
952               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
953               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
954               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
955               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
956               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
957               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
958               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
959               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
960               X.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer
     t .estimate       .se
1    1  8.691656 0.3159698
2    2  8.635672 0.4397937
3    3  8.579687 0.5298536
4    4  8.523703 0.6015113
5    5  8.467719 0.6607810
6    6  8.411734 0.7107686
7    7  8.355750 0.7533240
8    8  8.299766 0.7896498
9    9  8.243781 0.8205738
10  10  8.187797 0.8466880
11  11  8.131813 0.8684264
12  12  8.075828 0.8861112
13  13  8.019844 0.8999813
14  14  7.963860 0.9102111
15  15  7.907875 0.9169226
16  16  7.851891 0.9201926
17  17  7.795907 0.9200579
18  18  7.739922 0.9165169
19  19  7.683938 0.9095300
20  20  7.627954 0.8990167
21  21  7.571969 0.8848513
22  22  7.515985 0.8668549
23  23  7.460001 0.8447826
24  24  7.404016 0.8183047
25  25  7.348032 0.7869767
26  26  7.292048 0.7501911
27  27  7.236063 0.7070967
28  28  7.180079 0.6564522
29  29  7.124095 0.5963372
30  30  7.068110 0.5234990
31  31  7.012126 0.4315426
32  32  6.956142 0.3035649
33  33  6.915766 0.2655813
34  34  6.732872 0.2560580
35  35  6.572340 0.2538192
36  36  6.527248 0.2533019
37  37  6.728658 0.2531829
38  38  6.999058 0.2531555
39  39  7.348308 0.2531493
40  40  7.605265 0.2531478
41  41  7.868378 0.2531475
42  42  8.176820 0.2531474
43  43  8.520770 0.2531474
44  44  8.474517 0.2531474
45  45  8.305105 0.2531474
46  46  8.028577 0.2531474
47  47  7.812987 0.2531474
48  48  7.739228 0.2531474
49  49  7.634498 0.2531474
50  50  7.240628 0.2531474
51  51  7.192400 0.2531475
52  52  7.199210 0.2531479
53  53  7.104001 0.2531494
54  54  6.919095 0.2531563
55  55  6.722789 0.2531863
56  56  6.733826 0.2533165
57  57  6.492532 0.2538825
58  58  6.097799 0.2563313
59  59  5.664832 0.2667259
60  60  5.390382 0.3079011
61  61  5.334443 0.4447108
62  62  5.278505 0.5483906
63  63  5.222566 0.6353714
64  64  5.166628 0.7118015
65   1  6.579336 0.3131180
66   2  6.563771 0.4315582
67   3  6.548207 0.5143917
68   4  6.532643 0.5771584
69   5  6.517078 0.6259245
70   6  6.501514 0.6637830
71   7  6.485950 0.6925251
72   8  6.470386 0.7132537
73   9  6.454821 0.7266549
74  10  6.439257 0.7331306
75  11  6.423693 0.7328645
76  12  6.408129 0.7258491
77  13  6.392564 0.7118850
78  14  6.377000 0.6905507
79  15  6.361436 0.6611330
80  16  6.345872 0.6224872
81  17  6.330307 0.5727481
82  18  6.314743 0.5086719
83  19  6.299179 0.4238049
84  20  6.283615 0.3010494
85  21  6.204919 0.2649225
86  22  6.261238 0.2559012
87  23  6.365831 0.2537828
88  24  6.474940 0.2532935
89  25  6.405207 0.2531810
90  26  6.551341 0.2531551
91  27  6.658610 0.2531492
92  28  6.667251 0.2531478
93  29  6.732766 0.2531475
94  30  6.597180 0.2531474
95  31  6.346813 0.2531474
96  32  6.156689 0.2531474
97  33  5.949872 0.2531474
98  34  5.884095 0.2531474
99  35  5.603839 0.2531474
100 36  5.605120 0.2531474
101 37  5.581602 0.2531474
102 38  5.594359 0.2531474
103 39  5.922303 0.2531474
104 40  6.187605 0.2531474
105 41  6.294526 0.2531474
106 42  6.504959 0.2531474
107 43  6.721727 0.2531474
108 44  6.880454 0.2531474
109 45  6.903478 0.2531474
110 46  6.676269 0.2531474
111 47  6.547154 0.2531474
112 48  6.460341 0.2531474
113 49  6.501280 0.2531474
114 50  6.455777 0.2531474
115 51  6.446384 0.2531475
116 52  6.659948 0.2531479
117 53  6.727761 0.2531494
118 54  6.562663 0.2531563
119 55  6.497915 0.2531863
120 56  6.456434 0.2533165
121 57  6.445779 0.2538825
122 58  6.221046 0.2563313
123 59  5.934862 0.2667259
124 60  5.666902 0.3079011
125 61  5.651444 0.4447108
126 62  5.635986 0.5483906
127 63  5.620527 0.6353714
128 64  5.605069 0.7118015
129  1  7.188531 0.3150822
130  2  7.161682 0.4372392
131  3  7.134832 0.5250752
132  4  7.107983 0.5940154
133  5  7.081134 0.6500994
134  6  7.054284 0.6964399
135  7  7.027435 0.7348826
136  8  7.000586 0.7666165
137  9  6.973736 0.7924479
138 10  6.946887 0.8129398
139 11  6.920038 0.8284884
140 12  6.893188 0.8393685
141 13  6.866339 0.8457602
142 14  6.839490 0.8477652
143 15  6.812640 0.8454145
144 16  6.785791 0.8386717
145 17  6.758942 0.8274292
146 18  6.732092 0.8115002
147 19  6.705243 0.7906014
148 20  6.678394 0.7643253
149 21  6.651544 0.7320930
150 22  6.624695 0.6930740
151 23  6.597846 0.6460400
152 24  6.570996 0.5890740
153 25  6.544147 0.5189154
154 26  6.517298 0.4291419
155 27  6.490448 0.3027818
156 28  6.634471 0.2653758
157 29  6.560891 0.2560091
158 30  6.210184 0.2538078
159 31  6.068435 0.2532993
160 32  6.082425 0.2531823
161 33  5.969837 0.2531554
162 34  6.030775 0.2531492
163 35  5.880911 0.2531478
164 36  5.925324 0.2531475
165 37  5.864481 0.2531474
166 38  5.988624 0.2531474
167 39  6.019972 0.2531474
168 40  6.045262 0.2531474
169 41  6.418824 0.2531474
170 42  6.604498 0.2531474
171 43  6.655490 0.2531474
172 44  6.795376 0.2531474
173 45  6.666707 0.2531474
174 46  6.503024 0.2531474
175 47  6.117310 0.2531474
176 48  5.875039 0.2531474
177 49  5.615623 0.2531474
178 50  5.547053 0.2531474
179 51  5.867085 0.2531475
180 52  6.123900 0.2531479
181 53  6.110039 0.2531494
182 54  6.119708 0.2531563
183 55  5.987904 0.2531863
184 56  5.996395 0.2533165
185 57  5.853908 0.2538825
186 58  5.590120 0.2563313
187 59  5.317465 0.2667259
188 60  5.602762 0.3079011
189 61  5.575875 0.4447108
190 62  5.548987 0.5483906
191 63  5.522099 0.6353714
192 64  5.495211 0.7118015
193  1  8.259119 0.2221660
194  2  8.058096 0.2463731
195  3  8.116728 0.2516069
196  4  8.016463 0.2527943
197  5  7.966398 0.2530668
198  6  8.036891 0.2531311
199  7  8.192235 0.2531532
200  8  8.419730 0.2531905
201  9  8.403391 0.2533388
202 10  8.264098 0.2539803
203 11  8.139259 0.2567525
204 12  8.093156 0.2684832
205 13  7.848559 0.3144774
206 14  7.603962 0.2685305
207 15  7.571734 0.2569511
208 16  7.636954 0.2548210
209 17  7.880725 0.2569511
210 18  7.765620 0.2685305
211 19  7.554003 0.3144774
212 20  7.342386 0.2684832
213 21  6.723080 0.2567525
214 22  7.242345 0.2539803
215 23  7.472749 0.2533390
216 24  7.522900 0.2531914
217 25  7.616205 0.2531575
218 26  7.773046 0.2531497
219 27  8.051488 0.2531479
220 28  8.453809 0.2531475
221 29  8.593435 0.2531474
222 30  8.353578 0.2531474
223 31  7.962602 0.2531474
224 32  7.681123 0.2531474
225 33  7.408600 0.2531474
226 34  7.286807 0.2531474
227 35  7.011374 0.2531474
228 36  6.802505 0.2531474
229 37  6.829606 0.2531474
230 38  6.934470 0.2531474
231 39  6.992887 0.2531474
232 40  7.122526 0.2531474
233 41  7.615459 0.2531474
234 42  8.135890 0.2531474
235 43  8.332176 0.2531474
236 44  8.311774 0.2531474
237 45  7.870268 0.2531474
238 46  7.428389 0.2531474
239 47  7.125203 0.2531474
240 48  7.078981 0.2531474
241 49  7.293004 0.2531474
242 50  7.372139 0.2531474
243 51  7.681981 0.2531474
244 52  7.601553 0.2531475
245 53  7.743323 0.2531479
246 54  7.813801 0.2531494
247 55  7.600138 0.2531563
248 56  7.654528 0.2531863
249 57  7.410658 0.2533165
250 58  7.155115 0.2538825
251 59  7.021085 0.2563313
252 60  6.928960 0.2667259
253 61  7.085496 0.3079011
254 62  7.065920 0.4447108
255 63  7.046344 0.5483906
256 64  7.026768 0.6353714
257  1  7.051343 0.2221660
258  2  7.112453 0.2463731
259  3  7.211758 0.2516069
260  4  7.267042 0.2527943
261  5  7.512075 0.2530668
262  6  7.604493 0.2531311
263  7  7.641460 0.2531532
264  8  7.941480 0.2531904
265  9  7.897163 0.2533387
266 10  7.614877 0.2539800
267 11  7.534156 0.2567512
268 12  7.499663 0.2684779
269 13  7.410189 0.3144580
270 14  7.320716 0.2684779
271 15  7.149286 0.2567512
272 16  7.062082 0.2539800
273 17  7.090551 0.2533390
274 18  7.101203 0.2531914
275 19  7.118619 0.2531575
276 20  6.820593 0.2531497
277 21  6.430810 0.2531479
278 22  6.640506 0.2531475
279 23  6.681823 0.2531474
280 24  6.796261 0.2531474
281 25  6.973124 0.2531474
282 26  7.207683 0.2531474
283 27  7.532809 0.2531474
284 28  7.780381 0.2531474
285 29  7.850574 0.2531474
286 30  7.742287 0.2531474
287 31  7.289721 0.2531474
288 32  7.152401 0.2531474
289 33  6.996002 0.2531474
290 34  6.979407 0.2531474
291 35  6.617352 0.2531474
292 36  6.444774 0.2531474
293 37  6.028790 0.2531474
294 38  6.034403 0.2531474
295 39  6.082977 0.2531474
296 40  6.244447 0.2531474
297 41  6.204171 0.2531474
298 42  6.387833 0.2531474
299 43  6.545988 0.2531474
300 44  6.770800 0.2531474
301 45  6.594988 0.2531474
302 46  6.324980 0.2531474
303 47  6.139288 0.2531474
304 48  5.983290 0.2531474
305 49  6.319974 0.2531474
306 50  6.574005 0.2531474
307 51  6.650738 0.2531475
308 52  6.735095 0.2531479
309 53  6.875515 0.2531494
310 54  6.952096 0.2531563
311 55  7.022939 0.2531863
312 56  6.925135 0.2533165
313 57  6.704235 0.2538825
314 58  6.263279 0.2563313
315 59  5.949780 0.2667259
316 60  5.966395 0.3079011
317 61  5.948018 0.4447108
318 62  5.929641 0.5483906
319 63  5.911264 0.6353714
320 64  5.892888 0.7118015
321  1  7.820679 0.3023624
322  2  7.808493 0.3997036
323  3  7.796307 0.4527911
324  4  7.784121 0.4766482
325  5  7.771935 0.4756933
326  6  7.759749 0.4497684
327  7  7.747563 0.3939747
328  8  7.735377 0.2916461
329  9  7.517953 0.2626906
330 10  7.225499 0.2561758
331 11  7.129936 0.2572742
332 12  7.382255 0.2686155
333 13  7.274922 0.3145088
334 14  7.167588 0.2684916
335 15  7.074256 0.2567545
336 16  7.204796 0.2539808
337 17  7.409228 0.2533391
338 18  7.574038 0.2531915
339 19  7.609828 0.2531575
340 20  7.341054 0.2531497
341 21  6.818029 0.2531479
342 22  6.859296 0.2531475
343 23  6.988835 0.2531474
344 24  7.008091 0.2531475
345 25  7.000157 0.2531479
346 26  7.114192 0.2531497
347 27  7.515984 0.2531575
348 28  7.773062 0.2531914
349 29  7.804943 0.2533390
350 30  7.742294 0.2539800
351 31  7.447863 0.2567512
352 32  7.255404 0.2684779
353 33  7.282018 0.3144580
354 34  7.308633 0.2684779
355 35  6.969515 0.2567512
356 36  6.780301 0.2539800
357 37  6.514511 0.2533390
358 38  6.441385 0.2531914
359 39  6.389640 0.2531575
360 40  6.484015 0.2531497
361 41  6.757357 0.2531479
362 42  6.965800 0.2531475
363 43  7.118771 0.2531474
364 44  7.305190 0.2531474
365 45  7.011525 0.2531474
366 46  6.655813 0.2531474
367 47  6.256903 0.2531474
368 48  6.245187 0.2531474
369 49  6.670519 0.2531474
370 50  7.109496 0.2531474
371 51  7.639284 0.2531474
372 52  7.943691 0.2531477
373 53  8.274332 0.2531486
374 54  8.412520 0.2531525
375 55  8.466956 0.2531697
376 56  8.457270 0.2532447
377 57  8.256454 0.2535705
378 58  7.908768 0.2549837
379 59  7.654664 0.2610443
380 60  7.587259 0.2859265
381 61  7.453893 0.3754742
382 62  7.320527 0.4116098
383 63  7.187161 0.4087532
384 64  7.053795 0.3659925
385  1  8.554094 0.2974317
386  2  8.517681 0.3846211
387  3  8.481268 0.4223781
388  4  8.444854 0.4241170
389  5  8.408441 0.3903195
390  6  8.372028 0.3095571
391  7  8.524204 0.3303831
392  8  8.676379 0.2728382
393  9  8.513488 0.2578089
394 10  8.252235 0.2542415
395 11  8.235370 0.2534656
396 12  8.025026 0.2535086
397 13  7.958284 0.2544808
398 14  7.834793 0.2588462
399 15  7.613719 0.2770817
400 16  7.617137 0.3454105
401 17  7.620554 0.3454103
402 18  7.623972 0.2770808
403 19  7.544213 0.2588430
404 20  7.449623 0.2544675
405 21  7.308100 0.2534513
406 22  7.550371 0.2532173
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951 55  6.031737 0.2538825
952 56  6.112719 0.2563313
953 57  6.222687 0.2667259
954 58  6.276179 0.3079011
955 59  6.330472 0.4447108
956 60  6.384765 0.5483906
957 61  6.439058 0.6353714
958 62  6.493351 0.7118015
959 63  6.547643 0.7807854
960 64  6.601936 0.8441507

Finally, let’s look at the correlation plot.

Code

Q1 <- coef(m1, type = "matrix")$Q
corrmat1 <- diag(1/sqrt(diag(Q1))) %*% Q1 %*% diag(1/sqrt(diag(Q1)))
corrplot(corrmat1)

Hypothesis 2

Hypothesis two assumes that the four main DPCs form separate sub-populations. In this hypothesis we are utilizing 4 separate underlying states to model the observations from each of the main population groups. For each of the four models that we are comparing for this hypothesis, we are allowing the random walks to drift independent of one another based on the supplied “U” matrix.

\[ \text{Hypothesis Two}: \begin{bmatrix} y_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6\\ y_7\\ y_8\\ y_9\\ y_{10}\\ y_{11}\\ y_{12}\\ y_{13}\\ y_{14}\\ y_{15}\\ \end{bmatrix}_t= \begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1\\ \end{bmatrix}* \begin{bmatrix} x_1\\ x_2\\ x_3\\ x_4\\ \end{bmatrix}_t+ \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6\\ a_7\\ a_8\\ a_9\\ a_{10}\\ a_{11}\\ a_{12}\\ a_{13}\\ a_{14}\\ a_{15}\\ \end{bmatrix}+ \begin{bmatrix} w_1\\ w_2\\ w_3\\ w_4\\ w_5\\ w_6\\ w_7\\ w_8\\ w_9\\ w_{10}\\ w_{11}\\ w_{12}\\ w_{13}\\ w_{14}\\ w_{15}\\ \end{bmatrix}_t \]

\[ \text{Where }w \sim MVN \begin{pmatrix} \text{0,}\begin{bmatrix} R \end{bmatrix} \end{pmatrix} \]

Set up the U and Z matrices

Code

#give U values names to make it easier to read results
#this hypothesis has 4 hidden states based on major groups
U_mat2 <- matrix(c("Cascades","JohnDay","Walla","Yakima"),4,1)
#make Z matrix correspond to 4 hidden states
Z_mat2 <- matrix(c(rep(c(1,0,0,0),3),
                  rep(c(0,1,0,0),5),
                  rep(c(0,0,1,0),3),
                  rep(c(0,0,0,1),4)),15,4, byrow=TRUE)

###Hypothesis 2.1 The Q matrix for the variance of process errors is “diagonal and equal” meaning each state (x) model has the same process error but they are not correlated to each other.

Code

mod.list2.1 <- list(
  U = U_mat2,
  R = "diagonal and equal",
  Q = "diagonal and equal",
  Z = Z_mat2
)
m2.1 <- MARSS(dat, model = mod.list2.1)

Success! abstol and log-log tests passed at 125 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 125 iterations. 
Log-likelihood: -514.2653 
AIC: 1070.531   AICc: 1072.048   
 
                                                                               Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer      -0.88896
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter             -1.12228
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer -0.77474
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer    -0.34028
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer     -0.10687
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer     -1.17898
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                 1.76516
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer              0.63708
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                   -0.02276
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer               -0.84219
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer   -1.64351
R.diag                                                                          0.19787
U.Cascades                                                                     -0.02384
U.JohnDay                                                                      -0.01612
U.Walla                                                                        -0.00168
U.Yakima                                                                        0.04521
Q.diag                                                                          0.13118
x0.X1                                                                           7.55382
x0.X2                                                                           8.05157
x0.X3                                                                           5.48625
x0.X4                                                                           4.63376
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

The model converged with a better AICc than Hypothesis 1.

Code

autoplot(m2.1)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

There are larged ballooned CIs on the missing data in hidden states fitted CI have the same balloon shaped CIs on missing data.There also appears to be some cyclic structure in the residuals. Yakima (X4) is the only system with a positive drift value on the hidden state’s random walk. There is only one Q value output as all hidden states have the same one and there is no covariance/correlation between states. This model likely will not be a top contender when we evaluate based on AICs. Let’s look at the estimates.

Code

print(fit2.1_smooth<-tsSmooth(m2.1))

    .rownames  t .estimate       .se
1          X1  1  7.534868 0.3533879
2          X1  2  7.515915 0.4869991
3          X1  3  7.496963 0.5803924
4          X1  4  7.478010 0.6511104
5          X1  5  7.459058 0.7060002
6          X1  6  7.440105 0.7485518
7          X1  7  7.421153 0.7807852
8          X1  8  7.402200 0.8039424
9          X1  9  7.383247 0.8187938
10         X1 10  7.364295 0.8257877
11         X1 11  7.345342 0.8251240
12         X1 12  7.326390 0.8167838
13         X1 13  7.307437 0.8005275
14         X1 14  7.288485 0.7758574
15         X1 15  7.269532 0.7419349
16         X1 16  7.250580 0.6974110
17         X1 17  7.231627 0.6400774
18         X1 18  7.212675 0.5660548
19         X1 19  7.193722 0.4674817
20         X1 20  7.174770 0.3225870
21         X1 21  7.083255 0.2839633
22         X1 22  7.142996 0.2753813
23         X1 23  7.254560 0.2735359
24         X1 24  7.371268 0.2728823
25         X1 25  7.283177 0.2714025
26         X1 26  7.440895 0.2644156
27         X1 27  7.553439 0.2272896
28         X1 28  7.715273 0.2227543
29         X1 29  7.766615 0.2222125
30         X1 30  7.408640 0.2218900
31         X1 31  7.160209 0.2194956
32         X1 32  7.094684 0.1968361
33         X1 33  6.930902 0.1950968
34         X1 34  6.933032 0.1949705
35         X1 35  6.555657 0.1949614
36         X1 36  6.605411 0.1949608
37         X1 37  6.624167 0.1949607
38         X1 38  6.775403 0.1949607
39         X1 39  7.102769 0.1949607
40         X1 40  7.245681 0.1949607
41         X1 41  7.543980 0.1949607
42         X1 42  7.822339 0.1949607
43         X1 43  8.104223 0.1949607
44         X1 44  8.232257 0.1949607
45         X1 45  8.096734 0.1949607
46         X1 46  7.768808 0.1949607
47         X1 47  7.424908 0.1949607
48         X1 48  7.323222 0.1949607
49         X1 49  7.237882 0.1949607
50         X1 50  6.903469 0.1949607
51         X1 51  7.111413 0.1949607
52         X1 52  7.449938 0.1949607
53         X1 53  7.416309 0.1949607
54         X1 54  7.228192 0.1949607
55         X1 55  7.026148 0.1949608
56         X1 56  7.165223 0.1949614
57         X1 57  7.064999 0.1949706
58         X1 58  6.651242 0.1950975
59         X1 59  6.132436 0.1968451
60         X1 60  6.122926 0.2196067
61         X1 61  6.099084 0.4235670
62         X1 62  6.075241 0.5573069
63         X1 63  6.051399 0.6646600
64         X1 64  6.027557 0.7569378
65         X2  1  8.035449 0.2091516
66         X2  2  7.890334 0.2181526
67         X2  3  7.985295 0.1967221
68         X2  4  8.076309 0.1950062
69         X2  5  7.909146 0.1938239
70         X2  6  8.186306 0.1784508
71         X2  7  8.418015 0.1918147
72         X2  8  8.790832 0.1644013
73         X2  9  8.439890 0.1632643
74         X2 10  7.700598 0.1632237
75         X2 11  8.006868 0.1632995
76         X2 12  8.193261 0.1653154
77         X2 13  8.115058 0.2114055
78         X2 14  7.872122 0.1653355
79         X2 15  7.589920 0.1638086
80         X2 16  7.669605 0.1760634
81         X2 17  8.022818 0.1760818
82         X2 18  7.963683 0.1642156
83         X2 19  8.017074 0.1754812
84         X2 20  7.633942 0.1637037
85         X2 21  6.602794 0.1632375
86         X2 22  7.373261 0.1632196
87         X2 23  7.536066 0.1632190
88         X2 24  7.583630 0.1632189
89         X2 25  7.499459 0.1632189
90         X2 26  7.532227 0.1632189
91         X2 27  8.201664 0.1632189
92         X2 28  8.542266 0.1632189
93         X2 29  8.654963 0.1632190
94         X2 30  8.389947 0.1632196
95         X2 31  7.449096 0.1632374
96         X2 32  7.215709 0.1637027
97         X2 33  7.142982 0.1754571
98         X2 34  7.594092 0.1637027
99         X2 35  7.118468 0.1632374
100        X2 36  7.058651 0.1632196
101        X2 37  6.633692 0.1632190
102        X2 38  6.804781 0.1632189
103        X2 39  6.715705 0.1632189
104        X2 40  6.740319 0.1632189
105        X2 41  7.015840 0.1632189
106        X2 42  7.531813 0.1632189
107        X2 43  7.800550 0.1632189
108        X2 44  8.228155 0.1632189
109        X2 45  7.748717 0.1632189
110        X2 46  7.158482 0.1632189
111        X2 47  6.801163 0.1632189
112        X2 48  6.634353 0.1632189
113        X2 49  7.258688 0.1632189
114        X2 50  7.556570 0.1632189
115        X2 51  8.039138 0.1632189
116        X2 52  7.807702 0.1632189
117        X2 53  8.117254 0.1632189
118        X2 54  8.346447 0.1632189
119        X2 55  8.172071 0.1632189
120        X2 56  8.206859 0.1632189
121        X2 57  7.902473 0.1632193
122        X2 58  7.372746 0.1632271
123        X2 59  6.995562 0.1634337
124        X2 60  7.171945 0.1687572
125        X2 61  7.221308 0.2735464
126        X2 62  7.154227 0.3632258
127        X2 63  7.087146 0.3554352
128        X2 64  7.020064 0.2408969
129        X3  1  5.484574 0.3432659
130        X3  2  5.482903 0.4571238
131        X3  3  5.481231 0.5228689
132        X3  4  5.479560 0.5577819
133        X3  5  5.477889 0.5675814
134        X3  6  5.476217 0.5536025
135        X3  7  5.474546 0.5139084
136        X3  8  5.472875 0.4416188
137        X3  9  5.471203 0.3150421
138        X3 10  5.388338 0.2822278
139        X3 11  5.599148 0.2750307
140        X3 12  5.763582 0.2735346
141        X3 13  5.870631 0.2732274
142        X3 14  5.872518 0.2731645
143        X3 15  5.838919 0.2731517
144        X3 16  5.871408 0.2731490
145        X3 17  5.826006 0.2731485
146        X3 18  5.802248 0.2731484
147        X3 19  5.682038 0.2731483
148        X3 20  5.849675 0.2731483
149        X3 21  5.822694 0.2731482
150        X3 22  5.673967 0.2731477
151        X3 23  5.388804 0.2731453
152        X3 24  5.195295 0.2731333
153        X3 25  5.365512 0.2730747
154        X3 26  5.700978 0.2727881
155        X3 27  5.948093 0.2713835
156        X3 28  6.063164 0.2644121
157        X3 29  6.184791 0.2272895
158        X3 30  6.197868 0.2227575
159        X3 31  5.911918 0.2222422
160        X3 32  5.778123 0.2221553
161        X3 33  5.668230 0.2218837
162        X3 34  5.980248 0.2194949
163        X3 35  5.941698 0.1968361
164        X3 36  5.861117 0.1950968
165        X3 37  5.805540 0.1949705
166        X3 38  5.773789 0.1949614
167        X3 39  5.717150 0.1949608
168        X3 40  5.793967 0.1949607
169        X3 41  5.788374 0.1949607
170        X3 42  5.979663 0.1949607
171        X3 43  6.186346 0.1949607
172        X3 44  6.420112 0.1949607
173        X3 45  6.225710 0.1949607
174        X3 46  6.059635 0.1949607
175        X3 47  6.090954 0.1949607
176        X3 48  5.975718 0.1949607
177        X3 49  6.021047 0.1949607
178        X3 50  6.060137 0.1949607
179        X3 51  6.222050 0.1949607
180        X3 52  6.564561 0.1949607
181        X3 53  6.512909 0.1949607
182        X3 54  6.339326 0.1949607
183        X3 55  6.201491 0.1949607
184        X3 56  5.957020 0.1949608
185        X3 57  6.221388 0.1949618
186        X3 58  6.028905 0.1949756
187        X3 59  5.534490 0.1951669
188        X3 60  5.410836 0.1977954
189        X3 61  5.252750 0.2311222
190        X3 62  5.382095 0.3133692
191        X3 63  5.380416 0.4789386
192        X3 64  5.378736 0.6004700
193        X4  1  4.686722 0.3552359
194        X4  2  4.739687 0.4923476
195        X4  3  4.792652 0.5904582
196        X4  4  4.845617 0.6670058
197        X4  5  4.898582 0.7288166
198        X4  6  4.951547 0.7794047
199        X4  7  5.004512 0.8208477
200        X4  8  5.057477 0.8544771
201        X4  9  5.110442 0.8811881
202        X4 10  5.163407 0.9015958
203        X4 11  5.216372 0.9161215
204        X4 12  5.269337 0.9250423
205        X4 13  5.322302 0.9285199
206        X4 14  5.375267 0.9266154
207        X4 15  5.428232 0.9192955
208        X4 16  5.481197 0.9064289
209        X4 17  5.534162 0.8877745
210        X4 18  5.587127 0.8629571
211        X4 19  5.640093 0.8314249
212        X4 20  5.693058 0.7923768
213        X4 21  5.746023 0.7446313
214        X4 22  5.798988 0.6863756
215        X4 23  5.851953 0.6146287
216        X4 24  5.904918 0.5238764
217        X4 25  5.957883 0.4014295
218        X4 26  6.010848 0.1945915
219        X4 27  6.420953 0.1776865
220        X4 28  6.535918 0.1767801
221        X4 29  6.608963 0.1767337
222        X4 30  6.577876 0.1767313
223        X4 31  5.983138 0.1767312
224        X4 32  5.621145 0.1767312
225        X4 33  5.761960 0.1767312
226        X4 34  6.209068 0.1767312
227        X4 35  5.841370 0.1767312
228        X4 36  5.430386 0.1767312
229        X4 37  5.572386 0.1767312
230        X4 38  5.432808 0.1767312
231        X4 39  5.792098 0.1767312
232        X4 40  5.882392 0.1767312
233        X4 41  5.940084 0.1767312
234        X4 42  6.308583 0.1767312
235        X4 43  6.863374 0.1767312
236        X4 44  7.142373 0.1767312
237        X4 45  6.819503 0.1767312
238        X4 46  6.896515 0.1767312
239        X4 47  6.957085 0.1767312
240        X4 48  6.591429 0.1767312
241        X4 49  6.463417 0.1767312
242        X4 50  6.944253 0.1767312
243        X4 51  7.160561 0.1767312
244        X4 52  7.526500 0.1767312
245        X4 53  7.587335 0.1767312
246        X4 54  7.551244 0.1767314
247        X4 55  7.311761 0.1767339
248        X4 56  7.221288 0.1767832
249        X4 57  7.325625 0.1777475
250        X4 58  7.254980 0.1956808
251        X4 59  7.300189 0.4116709
252        X4 60  7.345397 0.5483200
253        X4 61  7.390605 0.6571429
254        X4 62  7.435813 0.7503458
255        X4 63  7.481022 0.8331871
256        X4 64  7.526230 0.9085057

And let’s look at corrplot, it should be very familiar.

Code

Q2.1 <- coef(m2.1, type = "matrix")$Q
corrmat2.1 <- diag(1/sqrt(diag(Q2.1))) %*% Q2.1 %*% diag(1/sqrt(diag(Q2.1)))
corrplot(corrmat2.1)

Code

#As expected output displays only diagonal as we told it diagonal and equal

The Confidence Intervals in the sections of the underlying states (x) were very large in the sections where data was missing. This was also reflected in the plots showing the fitted values. Some structuring may also be present in the residuals. Six of the river systems have multiple significant lags when examining the ACF plots indicating the residuals display autocorrelation. The corrplot is fairly uninformative as we forced the model to be equal variance with no correlation. One interesting thing to note is that we allowed the “U” values to varying independently of each other and the Yakima group is the only one with a positive U value, indicating it is the only system with positive growth or at least increasing number of adults counted over time.

Hypothesis 2.2

The Q matrix for the variance of process errors is “diagonal and unequal” meaning each of the four underlying states’ process error can be different but they are not correlated to each other.

Code

mod.list2.2 <- list(
  U = U_mat2,
  R = "diagonal and equal",
  Q = "diagonal and unequal",
  Z = Z_mat2
)
m2.2 <- MARSS(dat, model = mod.list2.2)

Success! abstol and log-log tests passed at 129 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 129 iterations. 
Log-likelihood: -504.4194 
AIC: 1056.839   AICc: 1058.819   
 
                                                                               Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer      -0.88829
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter             -1.11858
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer -0.77509
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer    -0.34298
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer     -0.10888
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer     -1.18031
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                 1.76167
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer              0.63346
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                   -0.02276
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer               -0.84219
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer   -1.64351
R.diag                                                                          0.19726
U.Cascades                                                                     -0.02343
U.JohnDay                                                                      -0.01653
U.Walla                                                                        -0.00221
U.Yakima                                                                        0.04458
Q.(X1,X1)                                                                       0.07752
Q.(X2,X2)                                                                       0.24414
Q.(X3,X3)                                                                       0.02414
Q.(X4,X4)                                                                       0.12012
x0.X1                                                                           7.55844
x0.X2                                                                           8.07225
x0.X3                                                                           5.59910
x0.X4                                                                           4.67230
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

Our model converged with an AICc value that is a little better than the model where Q was diagonal and equal, indicating that allowing Q to vary improved the model fits to data.

Code

autoplot(m2.2)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

We still see large CIs on the missing data in hidden states 1 (Cascades) and 4 (Yakima). The CI are tight in the John Day region, which is the most data rich stream. Again Yakima is the only positive drift value on the hidden state’s random walk

The 4 varied Q values indicate that the hidden states are uncorrelated, and the variance of the state variables varies over time.

Next let’s looks at the estimates:

Code

print(fit2.1_smooth<-tsSmooth(m2.2))

    .rownames  t .estimate       .se
1          X1  1  7.539618 0.2717632
2          X1  2  7.520799 0.3746837
3          X1  3  7.501981 0.4467636
4          X1  4  7.483163 0.5014828
5          X1  5  7.464345 0.5441043
6          X1  6  7.445527 0.5773139
7          X1  7  7.426708 0.6026695
8          X1  8  7.407890 0.6211336
9          X1  9  7.389072 0.6333094
10         X1 10  7.370254 0.6395560
11         X1 11  7.351436 0.6400471
12         X1 12  7.332617 0.6347961
13         X1 13  7.313799 0.6236579
14         X1 14  7.294981 0.6063081
15         X1 15  7.276163 0.5821918
16         X1 16  7.257344 0.5504201
17         X1 17  7.238526 0.5095632
18         X1 18  7.219708 0.4571917
19         X1 19  7.200890 0.3886788
20         X1 20  7.182072 0.2929088
21         X1 21  7.123374 0.2584322
22         X1 22  7.170362 0.2474501
23         X1 23  7.259085 0.2440030
24         X1 24  7.352898 0.2424470
25         X1 25  7.318359 0.2401126
26         X1 26  7.443612 0.2328375
27         X1 27  7.543419 0.2061996
28         X1 28  7.666284 0.2010398
29         X1 29  7.686870 0.2000137
30         X1 30  7.403871 0.1993380
31         X1 31  7.183774 0.1964497
32         X1 32  7.092335 0.1796554
33         X1 33  6.941601 0.1774389
34         X1 34  6.903543 0.1771592
35         X1 35  6.607482 0.1771241
36         X1 36  6.627420 0.1771197
37         X1 37  6.656647 0.1771192
38         X1 38  6.804410 0.1771191
39         X1 39  7.092490 0.1771191
40         X1 40  7.260833 0.1771191
41         X1 41  7.540863 0.1771191
42         X1 42  7.807117 0.1771191
43         X1 43  8.059232 0.1771191
44         X1 44  8.168827 0.1771191
45         X1 45  8.049133 0.1771191
46         X1 46  7.760990 0.1771191
47         X1 47  7.455880 0.1771191
48         X1 48  7.332575 0.1771191
49         X1 49  7.231704 0.1771191
50         X1 50  6.977626 0.1771191
51         X1 51  7.134176 0.1771191
52         X1 52  7.396681 0.1771191
53         X1 53  7.377520 0.1771191
54         X1 54  7.222773 0.1771192
55         X1 55  7.055098 0.1771201
56         X1 56  7.125472 0.1771268
57         X1 57  7.008854 0.1771808
58         X1 58  6.641913 0.1776107
59         X1 59  6.203423 0.1810048
60         X1 60  6.152227 0.2060977
61         X1 61  6.128796 0.3463999
62         X1 62  6.105364 0.4444204
63         X1 63  6.081933 0.5244293
64         X1 64  6.058502 0.5937530
65         X2  1  8.055723 0.2398874
66         X2  2  7.848129 0.2448836
67         X2  3  7.983823 0.2146833
68         X2  4  8.104605 0.2136039
69         X2  5  7.846381 0.2129334
70         X2  6  8.182530 0.1924605
71         X2  7  8.410569 0.2118197
72         X2  8  8.872309 0.1759647
73         X2  9  8.479183 0.1753562
74         X2 10  7.596300 0.1753471
75         X2 11  8.012026 0.1753648
76         X2 12  8.227970 0.1765268
77         X2 13  8.156821 0.2403171
78         X2 14  7.877381 0.1765307
79         X2 15  7.549274 0.1756190
80         X2 16  7.637322 0.1911718
81         X2 17  8.070774 0.1911763
82         X2 18  7.966406 0.1758506
83         X2 19  8.081063 0.1908555
84         X2 20  7.689671 0.1755955
85         X2 21  6.425470 0.1753505
86         X2 22  7.419428 0.1753467
87         X2 23  7.556739 0.1753466
88         X2 24  7.598937 0.1753466
89         X2 25  7.482021 0.1753466
90         X2 26  7.467596 0.1753466
91         X2 27  8.233900 0.1753466
92         X2 28  8.577882 0.1753466
93         X2 29  8.708894 0.1753466
94         X2 30  8.460605 0.1753467
95         X2 31  7.379961 0.1753505
96         X2 32  7.184331 0.1755954
97         X2 33  7.086551 0.1908504
98         X2 34  7.682779 0.1755954
99         X2 35  7.089818 0.1753505
100        X2 36  7.087923 0.1753467
101        X2 37  6.577592 0.1753466
102        X2 38  6.824937 0.1753466
103        X2 39  6.703410 0.1753466
104        X2 40  6.710137 0.1753466
105        X2 41  6.990589 0.1753466
106        X2 42  7.555775 0.1753466
107        X2 43  7.799782 0.1753466
108        X2 44  8.327716 0.1753466
109        X2 45  7.770519 0.1753466
110        X2 46  7.133484 0.1753466
111        X2 47  6.768676 0.1753466
112        X2 48  6.550578 0.1753466
113        X2 49  7.283122 0.1753466
114        X2 50  7.549515 0.1753466
115        X2 51  8.109135 0.1753466
116        X2 52  7.761052 0.1753466
117        X2 53  8.126335 0.1753466
118        X2 54  8.389887 0.1753466
119        X2 55  8.160942 0.1753466
120        X2 56  8.245701 0.1753466
121        X2 57  7.929720 0.1753467
122        X2 58  7.353755 0.1753474
123        X2 59  6.937139 0.1753939
124        X2 60  7.184528 0.1783962
125        X2 61  7.264739 0.3176519
126        X2 62  7.181316 0.4669904
127        X2 63  7.097893 0.4525327
128        X2 64  7.014470 0.2470222
129        X3  1  5.596926 0.1484153
130        X3  2  5.594755 0.1995468
131        X3  3  5.592585 0.2310312
132        X3  4  5.590414 0.2503930
133        X3  5  5.588243 0.2603508
134        X3  6  5.586072 0.2619792
135        X3  7  5.583901 0.2554377
136        X3  8  5.581730 0.2400592
137        X3  9  5.579560 0.2139464
138        X3 10  5.575234 0.1996585
139        X3 11  5.647579 0.1921399
140        X3 12  5.716863 0.1882799
141        X3 13  5.769406 0.1863258
142        X3 14  5.789716 0.1853438
143        X3 15  5.792911 0.1848519
144        X3 16  5.802250 0.1846053
145        X3 17  5.788315 0.1844803
146        X3 18  5.773336 0.1844141
147        X3 19  5.736582 0.1843735
148        X3 20  5.759222 0.1843378
149        X3 21  5.734432 0.1842891
150        X3 22  5.675933 0.1842028
151        X3 23  5.592057 0.1840352
152        X3 24  5.549555 0.1837014
153        X3 25  5.617143 0.1830314
154        X3 26  5.745615 0.1816805
155        X3 27  5.862811 0.1789401
156        X3 28  5.944761 0.1733119
157        X3 29  6.013003 0.1614307
158        X3 30  6.018720 0.1566917
159        X3 31  5.924943 0.1547501
160        X3 32  5.862021 0.1536797
161        X3 33  5.823623 0.1523772
162        X3 34  5.900737 0.1494569
163        X3 35  5.893233 0.1416235
164        X3 36  5.859302 0.1391632
165        X3 37  5.828451 0.1384092
166        X3 38  5.809543 0.1381800
167        X3 39  5.798298 0.1381104
168        X3 40  5.840623 0.1380894
169        X3 41  5.883994 0.1380830
170        X3 42  5.997959 0.1380810
171        X3 43  6.120615 0.1380805
172        X3 44  6.223265 0.1380803
173        X3 45  6.173706 0.1380802
174        X3 46  6.109412 0.1380802
175        X3 47  6.098971 0.1380802
176        X3 48  6.063546 0.1380802
177        X3 49  6.089148 0.1380802
178        X3 50  6.137736 0.1380803
179        X3 51  6.236627 0.1380805
180        X3 52  6.373344 0.1380811
181        X3 53  6.366199 0.1380830
182        X3 54  6.281800 0.1380895
183        X3 55  6.182006 0.1381110
184        X3 56  6.054499 0.1381817
185        X3 57  6.055865 0.1384149
186        X3 58  5.911230 0.1391820
187        X3 59  5.666768 0.1416844
188        X3 60  5.538463 0.1496475
189        X3 61  5.449798 0.1733542
190        X3 62  5.461998 0.2129934
191        X3 63  5.459787 0.2636493
192        X3 64  5.457575 0.3060322
193        X4  1  4.724073 0.3399321
194        X4  2  4.775846 0.4711450
195        X4  3  4.827618 0.5650409
196        X4  4  4.879391 0.6383062
197        X4  5  4.931164 0.6974727
198        X4  6  4.982937 0.7459032
199        X4  7  5.034709 0.7855857
200        X4  8  5.086482 0.8177946
201        X4  9  5.138255 0.8433867
202        X4 10  5.190027 0.8629508
203        X4 11  5.241800 0.8768905
204        X4 12  5.293573 0.8854715
205        X4 13  5.345346 0.8888490
206        X4 14  5.397118 0.8870824
207        X4 15  5.448891 0.8801407
208        X4 16  5.500664 0.8678998
209        X4 17  5.552436 0.8501308
210        X4 18  5.604209 0.8264772
211        X4 19  5.655982 0.7964149
212        X4 20  5.707755 0.7591830
213        X4 21  5.759527 0.7136601
214        X4 22  5.811300 0.6581281
215        X4 23  5.863073 0.5897664
216        X4 24  5.914845 0.5033747
217        X4 25  5.966618 0.3870628
218        X4 26  6.018391 0.1926633
219        X4 27  6.416585 0.1752796
220        X4 28  6.533054 0.1742410
221        X4 29  6.604047 0.1741819
222        X4 30  6.567423 0.1741785
223        X4 31  5.987617 0.1741783
224        X4 32  5.632509 0.1741783
225        X4 33  5.766929 0.1741783
226        X4 34  6.194798 0.1741783
227        X4 35  5.839490 0.1741783
228        X4 36  5.439849 0.1741783
229        X4 37  5.571196 0.1741783
230        X4 38  5.441013 0.1741783
231        X4 39  5.789045 0.1741783
232        X4 40  5.882568 0.1741783
233        X4 41  5.946580 0.1741783
234        X4 42  6.311905 0.1741783
235        X4 43  6.856442 0.1741783
236        X4 44  7.130780 0.1741783
237        X4 45  6.824052 0.1741783
238        X4 46  6.895713 0.1741783
239        X4 47  6.950323 0.1741783
240        X4 48  6.596971 0.1741783
241        X4 49  6.475406 0.1741783
242        X4 50  6.942293 0.1741783
243        X4 51  7.161438 0.1741783
244        X4 52  7.520161 0.1741783
245        X4 53  7.583207 0.1741783
246        X4 54  7.547176 0.1741785
247        X4 55  7.314424 0.1741821
248        X4 56  7.225032 0.1742451
249        X4 57  7.323705 0.1753519
250        X4 58  7.256983 0.1938185
251        X4 59  7.301560 0.3970951
252        X4 60  7.346138 0.5270706
253        X4 61  7.390715 0.6308108
254        X4 62  7.435292 0.7197507
255        X4 63  7.479869 0.7988492
256        X4 64  7.524446 0.8707921

Lets look at the correlation plots

Code

Q2.2 <- coef(m2.2, type = "matrix")$Q
corrmat2.2 <- diag(1/sqrt(diag(Q2.2))) %*% Q2.2 %*% diag(1/sqrt(diag(Q2.2)))
corrplot(corrmat2.2)

Again we see large balloon shaped confidence intervals in the sections of rivers that are missing data. This appears in both the estimated underlying states and the fitted value plots. The Walla Walla groups seems to have the widest confidence intervals of all of the river systems. Again six of the rivers have ACF plots with multiple significant lags. The Yakima Population group continues to be the only group with an underlying state which is estimated to have a positive drift. The variance-covariance matrix was allowed to vary the variance of each underlying state independently of one another, but we see that it estimated each to be equal.

Hypothesis 2.3

The Q matrix for the variance of process errors is “equal variance and covariance” so they each have equal variance and they are all correlated equally to one another.

Code

mod.list2.3 <- list(
  U = U_mat2,
  R = "diagonal and equal",
  Q = "equalvarcov",
  Z = Z_mat2
)
m2.3 <- MARSS(dat, model = mod.list2.3)

Success! abstol and log-log tests passed at 124 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 124 iterations. 
Log-likelihood: -493.5994 
AIC: 1031.199   AICc: 1032.863   
 
                                                                                Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer      -0.888148
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter             -1.138376
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer -0.776205
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer    -0.342198
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer     -0.108360
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer     -1.180730
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                 1.762920
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer              0.648925
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                   -0.022687
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer               -0.842115
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer   -1.643436
R.diag                                                                          0.203477
U.Cascades                                                                     -0.027075
U.JohnDay                                                                      -0.016191
U.Walla                                                                        -0.000538
U.Yakima                                                                        0.029454
Q.diag                                                                          0.131007
Q.offdiag                                                                       0.107523
x0.X1                                                                           7.702035
x0.X2                                                                           8.051674
x0.X3                                                                           5.371496
x0.X4                                                                           4.934851
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model seemed to improve performance. Let’s take a look at some plots:

Code

autoplot(m2.3)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model fits each of the four states quite well, which indicates that the variability and the relationships between the different state variables are correlated.

Let’s looks at the estimates

Code

print(fit2.1_smooth<-tsSmooth(m2.3))

    .rownames  t .estimate       .se
1          X1  1  7.679999 0.2650985
2          X1  2  7.554402 0.3297505
3          X1  3  7.622308 0.3657975
4          X1  4  7.687265 0.3996797
5          X1  5  7.544701 0.4256157
6          X1  6  7.761073 0.4407865
7          X1  7  7.941290 0.4580768
8          X1  8  8.231850 0.4583931
9          X1  9  7.919789 0.4554334
10         X1 10  7.322210 0.4588414
11         X1 11  7.632272 0.4602276
12         X1 12  7.816635 0.4587336
13         X1 13  7.801844 0.4623177
14         X1 14  7.622790 0.4454760
15         X1 15  7.403017 0.4332599
16         X1 16  7.475021 0.4192874
17         X1 17  7.698017 0.3983516
18         X1 18  7.641013 0.3688196
19         X1 19  7.619257 0.3347281
20         X1 20  7.435176 0.2678647
21         X1 21  6.719368 0.2460981
22         X1 22  7.214218 0.2367835
23         X1 23  7.241020 0.2322616
24         X1 24  7.229720 0.2294442
25         X1 25  7.107792 0.2260253
26         X1 26  7.224311 0.2121113
27         X1 27  7.669985 0.1914103
28         X1 28  7.830052 0.1868241
29         X1 29  7.920000 0.1836535
30         X1 30  7.716768 0.1824754
31         X1 31  7.039661 0.1800981
32         X1 32  6.899473 0.1671936
33         X1 33  6.783124 0.1661814
34         X1 34  7.216908 0.1649353
35         X1 35  6.690985 0.1638188
36         X1 36  6.650792 0.1637458
37         X1 37  6.626149 0.1637050
38         X1 38  6.772343 0.1636900
39         X1 39  7.052139 0.1636848
40         X1 40  7.176321 0.1636831
41         X1 41  7.381858 0.1636825
42         X1 42  7.770564 0.1636823
43         X1 43  8.104614 0.1636823
44         X1 44  8.432610 0.1636822
45         X1 45  8.019875 0.1636822
46         X1 46  7.672487 0.1636822
47         X1 47  7.429594 0.1636822
48         X1 48  7.108466 0.1636823
49         X1 49  7.149351 0.1636823
50         X1 50  7.043170 0.1636825
51         X1 51  7.270280 0.1636833
52         X1 52  7.452399 0.1636857
53         X1 53  7.461205 0.1636938
54         X1 54  7.356775 0.1637206
55         X1 55  7.073927 0.1638092
56         X1 56  7.098320 0.1640990
57         X1 57  7.108591 0.1650029
58         X1 58  6.671678 0.1672115
59         X1 59  6.055079 0.1761959
60         X1 60  6.142143 0.1952092
61         X1 61  6.046543 0.3178464
62         X1 62  6.084194 0.4055980
63         X1 63  6.026514 0.4763777
64         X1 64  5.968835 0.4916565
65         X2  1  8.035477 0.2105232
66         X2  2  7.893099 0.2198613
67         X2  3  7.986489 0.1985492
68         X2  4  8.076284 0.1967827
69         X2  5  7.913234 0.1955579
70         X2  6  8.187515 0.1802085
71         X2  7  8.417743 0.1934622
72         X2  8  8.782414 0.1659005
73         X2  9  8.412848 0.1604277
74         X2 10  7.673789 0.1603737
75         X2 11  8.008154 0.1602917
76         X2 12  8.195929 0.1620392
77         X2 13  8.139262 0.2027759
78         X2 14  7.894870 0.1619881
79         X2 15  7.615286 0.1605936
80         X2 16  7.704351 0.1716713
81         X2 17  8.007474 0.1716844
82         X2 18  7.951137 0.1609358
83         X2 19  7.941633 0.1710309
84         X2 20  7.658223 0.1571733
85         X2 21  6.726006 0.1566277
86         X2 22  7.403991 0.1563886
87         X2 23  7.507776 0.1562679
88         X2 24  7.536882 0.1561995
89         X2 25  7.456583 0.1560761
90         X2 26  7.586937 0.1499994
91         X2 27  8.220858 0.1474178
92         X2 28  8.488518 0.1468510
93         X2 29  8.609111 0.1454430
94         X2 30  8.381163 0.1453711
95         X2 31  7.467425 0.1453824
96         X2 32  7.221325 0.1448887
97         X2 33  7.115969 0.1522491
98         X2 34  7.662973 0.1447902
99         X2 35  7.097467 0.1434237
100        X2 36  6.957747 0.1433133
101        X2 37  6.702468 0.1432720
102        X2 38  6.738727 0.1432604
103        X2 39  6.796580 0.1432574
104        X2 40  6.815921 0.1432566
105        X2 41  7.001888 0.1432564
106        X2 42  7.465437 0.1432564
107        X2 43  7.811301 0.1432564
108        X2 44  8.212619 0.1432564
109        X2 45  7.683851 0.1432564
110        X2 46  7.224817 0.1432564
111        X2 47  6.986235 0.1432564
112        X2 48  6.766593 0.1432564
113        X2 49  7.195513 0.1432564
114        X2 50  7.459095 0.1432564
115        X2 51  7.890082 0.1432565
116        X2 52  7.989030 0.1432567
117        X2 53  8.178417 0.1432580
118        X2 54  8.288222 0.1432640
119        X2 55  8.074659 0.1432906
120        X2 56  8.073611 0.1434078
121        X2 57  7.988326 0.1438807
122        X2 58  7.506660 0.1451158
123        X2 59  6.961352 0.1503533
124        X2 60  7.135457 0.1574503
125        X2 61  7.080571 0.2480325
126        X2 62  7.122422 0.3251280
127        X2 63  7.068943 0.3426167
128        X2 64  7.015464 0.2415542
129        X3  1  5.371011 0.2620614
130        X3  2  5.266964 0.3198927
131        X3  3  5.356421 0.3455408
132        X3  4  5.442928 0.3662312
133        X3  5  5.321914 0.3756477
134        X3  6  5.559837 0.3695945
135        X3  7  5.761604 0.3620747
136        X3  8  6.073714 0.3262558
137        X3  9  5.783203 0.2663743
138        X3 10  5.228790 0.2481150
139        X3 11  5.614471 0.2411266
140        X3 12  5.867890 0.2379654
141        X3 13  5.923945 0.2473907
142        X3 14  5.803326 0.2362413
143        X3 15  5.627565 0.2361897
144        X3 16  5.730435 0.2385645
145        X3 17  5.954209 0.2384201
146        X3 18  5.916290 0.2356233
147        X3 19  5.909731 0.2376566
148        X3 20  5.816978 0.2303588
149        X3 21  5.137070 0.2303613
150        X3 22  5.582810 0.2300795
151        X3 23  5.508637 0.2297083
152        X3 24  5.426463 0.2292237
153        X3 25  5.375298 0.2280932
154        X3 26  5.569574 0.2192922
155        X3 27  6.094969 0.2144689
156        X3 28  6.294744 0.2086529
157        X3 29  6.446516 0.1890842
158        X3 30  6.383981 0.1850527
159        X3 31  5.749788 0.1837072
160        X3 32  5.639067 0.1818837
161        X3 33  5.610449 0.1818937
162        X3 34  6.215866 0.1787332
163        X3 35  5.824218 0.1661419
164        X3 36  5.751968 0.1643331
165        X3 37  5.654386 0.1638739
166        X3 38  5.658578 0.1637399
167        X3 39  5.725422 0.1636996
168        X3 40  5.715262 0.1636875
169        X3 41  5.744699 0.1636838
170        X3 42  6.038934 0.1636827
171        X3 43  6.321901 0.1636824
172        X3 44  6.679984 0.1636823
173        X3 45  6.292162 0.1636822
174        X3 46  6.060606 0.1636822
175        X3 47  6.007249 0.1636822
176        X3 48  5.780247 0.1636822
177        X3 49  5.949915 0.1636823
178        X3 50  6.023690 0.1636825
179        X3 51  6.299605 0.1636831
180        X3 52  6.513956 0.1636852
181        X3 53  6.535664 0.1636919
182        X3 54  6.445011 0.1637142
183        X3 55  6.179534 0.1637875
184        X3 56  6.122078 0.1640250
185        X3 57  6.263928 0.1647499
186        X3 58  5.936153 0.1663259
187        X3 59  5.358101 0.1727617
188        X3 60  5.428180 0.1784394
189        X3 61  5.314320 0.2214705
190        X3 62  5.399328 0.2961709
191        X3 63  5.368185 0.4046793
192        X3 64  5.337043 0.4383811
193        X4  1  4.968000 0.2655764
194        X4  2  4.897588 0.3312851
195        X4  3  5.020679 0.3689042
196        X4  4  5.140821 0.4047240
197        X4  5  5.053442 0.4329998
198        X4  6  5.324999 0.4510238
199        X4  7  5.560401 0.4714453
200        X4  8  5.906146 0.4757664
201        X4  9  5.649270 0.4774463
202        X4 10  5.106876 0.4856858
203        X4 11  5.472123 0.4924413
204        X4 12  5.711671 0.4969650
205        X4 13  5.752064 0.5065872
206        X4 14  5.628196 0.4982696
207        X4 15  5.463608 0.4949762
208        X4 16  5.590796 0.4910321
209        X4 17  5.868978 0.4823767
210        X4 18  5.867159 0.4686506
211        X4 19  5.900587 0.4549984
212        X4 20  5.771692 0.4294378
213        X4 21  5.093553 0.4093963
214        X4 22  5.641645 0.3861658
215        X4 23  5.704353 0.3581628
216        X4 24  5.729533 0.3233829
217        X4 25  5.695979 0.2758431
218        X4 26  5.878230 0.1745965
219        X4 27  6.423379 0.1597556
220        X4 28  6.598968 0.1567423
221        X4 29  6.720779 0.1546361
222        X4 30  6.620543 0.1543396
223        X4 31  5.879602 0.1542316
224        X4 32  5.666074 0.1531829
225        X4 33  5.671142 0.1538983
226        X4 34  6.261813 0.1530866
227        X4 35  5.744849 0.1521346
228        X4 36  5.542098 0.1520922
229        X4 37  5.494209 0.1520633
230        X4 38  5.510089 0.1520536
231        X4 39  5.754283 0.1520507
232        X4 40  5.830006 0.1520499
233        X4 41  5.955980 0.1520496
234        X4 42  6.387962 0.1520495
235        X4 43  6.853467 0.1520495
236        X4 44  7.288771 0.1520495
237        X4 45  6.924455 0.1520495
238        X4 46  6.805447 0.1520495
239        X4 47  6.783949 0.1520495
240        X4 48  6.488091 0.1520495
241        X4 49  6.621212 0.1520496
242        X4 50  6.861908 0.1520500
243        X4 51  7.210833 0.1520514
244        X4 52  7.488076 0.1520566
245        X4 53  7.586880 0.1520759
246        X4 54  7.568503 0.1521474
247        X4 55  7.303963 0.1524140
248        X4 56  7.294868 0.1534136
249        X4 57  7.406147 0.1572435
250        X4 58  7.118231 0.1735254
251        X4 59  6.620667 0.2716668
252        X4 60  6.766655 0.3264219
253        X4 61  6.727584 0.4020779
254        X4 62  6.821763 0.4727441
255        X4 63  6.820613 0.5343203
256        X4 64  6.819462 0.5476073

And the corrplot:

Code

Q2.3 <- coef(m2.3, type = "matrix")$Q
corrmat2.3 <- diag(1/sqrt(diag(Q2.3))) %*% Q2.3 %*% diag(1/sqrt(diag(Q2.3)))
corrplot(corrmat2.3)

Code

#corrplot mirrors what we told MARSS to use as a Q matrix (equal variance and covariance)

This is the first set of models in the Hypothesis Two group that has not generated balloon shaped confidence intervals on the underlying states or the fitted values plots. Indicating that we have a better model when we allow the models to be correlated with one another. We potentially see some structuring in the residuals. Fewer of the ACF plots show strong structuring in the residuals, some of the plots with significant lags onlt have a few and we see less of the sine wave shaped plots than in the previous Hypothesis Two models. The variance covariate plot was forced to be equal, but we see that correlation between the major groups is estimated to be quite high.

Hypothesis 2.4

The Q matrix for the variance of process errors is “unconstrained”. Meaning that each hidden state is allowed to vary separately as is the correlation between the underlying states.

Code

mod.list2.4 <- list(
  U = U_mat2,
  R = "diagonal and equal",
  Q = "unconstrained",
  Z = Z_mat2
)
m2.4 <- MARSS(dat, model = mod.list2.4, method="BFGS")

Success! Converged in 276 iterations.
Function MARSSkfas used for likelihood calculation.

MARSS fit is
Estimation method: BFGS 
Estimation converged in 276 iterations. 
Log-likelihood: -472.9633 
AIC: 1005.927   AICc: 1009.027   
 
                                                                               Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer      -0.88369
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter             -1.13428
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer -0.77827
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer    -0.34595
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer     -0.11152
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer     -1.18329
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                 1.72948
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer              0.64196
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                   -0.02319
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer               -0.84264
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer   -1.64395
R.diag                                                                          0.20323
U.Cascades                                                                     -0.02396
U.JohnDay                                                                      -0.01680
U.Walla                                                                        -0.00129
U.Yakima                                                                        0.03104
Q.(1,1)                                                                         0.08502
Q.(2,1)                                                                         0.10316
Q.(3,1)                                                                         0.04997
Q.(4,1)                                                                         0.08806
Q.(2,2)                                                                         0.21631
Q.(3,2)                                                                         0.07749
Q.(4,2)                                                                         0.15904
Q.(3,3)                                                                         0.03249
Q.(4,3)                                                                         0.06141
Q.(4,4)                                                                         0.12110
x0.X1                                                                           7.40935
x0.X2                                                                           8.06903
x0.X3                                                                           5.71073
x0.X4                                                                           4.80279
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model, with the unconstrained Q matrix, had a hard time converging, thus the Broyden-Fletcher-Goldfarb-Shanno method was used to help with optimization. Thus far, this model has the lowest AICc.

Code

autoplot(m2.4)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

These model fits looks pretty well, with tight confidence intervals and the model is fitting the data well. All of the residuals looks like they are normal, and standardized residuals may show a bit of structure in a few graphs but otherwise white noise. Again X4 (Yakima) is the only positive drift value. On the hidden state’s random walk John Day and Yakima have strongest correlation and then John Day and Cascades I expect this to be seen in the corrplot.

Lets look at estimates:

Code

print(fit2.1_smooth<-tsSmooth(m2.4))

    .rownames  t .estimate        .se
1          X1  1  7.385345 0.21476146
2          X1  2  7.277819 0.27458508
3          X1  3  7.321792 0.31027387
4          X1  4  7.359836 0.33938396
5          X1  5  7.231228 0.35963874
6          X1  6  7.368759 0.37010615
7          X1  7  7.462571 0.37846826
8          X1  8  7.656250 0.37388232
9          X1  9  7.449173 0.36376338
10         X1 10  7.036249 0.35792488
11         X1 11  7.275639 0.35307567
12         X1 12  7.424726 0.34837298
13         X1 13  7.441529 0.34879991
14         X1 14  7.338854 0.33823956
15         X1 15  7.205429 0.33177698
16         X1 16  7.267186 0.32461019
17         X1 17  7.454489 0.31343325
18         X1 18  7.414468 0.29652331
19         X1 19  7.442327 0.27544268
20         X1 20  7.318096 0.23393681
21         X1 21  6.773208 0.21873060
22         X1 22  7.172334 0.21211017
23         X1 23  7.202546 0.20886472
24         X1 24  7.219120 0.20648012
25         X1 25  7.154770 0.20257520
26         X1 26  7.286097 0.18428765
27         X1 27  7.643514 0.16929659
28         X1 28  7.764405 0.16565017
29         X1 29  7.827422 0.16288196
30         X1 30  7.660195 0.16187541
31         X1 31  7.106520 0.15997551
32         X1 32  6.955777 0.15079386
33         X1 33  6.857928 0.14936964
34         X1 34  7.184499 0.14875863
35         X1 35  6.752368 0.14736809
36         X1 36  6.690761 0.14719083
37         X1 37  6.670119 0.14715174
38         X1 38  6.787869 0.14714218
39         X1 39  7.036715 0.14713980
40         X1 40  7.152745 0.14713920
41         X1 41  7.334489 0.14713906
42         X1 42  7.678443 0.14713902
43         X1 43  7.993513 0.14713901
44         X1 44  8.284049 0.14713901
45         X1 45  7.968828 0.14713901
46         X1 46  7.715970 0.14713901
47         X1 47  7.520754 0.14713901
48         X1 48  7.221997 0.14713901
49         X1 49  7.204344 0.14713901
50         X1 50  7.121700 0.14713903
51         X1 51  7.320153 0.14713910
52         X1 52  7.476780 0.14713938
53         X1 53  7.479820 0.14714050
54         X1 54  7.352149 0.14714498
55         X1 55  7.065461 0.14716289
56         X1 56  7.055895 0.14723465
57         X1 57  7.026528 0.14752546
58         X1 58  6.638505 0.14881267
59         X1 59  6.077321 0.16061937
60         X1 60  6.078157 0.17802625
61         X1 61  5.942462 0.26903136
62         X1 62  5.912932 0.34756233
63         X1 63  5.894156 0.40562762
64         X1 64  5.875380 0.42499966
65         X2  1  8.052222 0.23629095
66         X2  2  7.860272 0.24223974
67         X2  3  7.985995 0.21372325
68         X2  4  8.099287 0.21246962
69         X2  5  7.863130 0.21167423
70         X2  6  8.185033 0.19213872
71         X2  7  8.415261 0.21034651
72         X2  8  8.854896 0.17596283
73         X2  9  8.454201 0.17372945
74         X2 10  7.586722 0.17371819
75         X2 11  8.005137 0.17374387
76         X2 12  8.225963 0.17503934
77         X2 13  8.171307 0.23324755
78         X2 14  7.891189 0.17503137
79         X2 15  7.568357 0.17401329
80         X2 16  7.664595 0.18857281
81         X2 17  8.060585 0.18856276
82         X2 18  7.969773 0.17423742
83         X2 19  8.034681 0.18817893
84         X2 20  7.695725 0.17237526
85         X2 21  6.508551 0.17202292
86         X2 22  7.413741 0.17191400
87         X2 23  7.527911 0.17185602
88         X2 24  7.561927 0.17182151
89         X2 25  7.444617 0.17159669
90         X2 26  7.563667 0.15324449
91         X2 27  8.308817 0.15293291
92         X2 28  8.572261 0.15273076
93         X2 29  8.709822 0.15208565
94         X2 30  8.509600 0.15199352
95         X2 31  7.412962 0.15188322
96         X2 32  7.133278 0.15183911
97         X2 33  7.071631 0.16085563
98         X2 34  7.804746 0.15165069
99         X2 35  7.111383 0.15094333
100        X2 36  6.925811 0.15092453
101        X2 37  6.627376 0.15091812
102        X2 38  6.667187 0.15091636
103        X2 39  6.740224 0.15091591
104        X2 40  6.718918 0.15091580
105        X2 41  6.877080 0.15091577
106        X2 42  7.404853 0.15091576
107        X2 43  7.808958 0.15091576
108        X2 44  8.322028 0.15091576
109        X2 45  7.687611 0.15091576
110        X2 46  7.244139 0.15091576
111        X2 47  7.029712 0.15091576
112        X2 48  6.699442 0.15091576
113        X2 49  7.171283 0.15091576
114        X2 50  7.514692 0.15091576
115        X2 51  8.005832 0.15091578
116        X2 52  8.021065 0.15091583
117        X2 53  8.249096 0.15091604
118        X2 54  8.378383 0.15091689
119        X2 55  8.069200 0.15092026
120        X2 56  8.092236 0.15093362
121        X2 57  8.001672 0.15098513
122        X2 58  7.488129 0.15127544
123        X2 59  6.877173 0.16807349
124        X2 60  7.115984 0.17242630
125        X2 61  7.030345 0.29382288
126        X2 62  7.007748 0.41854178
127        X2 63  7.001829 0.42338072
128        X2 64  6.995910 0.24856107
129        X3  1  5.709433 0.10758928
130        X3  2  5.645386 0.12536917
131        X3  3  5.695147 0.13111978
132        X3  4  5.740455 0.14012400
133        X3  5  5.660571 0.14652392
134        X3  6  5.780615 0.14730996
135        X3  7  5.867816 0.15244573
136        X3  8  6.030037 0.14642240
137        X3  9  5.891207 0.14159404
138        X3 10  5.591238 0.13951885
139        X3 11  5.766136 0.13789426
140        X3 12  5.871674 0.13638990
141        X3 13  5.878191 0.14397185
142        X3 14  5.799592 0.13307431
143        X3 15  5.701920 0.13092971
144        X3 16  5.752685 0.13041221
145        X3 17  5.904516 0.12686411
146        X3 18  5.883702 0.11957259
147        X3 19  5.916357 0.11469629
148        X3 20  5.819042 0.09717085
149        X3 21  5.411987 0.09264785
150        X3 22  5.734975 0.09082354
151        X3 23  5.777600 0.08994520
152        X3 24  5.800440 0.08928906
153        X3 25  5.765887 0.08800156
154        X3 26  5.846142 0.07661352
155        X3 27  6.124367 0.07124654
156        X3 28  6.227550 0.07035147
157        X3 29  6.286419 0.06927177
158        X3 30  6.199157 0.06904153
159        X3 31  5.805663 0.06858812
160        X3 32  5.709686 0.06528808
161        X3 33  5.673265 0.06626322
162        X3 34  5.938060 0.06484371
163        X3 35  5.663332 0.06417165
164        X3 36  5.617144 0.06413062
165        X3 37  5.564952 0.06412318
166        X3 38  5.625613 0.06412143
167        X3 39  5.740039 0.06412099
168        X3 40  5.788766 0.06412089
169        X3 41  5.894570 0.06412086
170        X3 42  6.127683 0.06412085
171        X3 43  6.327422 0.06412085
172        X3 44  6.538411 0.06412085
173        X3 45  6.317054 0.06412085
174        X3 46  6.153681 0.06412085
175        X3 47  6.053625 0.06412085
176        X3 48  5.894526 0.06412085
177        X3 49  5.985949 0.06412085
178        X3 50  6.030001 0.06412085
179        X3 51  6.203487 0.06412087
180        X3 52  6.273721 0.06412092
181        X3 53  6.327545 0.06412112
182        X3 54  6.315623 0.06412194
183        X3 55  6.164810 0.06412520
184        X3 56  6.176124 0.06413834
185        X3 57  6.159226 0.06419309
186        X3 58  5.933793 0.06450285
187        X3 59  5.627440 0.07224816
188        X3 60  5.682459 0.07652311
189        X3 61  5.627852 0.12808587
190        X3 62  5.623472 0.17574581
191        X3 63  5.626083 0.19504169
192        X3 64  5.628695 0.17203601
193        X4  1  4.833806 0.18458545
194        X4  2  4.736050 0.19733531
195        X4  3  4.871863 0.18623811
196        X4  4  4.998536 0.19133763
197        X4  5  4.868277 0.19514972
198        X4  6  5.148331 0.18685147
199        X4  7  5.360982 0.19846691
200        X4  8  5.727599 0.17901389
201        X4  9  5.476364 0.17268352
202        X4 10  4.887642 0.17102224
203        X4 11  5.257701 0.16988712
204        X4 12  5.483822 0.16925301
205        X4 13  5.507086 0.20042777
206        X4 14  5.360501 0.16705555
207        X4 15  5.178971 0.16534006
208        X4 16  5.303967 0.17148168
209        X4 17  5.643419 0.16928282
210        X4 18  5.626599 0.15812125
211        X4 19  5.722094 0.16141900
212        X4 20  5.534467 0.14003719
213        X4 21  4.717683 0.13757115
214        X4 22  5.420945 0.13677086
215        X4 23  5.545450 0.13640066
216        X4 24  5.619407 0.13610714
217        X4 25  5.579116 0.13516927
218        X4 26  5.740903 0.11554605
219        X4 27  6.338303 0.11152776
220        X4 28  6.579209 0.11127518
221        X4 29  6.728296 0.11019088
222        X4 30  6.605441 0.11013800
223        X4 31  5.837498 0.11002121
224        X4 32  5.674767 0.10760895
225        X4 33  5.654921 0.11274062
226        X4 34  6.234908 0.10752385
227        X4 35  5.739332 0.10663252
228        X4 36  5.660890 0.10662007
229        X4 37  5.531798 0.10661942
230        X4 38  5.643584 0.10661930
231        X4 39  5.819110 0.10661928
232        X4 40  5.895315 0.10661927
233        X4 41  6.096695 0.10661927
234        X4 42  6.565036 0.10661927
235        X4 43  6.952715 0.10661927
236        X4 44  7.394416 0.10661927
237        X4 45  6.972468 0.10661927
238        X4 46  6.681124 0.10661927
239        X4 47  6.540588 0.10661927
240        X4 48  6.298405 0.10661927
241        X4 49  6.611380 0.10661927
242        X4 50  6.828648 0.10661927
243        X4 51  7.226385 0.10661927
244        X4 52  7.337359 0.10661927
245        X4 53  7.517790 0.10661928
246        X4 54  7.597098 0.10661933
247        X4 55  7.371106 0.10661953
248        X4 56  7.429860 0.10662036
249        X4 57  7.416816 0.10662542
250        X4 58  7.039246 0.10678487
251        X4 59  6.546836 0.12136884
252        X4 60  6.732688 0.12386670
253        X4 61  6.686196 0.21923225
254        X4 62  6.712012 0.31225201
255        X4 63  6.751047 0.32794004
256        X4 64  6.790081 0.22777820

look at corrplot:

Code

Q2.4 <- coef(m2.4, type = "matrix")$Q
corrmat2.4 <- diag(1/sqrt(diag(Q2.4))) %*% Q2.4 %*% diag(1/sqrt(diag(Q2.4)))
corrplot(corrmat2.4)

All groups are highly correlated with each other, which means there is likely a lot of connectivity between these four DPCs.

Again we see that confidence intervals have some shape to them and fit the predicted values better than the balloon shaped confidence intervals seen in previous Hypotheis Two plots. Plots of the residuals look very similar to the other plots from this Hypothesis group. ACF plots are similar to the Hypothesis 2.3 and do not show as much structuring as previous plots.

The Q matrix was allowed to be unconstrained. We see very different variances estimated by the MARSS model, ranging from 0.03 to .21. Covariance between the underlying states allowed for better fits to data and realistic estimates for streams with missing data.

Hypothesis 3

Description of H3: There are two underlying states, one representing the northern area (Walla Walla and Yakima) and one representing the southern area (John Day and Cascades).

\[ \text{Hypothesis Three}: \begin{bmatrix} y_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6\\ y_7\\ y_8\\ y_9\\ y_{10}\\ y_{11}\\ y_{12}\\ y_{13}\\ y_{14}\\ y_{15}\\ \end{bmatrix}_t= \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ \end{bmatrix}* \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix}_t+ \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6\\ a_7\\ a_8\\ a_9\\ a_{10}\\ a_{11}\\ a_{12}\\ a_{13}\\ a_{14}\\ a_{15}\\ \end{bmatrix}+ \begin{bmatrix} w_1\\ w_2\\ w_3\\ w_4\\ w_5\\ w_6\\ w_7\\ w_8\\ w_9\\ w_{10}\\ w_{11}\\ w_{12}\\ w_{13}\\ w_{14}\\ w_{15}\\ \end{bmatrix}_t \]

\[ \text{Where }w \sim MVN \begin{pmatrix} \text{0,}\begin{bmatrix} R \end{bmatrix} \end{pmatrix} \]

We start by establishing our U matrix and our Z matrix.

Code

U_mat3 <- matrix(c("North","South"),2,1)
#make Z matrix correspond to 2 hidden states
Z_mat3 <- matrix(c(rep(c(0,1),8),
                     rep(c(1,0),7)),15,2, byrow=TRUE)

Hypothesis 3.1

The Q matrix for the variance of process errors is “diagonal and equal” meaning each state (x) model has the same process error but they are not correlated to each other.

Code

mod.list3.1 <- list(
  U = U_mat3,
  R = "diagonal and equal",
  Q = "diagonal and equal",
  Z = Z_mat3
)
m3.1 <- MARSS(dat, model = mod.list3.1)

Success! abstol and log-log tests passed at 78 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 78 iterations. 
Log-likelihood: -544.2628 
AIC: 1126.526   AICc: 1127.77   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                  -0.91016
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                         -1.16869
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer  0.27060
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer             -0.50637
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                -0.07071
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                  0.16262
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                 -0.90975
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                             1.77502
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          0.58813
A.Steelhead (Middle Columbia River DPS) Naches River - summer                               0.51010
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                0.48692
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                           -0.33251
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer               -1.13383
R.diag                                                                                      0.25108
U.North                                                                                    -0.00129
U.South                                                                                    -0.01638
Q.diag                                                                                      0.12901
x0.X1                                                                                       5.47856
x0.X2                                                                                       7.77580
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This AICc is bad compared to Hypothesis 2. This is either because there is coorelation between the hidden states, as supported by hypothesis two, or the assumption that there are only two underlying states is incorrect. Plots

Code

autoplot(m3.1)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

The southern area (John Day and Cascades) are more informed by data and the fits look ok. The northern area (Yakima and Walla Walla) have large confidence intercals in the early period. The models are fitting data pretty well for individual streams, and while residuals don’t seem to have too much structure, some outliers seem to be present for the southern area. The Residuals normality tests are a little wobbly, which is troubling.

Let’s look at estimates:

Code

print(fit3.1_smooth<-tsSmooth(m3.1))

    .rownames  t .estimate       .se
1          X1  1  5.477532 0.3407100
2          X1  2  5.476506 0.4542230
3          X1  3  5.475479 0.5202938
4          X1  4  5.474453 0.5560977
5          X1  5  5.473426 0.5673937
6          X1  6  5.472400 0.5556783
7          X1  7  5.471374 0.5193969
8          X1  8  5.470347 0.4526806
9          X1  9  5.469321 0.3379619
10         X1 10  5.409468 0.3032387
11         X1 11  5.593148 0.2940892
12         X1 12  5.742870 0.2917994
13         X1 13  5.842543 0.2912346
14         X1 14  5.851348 0.2910958
15         X1 15  5.826841 0.2910617
16         X1 16  5.852416 0.2910533
17         X1 17  5.812934 0.2910510
18         X1 18  5.788579 0.2910496
19         X1 19  5.687514 0.2910459
20         X1 20  5.817422 0.2910314
21         X1 21  5.784991 0.2909723
22         X1 22  5.643900 0.2907315
23         X1 23  5.386686 0.2897482
24         X1 24  5.204488 0.2857076
25         X1 25  5.313979 0.2686194
26         X1 26  5.530135 0.1832141
27         X1 27  5.931645 0.1775040
28         X1 28  6.040779 0.1766790
29         X1 29  6.146173 0.1666108
30         X1 30  6.148702 0.1661743
31         X1 31  5.602851 0.1661559
32         X1 32  5.308922 0.1661545
33         X1 33  5.352153 0.1661401
34         X1 34  5.845502 0.1657966
35         X1 35  5.602765 0.1574038
36         X1 36  5.308721 0.1571094
37         X1 37  5.388751 0.1570993
38         X1 38  5.276800 0.1570990
39         X1 39  5.473139 0.1570990
40         X1 40  5.560416 0.1570990
41         X1 41  5.564462 0.1570990
42         X1 42  5.872493 0.1570990
43         X1 43  6.303023 0.1570990
44         X1 44  6.596640 0.1570990
45         X1 45  6.270092 0.1570990
46         X1 46  6.242309 0.1570990
47         X1 47  6.319448 0.1570990
48         X1 48  6.022024 0.1570990
49         X1 49  5.961986 0.1570990
50         X1 50  6.275040 0.1570990
51         X1 51  6.461784 0.1570990
52         X1 52  6.865492 0.1570990
53         X1 53  6.860231 0.1570990
54         X1 54  6.754002 0.1570990
55         X1 55  6.546166 0.1570990
56         X1 56  6.356627 0.1571012
57         X1 57  6.578965 0.1571650
58         X1 58  6.355875 0.1590227
59         X1 59  5.672335 0.2061349
60         X1 60  5.468549 0.2138452
61         X1 61  5.306958 0.2503255
62         X1 62  5.398691 0.3355052
63         X1 63  5.397404 0.4914989
64         X1 64  5.396116 0.6087516
65         X2  1  7.759752 0.2202349
66         X2  2  7.637348 0.2324296
67         X2  3  7.717964 0.2123216
68         X2  4  7.798715 0.2101655
69         X2  5  7.666740 0.2085890
70         X2  6  7.920741 0.1935479
71         X2  7  8.147444 0.2058659
72         X2  8  8.481707 0.1788469
73         X2  9  8.153114 0.1772968
74         X2 10  7.479031 0.1772216
75         X2 11  7.738313 0.1773593
76         X2 12  7.906992 0.1800140
77         X2 13  7.827395 0.2241976
78         X2 14  7.601318 0.1800538
79         X2 15  7.341338 0.1780793
80         X2 16  7.415448 0.1902864
81         X2 17  7.732970 0.1903161
82         X2 18  7.688971 0.1785688
83         X2 19  7.718670 0.1888305
84         X2 20  7.349959 0.1671700
85         X2 21  6.464567 0.1661979
86         X2 22  7.101052 0.1661569
87         X2 23  7.267216 0.1661552
88         X2 24  7.345209 0.1661545
89         X2 25  7.217818 0.1661401
90         X2 26  7.295775 0.1657966
91         X2 27  7.831481 0.1574038
92         X2 28  8.142726 0.1571094
93         X2 29  8.262019 0.1570990
94         X2 30  7.928431 0.1570908
95         X2 31  7.161937 0.1568585
96         X2 32  7.007047 0.1498967
97         X2 33  6.886865 0.1566127
98         X2 34  7.225984 0.1496821
99         X2 35  6.716098 0.1494812
100        X2 36  6.730607 0.1494755
101        X2 37  6.445811 0.1494753
102        X2 38  6.625537 0.1494753
103        X2 39  6.706132 0.1494753
104        X2 40  6.753399 0.1494753
105        X2 41  7.047831 0.1494753
106        X2 42  7.488069 0.1494753
107        X2 43  7.769284 0.1494753
108        X2 44  8.115274 0.1494753
109        X2 45  7.741188 0.1494753
110        X2 46  7.222219 0.1494753
111        X2 47  6.850099 0.1494753
112        X2 48  6.710536 0.1494753
113        X2 49  7.105604 0.1494753
114        X2 50  7.114939 0.1494753
115        X2 51  7.536639 0.1494753
116        X2 52  7.528176 0.1494753
117        X2 53  7.710539 0.1494753
118        X2 54  7.777051 0.1494753
119        X2 55  7.560708 0.1494753
120        X2 56  7.681707 0.1494753
121        X2 57  7.458852 0.1494754
122        X2 58  6.943258 0.1494796
123        X2 59  6.468715 0.1496290
124        X2 60  6.640272 0.1548093
125        X2 61  6.791326 0.2822300
126        X2 62  6.770239 0.3682749
127        X2 63  6.749153 0.3644883
128        X2 64  6.728066 0.2670760

Our corrplots are as expected.

Code

Q3.1 <- coef(m3.1, type = "matrix")$Q
corrmat3.1 <- diag(1/sqrt(diag(Q3.1))) %*% Q3.1 %*% diag(1/sqrt(diag(Q3.1)))
corrplot(corrmat3.1)

This model probably isn’t it. It’s failing our normality tests, and while it seems to fit the data ok, it didn’t perform as well as hypothesis 2. Let’s see how our models improve with different, but uncorrelated process errors.

Hypothesis 3.2

The Q matrix for the variance of process errors is “diagonal and unequal” meaning each of the four underlying states’ process error can be different but they are not correlated to each other.

Code

mod.list3.2 <- list(
  U = U_mat3,
  R = "diagonal and equal",
  Q = "diagonal and unequal",
  Z = Z_mat3
)
m3.2 <- MARSS(dat, model = mod.list3.2)

Success! abstol and log-log tests passed at 81 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 81 iterations. 
Log-likelihood: -542.9455 
AIC: 1125.891   AICc: 1127.268   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                  -0.91086
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                         -1.17082
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer  0.26954
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer             -0.50751
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                -0.07285
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                  0.16063
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                 -0.91143
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                             1.77489
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          0.58698
A.Steelhead (Middle Columbia River DPS) Naches River - summer                               0.51639
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                0.49321
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                           -0.32622
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer               -1.12754
R.diag                                                                                      0.25075
U.North                                                                                    -0.00136
U.South                                                                                    -0.01641
Q.(X1,X1)                                                                                   0.08331
Q.(X2,X2)                                                                                   0.16535
x0.X1                                                                                       5.50140
x0.X2                                                                                       7.78310
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model also has a higher AICc than some of the other models, indicating that the assumption that process errors ARE correlated is likely a better assumption than the diagonal and unequal assumption for the Q matrix.

Let’s look at some plots:

Code

autoplot(m3.1)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model isn’t performing great, which is unsurprising given the lack of correlation in the process errors. The curvy QQ plots remain, and the model CIs are high where there is a lack of data.

What are the estimates

Code

print(fit3.2_smooth<-tsSmooth(m3.2))

    .rownames  t .estimate       .se
1          X1  1  5.500332 0.2742685
2          X1  2  5.499266 0.3664410
3          X1  3  5.498199 0.4209126
4          X1  4  5.497133 0.4515401
5          X1  5  5.496067 0.4630790
6          X1  6  5.495000 0.4569775
7          X1  7  5.493934 0.4324897
8          X1  8  5.492867 0.3861332
9          X1  9  5.491801 0.3081931
10         X1 10  5.460124 0.2786224
11         X1 11  5.602698 0.2684516
12         X1 12  5.726445 0.2651077
13         X1 13  5.812331 0.2640265
14         X1 14  5.829385 0.2636788
15         X1 15  5.817562 0.2635670
16         X1 16  5.834995 0.2635307
17         X1 17  5.804533 0.2635174
18         X1 18  5.781017 0.2635080
19         X1 19  5.705348 0.2634890
20         X1 20  5.784903 0.2634330
21         X1 21  5.748648 0.2632594
22         X1 22  5.630018 0.2627178
23         X1 23  5.431652 0.2610216
24         X1 24  5.296685 0.2556594
25         X1 25  5.380906 0.2381627
26         X1 26  5.556286 0.1725567
27         X1 27  5.903273 0.1654518
28         X1 28  6.022408 0.1641138
29         X1 29  6.116927 0.1559043
30         X1 30  6.094959 0.1552958
31         X1 31  5.619612 0.1552517
32         X1 32  5.348883 0.1552460
33         X1 33  5.384132 0.1552109
34         X1 34  5.782483 0.1547257
35         X1 35  5.587600 0.1478297
36         X1 36  5.332215 0.1474053
37         X1 37  5.381273 0.1473798
38         X1 38  5.296750 0.1473782
39         X1 39  5.465891 0.1473781
40         X1 40  5.555599 0.1473781
41         X1 41  5.588227 0.1473781
42         X1 42  5.880610 0.1473781
43         X1 43  6.279009 0.1473781
44         X1 44  6.540975 0.1473781
45         X1 45  6.280441 0.1473781
46         X1 46  6.245085 0.1473781
47         X1 47  6.291962 0.1473781
48         X1 48  6.040674 0.1473781
49         X1 49  5.994183 0.1473781
50         X1 50  6.271738 0.1473781
51         X1 51  6.467882 0.1473781
52         X1 52  6.826429 0.1473781
53         X1 53  6.837641 0.1473781
54         X1 54  6.738971 0.1473782
55         X1 55  6.547584 0.1473787
56         X1 56  6.379265 0.1473882
57         X1 57  6.537841 0.1475465
58         X1 58  6.320714 0.1501573
59         X1 59  5.732040 0.1883841
60         X1 60  5.512645 0.1986660
61         X1 61  5.364056 0.2330335
62         X1 62  5.416816 0.3051711
63         X1 63  5.415452 0.4200432
64         X1 64  5.414088 0.5096500
65         X2  1  7.766962 0.2351280
66         X2  2  7.622386 0.2453036
67         X2  3  7.717172 0.2212643
68         X2  4  7.808000 0.2193272
69         X2  5  7.641634 0.2179910
70         X2  6  7.918381 0.2007342
71         X2  7  8.150000 0.2157196
72         X2  8  8.522150 0.1849422
73         X2  9  8.171678 0.1836566
74         X2 10  7.433739 0.1836104
75         X2 11  7.739025 0.1836966
76         X2 12  7.925102 0.1859744
77         X2 13  7.846976 0.2376917
78         X2 14  7.604267 0.1859973
79         X2 15  7.322485 0.1842727
80         X2 16  7.402016 0.1980347
81         X2 17  7.754528 0.1980546
82         X2 18  7.695965 0.1847102
83         X2 19  7.749760 0.1968408
84         X2 20  7.369675 0.1724763
85         X2 21  6.398769 0.1716947
86         X2 22  7.114896 0.1716713
87         X2 23  7.276464 0.1716706
88         X2 24  7.355724 0.1716703
89         X2 25  7.207262 0.1716623
90         X2 26  7.274827 0.1713939
91         X2 27  7.841225 0.1621584
92         X2 28  8.157388 0.1619307
93         X2 29  8.287742 0.1619252
94         X2 30  7.948475 0.1619208
95         X2 31  7.139006 0.1617414
96         X2 32  7.001448 0.1540665
97         X2 33  6.871753 0.1615547
98         X2 34  7.256012 0.1539051
99         X2 35  6.701961 0.1537510
100        X2 36  6.739168 0.1537480
101        X2 37  6.430277 0.1537480
102        X2 38  6.628441 0.1537480
103        X2 39  6.707556 0.1537480
104        X2 40  6.744264 0.1537480
105        X2 41  7.042841 0.1537480
106        X2 42  7.495064 0.1537480
107        X2 43  7.772957 0.1537480
108        X2 44  8.146208 0.1537480
109        X2 45  7.751391 0.1537480
110        X2 46  7.217308 0.1537480
111        X2 47  6.838627 0.1537480
112        X2 48  6.690731 0.1537480
113        X2 49  7.117334 0.1537480
114        X2 50  7.103633 0.1537480
115        X2 51  7.552358 0.1537480
116        X2 52  7.524800 0.1537480
117        X2 53  7.717185 0.1537480
118        X2 54  7.788801 0.1537480
119        X2 55  7.552251 0.1537480
120        X2 56  7.696871 0.1537480
121        X2 57  7.473047 0.1537480
122        X2 58  6.941272 0.1537496
123        X2 59  6.444554 0.1538332
124        X2 60  6.642405 0.1580316
125        X2 61  6.818049 0.3027063
126        X2 62  6.789694 0.4061910
127        X2 63  6.761339 0.3986559
128        X2 64  6.732984 0.2709930

And the corr plot is as expected.

Code

Q3.2 <- coef(m3.2, type = "matrix")$Q
corrmat3.2 <- diag(1/sqrt(diag(Q3.2))) %*% Q3.2 %*% diag(1/sqrt(diag(Q3.2)))
corrplot(corrmat3.2)

Hypothesis 3.3

The Q matrix for the variance of process errors is “equal variance and covariance” so they each have equal variance and they are all correlated equally to one another. I would guess this model performs better than the other too, let’s see!

Code

mod.list3.3 <- list(
  U = U_mat3,
  R = "diagonal and equal",
  Q = "equalvarcov",
  Z = Z_mat3
)
m3.3 <- MARSS(dat, model = mod.list3.3)

Success! abstol and log-log tests passed at 83 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 83 iterations. 
Log-likelihood: -535.7385 
AIC: 1111.477   AICc: 1112.854   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                  -0.91224
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                         -1.16892
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer  0.27394
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer             -0.50618
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                -0.07051
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                  0.16385
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                 -0.90916
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                             1.79638
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          0.59791
A.Steelhead (Middle Columbia River DPS) Naches River - summer                               0.51607
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                0.49288
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                           -0.32654
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer               -1.12786
R.diag                                                                                      0.25814
U.North                                                                                     0.00518
U.South                                                                                    -0.01709
Q.diag                                                                                      0.12193
Q.offdiag                                                                                   0.11142
x0.X1                                                                                       5.34577
x0.X2                                                                                       7.77382
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

The AICc is a little better! It seems our correlation hunch is further supported! Let’s look at plots:

Code

autoplot(m3.3)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model looks ok! The confidence intervals are wider where there is a lack of data, but we are not seeing them balloon out! The residuals don’t seem to have clear structure, but there are some outliers, and generally the qq plots by stream appear to be pretty normal, but the two states have lifting at the left tail.

Let’s look at estimates:

Code

print(fit3.3_smooth<-tsSmooth(m3.3))

    .rownames  t .estimate       .se
1          X1  1  5.351407 0.2412707
2          X1  2  5.266779 0.2799122
3          X1  3  5.357551 0.2886001
4          X1  4  5.449578 0.3020571
5          X1  5  5.360014 0.3100310
6          X1  6  5.605853 0.3055782
7          X1  7  5.829720 0.3093909
8          X1  8  6.138486 0.2850231
9          X1  9  5.838832 0.2507264
10         X1 10  5.279523 0.2407967
11         X1 11  5.609977 0.2362207
12         X1 12  5.837270 0.2336443
13         X1 13  5.860791 0.2480968
14         X1 14  5.719186 0.2313194
15         X1 15  5.535996 0.2307949
16         X1 16  5.641896 0.2346076
17         X1 17  5.885416 0.2342537
18         X1 18  5.862653 0.2297497
19         X1 19  5.850084 0.2335542
20         X1 20  5.679782 0.2254517
21         X1 21  4.982414 0.2246429
22         X1 22  5.439854 0.2232919
23         X1 23  5.439488 0.2209958
24         X1 24  5.395131 0.2168108
25         X1 25  5.323457 0.2069917
26         X1 26  5.446944 0.1603950
27         X1 27  5.962044 0.1540204
28         X1 28  6.159541 0.1524173
29         X1 29  6.297717 0.1456389
30         X1 30  6.193282 0.1450369
31         X1 31  5.500153 0.1449027
32         X1 32  5.319753 0.1434734
33         X1 33  5.314327 0.1447162
34         X1 34  5.883725 0.1429597
35         X1 35  5.473143 0.1376154
36         X1 36  5.341949 0.1372639
37         X1 37  5.261609 0.1371955
38         X1 38  5.283662 0.1371752
39         X1 39  5.436894 0.1371686
40         X1 40  5.480690 0.1371663
41         X1 41  5.583503 0.1371656
42         X1 42  5.960942 0.1371653
43         X1 43  6.342470 0.1371652
44         X1 44  6.734310 0.1371652
45         X1 45  6.372694 0.1371652
46         X1 46  6.179288 0.1371652
47         X1 47  6.120074 0.1371652
48         X1 48  5.875766 0.1371652
49         X1 49  6.045535 0.1371653
50         X1 50  6.215955 0.1371656
51         X1 51  6.534057 0.1371665
52         X1 52  6.761419 0.1371689
53         X1 53  6.833614 0.1371764
54         X1 54  6.792080 0.1371984
55         X1 55  6.536485 0.1372640
56         X1 56  6.498104 0.1374625
57         X1 57  6.573531 0.1381062
58         X1 58  6.212645 0.1409725
59         X1 59  5.519559 0.1721202
60         X1 60  5.551588 0.1814538
61         X1 61  5.479609 0.2308747
62         X1 62  5.553883 0.3016721
63         X1 63  5.615950 0.3797782
64         X1 64  5.678016 0.3799765
65         X2  1  7.756908 0.2181015
66         X2  2  7.641215 0.2311551
67         X2  3  7.717470 0.2120982
68         X2  4  7.795097 0.2098575
69         X2  5  7.674003 0.2081810
70         X2  6  7.919954 0.1936586
71         X2  7  8.141859 0.2052310
72         X2  8  8.456673 0.1787285
73         X2  9  8.105670 0.1708058
74         X2 10  7.446948 0.1707942
75         X2 11  7.730770 0.1708981
76         X2 12  7.904078 0.1730917
77         X2 13  7.852826 0.2096064
78         X2 14  7.632574 0.1730308
79         X2 15  7.381793 0.1713448
80         X2 16  7.462041 0.1817109
81         X2 17  7.719821 0.1817115
82         X2 18  7.675684 0.1716658
83         X2 19  7.647297 0.1803186
84         X2 20  7.392278 0.1616181
85         X2 21  6.597330 0.1606881
86         X2 22  7.150785 0.1605891
87         X2 23  7.238095 0.1605235
88         X2 24  7.258492 0.1604774
89         X2 25  7.182506 0.1603449
90         X2 26  7.299760 0.1489555
91         X2 27  7.846425 0.1427498
92         X2 28  8.077230 0.1420697
93         X2 29  8.189245 0.1402641
94         X2 30  7.971439 0.1401209
95         X2 31  7.173748 0.1398115
96         X2 32  6.944035 0.1349607
97         X2 33  6.840449 0.1394330
98         X2 34  7.306709 0.1346399
99         X2 35  6.799289 0.1331625
100        X2 36  6.678813 0.1331141
101        X2 37  6.527382 0.1330765
102        X2 38  6.591068 0.1330617
103        X2 39  6.738357 0.1330566
104        X2 40  6.800931 0.1330548
105        X2 41  6.984804 0.1330542
106        X2 42  7.402703 0.1330540
107        X2 43  7.747260 0.1330540
108        X2 44  8.097587 0.1330540
109        X2 45  7.639435 0.1330540
110        X2 46  7.249841 0.1330540
111        X2 47  7.022615 0.1330540
112        X2 48  6.769316 0.1330540
113        X2 49  7.012374 0.1330541
114        X2 50  7.139664 0.1330543
115        X2 51  7.480111 0.1330549
116        X2 52  7.637415 0.1330569
117        X2 53  7.750222 0.1330626
118        X2 54  7.766526 0.1330798
119        X2 55  7.538017 0.1331304
120        X2 56  7.545266 0.1332777
121        X2 57  7.519562 0.1336695
122        X2 58  7.088863 0.1343248
123        X2 59  6.451847 0.1410203
124        X2 60  6.565160 0.1462899
125        X2 61  6.533752 0.2409667
126        X2 62  6.590072 0.3118246
127        X2 63  6.635237 0.3398633
128        X2 64  6.680403 0.2649879

Let’s look at the corrplots:

Code

Q3.3 <- coef(m3.3, type = "matrix")$Q
corrmat3.3 <- diag(1/sqrt(diag(Q3.3))) %*% Q3.3 %*% diag(1/sqrt(diag(Q3.3)))
corrplot(corrmat3.3)

Hypothesis 3.4

The Q matrix for the variance of process errors is “unconstrained”. Meaning that each hidden state is allowed to vary separately as is the correlation between the underlying states.

I’d expect this model to be the best of hypothesis three, as allowing correlation between the two states seems to improve performance.

Code

mod.list3.4 <- list(
  U = U_mat3,
  R = "diagonal and equal",
  Q = "unconstrained",
  Z = Z_mat3
)
m3.4 <- MARSS(dat, model = mod.list3.4)

Success! abstol and log-log tests passed at 156 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 156 iterations. 
Log-likelihood: -532.1349 
AIC: 1106.27   AICc: 1107.787   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                  -0.91355
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                         -1.17337
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer  0.27329
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer             -0.50940
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                -0.07454
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                  0.16051
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                 -0.91259
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                             1.81473
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          0.60088
A.Steelhead (Middle Columbia River DPS) Naches River - summer                               0.53840
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                0.51522
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                           -0.30421
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer               -1.10553
R.diag                                                                                      0.25947
U.North                                                                                     0.00786
U.South                                                                                    -0.01749
Q.(1,1)                                                                                     0.07814
Q.(2,1)                                                                                     0.10613
Q.(2,2)                                                                                     0.15103
x0.X1                                                                                       5.41086
x0.X2                                                                                       7.78119
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model performs the best of hypothesis 3 in terms of AICc.

Let’s look at plots.

Code

autoplot(m3.4)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model isn’t fitting all the data particularly well, and there is CLEAR structure in the residuals in X2 (John Day and Cascades). This model isn’t gonna cut it, but let’s look at the estimates:

Code

print(fit3.4_smooth<-tsSmooth(m3.4))

    .rownames  t .estimate       .se
1          X1  1  5.418442 0.1727225
2          X1  2  5.344432 0.1882528
3          X1  3  5.425942 0.1812055
4          X1  4  5.506016 0.1857493
5          X1  5  5.421226 0.1894500
6          X1  6  5.627329 0.1838281
7          X1  7  5.807461 0.1934857
8          X1  8  6.072003 0.1782095
9          X1  9  5.833735 0.1697881
10         X1 10  5.353216 0.1688755
11         X1 11  5.609680 0.1687191
12         X1 12  5.775760 0.1691062
13         X1 13  5.772538 0.1898016
14         X1 14  5.630974 0.1688991
15         X1 15  5.463751 0.1683718
16         X1 16  5.546488 0.1735483
17         X1 17  5.770454 0.1731943
18         X1 18  5.747545 0.1674021
19         X1 19  5.757602 0.1719044
20         X1 20  5.576753 0.1614083
21         X1 21  4.967975 0.1602269
22         X1 22  5.410376 0.1589139
23         X1 23  5.471418 0.1572075
24         X1 24  5.487648 0.1548610
25         X1 25  5.423907 0.1505070
26         X1 26  5.500080 0.1294593
27         X1 27  5.940052 0.1240356
28         X1 28  6.135302 0.1225602
29         X1 29  6.252990 0.1185448
30         X1 30  6.119486 0.1177983
31         X1 31  5.525333 0.1174224
32         X1 32  5.388732 0.1149499
33         X1 33  5.354597 0.1169278
34         X1 34  5.781525 0.1143481
35         X1 35  5.420365 0.1115277
36         X1 36  5.354754 0.1112398
37         X1 37  5.251513 0.1111273
38         X1 38  5.318872 0.1110646
39         X1 39  5.446545 0.1110275
40         X1 40  5.502383 0.1110056
41         X1 41  5.652753 0.1109927
42         X1 42  6.003675 0.1109854
43         X1 43  6.316871 0.1109816
44         X1 44  6.651722 0.1109805
45         X1 45  6.356882 0.1109816
46         X1 46  6.131226 0.1109853
47         X1 47  6.021661 0.1109926
48         X1 48  5.853327 0.1110054
49         X1 49  6.063305 0.1110272
50         X1 50  6.181036 0.1110640
51         X1 51  6.481281 0.1111257
52         X1 52  6.625285 0.1112290
53         X1 53  6.718413 0.1114017
54         X1 54  6.715974 0.1116902
55         X1 55  6.514381 0.1121720
56         X1 56  6.518943 0.1129788
57         X1 57  6.512852 0.1143713
58         X1 58  6.179792 0.1173383
59         X1 59  5.681432 0.1342986
60         X1 60  5.753569 0.1401443
61         X1 61  5.693915 0.1938515
62         X1 62  5.736096 0.2523263
63         X1 63  5.825363 0.2860757
64         X1 64  5.914631 0.2440031
65         X2  1  7.763682 0.2315774
66         X2  2  7.630067 0.2429479
67         X2  3  7.717767 0.2205092
68         X2  4  7.803422 0.2184182
69         X2  5  7.654466 0.2169355
70         X2  6  7.919471 0.2005271
71         X2  7  8.147517 0.2143797
72         X2  8  8.495685 0.1847808
73         X2  9  8.128317 0.1782024
74         X2 10  7.410558 0.1780712
75         X2 11  7.732657 0.1781893
76         X2 12  7.926325 0.1802508
77         X2 13  7.879548 0.2237796
78         X2 14  7.639940 0.1802776
79         X2 15  7.368606 0.1787450
80         X2 16  7.457391 0.1907147
81         X2 17  7.754819 0.1907242
82         X2 18  7.700404 0.1791125
83         X2 19  7.695842 0.1895497
84         X2 20  7.408694 0.1681428
85         X2 21  6.522656 0.1673684
86         X2 22  7.151812 0.1673222
87         X2 23  7.246456 0.1673037
88         X2 24  7.272053 0.1672981
89         X2 25  7.166472 0.1670758
90         X2 26  7.253030 0.1552215
91         X2 27  7.850214 0.1484565
92         X2 28  8.106436 0.1481033
93         X2 29  8.240223 0.1460178
94         X2 30  8.004963 0.1458888
95         X2 31  7.135422 0.1456365
96         X2 32  6.913538 0.1400904
97         X2 33  6.812346 0.1453332
98         X2 34  7.346920 0.1398646
99         X2 35  6.792440 0.1380915
100        X2 36  6.682172 0.1380702
101        X2 37  6.494634 0.1380432
102        X2 38  6.572670 0.1380243
103        X2 39  6.715855 0.1380128
104        X2 40  6.761353 0.1380059
105        X2 41  6.949232 0.1380019
106        X2 42  7.395682 0.1379996
107        X2 43  7.762648 0.1379984
108        X2 44  8.160508 0.1379981
109        X2 45  7.675061 0.1379984
110        X2 46  7.261642 0.1379996
111        X2 47  7.021384 0.1380019
112        X2 48  6.749146 0.1380059
113        X2 49  7.026065 0.1380127
114        X2 50  7.140479 0.1380242
115        X2 51  7.529024 0.1380436
116        X2 52  7.681094 0.1380759
117        X2 53  7.805863 0.1381301
118        X2 54  7.814599 0.1382207
119        X2 55  7.545744 0.1383719
120        X2 56  7.570675 0.1386212
121        X2 57  7.539414 0.1390011
122        X2 58  7.065558 0.1393636
123        X2 59  6.394764 0.1464324
124        X2 60  6.516058 0.1499635
125        X2 61  6.430498 0.2576196
126        X2 62  6.464893 0.3425179
127        X2 63  6.563245 0.3690075
128        X2 64  6.661598 0.2702812

And finally the corrplots:

Code

Q3.4 <- coef(m3.4, type = "matrix")$Q
corrmat3.4 <- diag(1/sqrt(diag(Q3.4))) %*% Q3.4 %*% diag(1/sqrt(diag(Q3.4)))
corrplot(corrmat3.4)

This hypothesis didn’t perform as well as anticipated. The two underlying states of nature didn’t seem to inform each other very well, and while model fits seemed to improve when the process errors were allowed to correlate, some of the residuals had structure, and residuals didn’t appear to be normal. While this hypothesis explored the north and south areas as being separate, hypothesis 2, with four distinct DPCs performed better.

Hypothesis 4:

Salmon of the Yakima group have to swim the furthest to reach their spawning ground, including a large bend in the river that heads back west. They are the most isolated group and thus may have their own hidden state while the other 3 major population groups maybe more closely linked to each other due to their closer geographic proximity. Thus, we hypothesize that there may be two underlying states describing the entire system. The first describing just the Yakima group while the second describes the Cascades, John Day, and Walla Walla groups.

\[ \text{Hypothesis Four}: \begin{bmatrix} y_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6\\ y_7\\ y_8\\ y_9\\ y_{10}\\ y_{11}\\ y_{12}\\ y_{13}\\ y_{14}\\ y_{15}\\ \end{bmatrix}_t= \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ \end{bmatrix}* \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix}_t+ \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6\\ a_7\\ a_8\\ a_9\\ a_{10}\\ a_{11}\\ a_{12}\\ a_{13}\\ a_{14}\\ a_{15}\\ \end{bmatrix}+ \begin{bmatrix} w_1\\ w_2\\ w_3\\ w_4\\ w_5\\ w_6\\ w_7\\ w_8\\ w_9\\ w_{10}\\ w_{11}\\ w_{12}\\ w_{13}\\ w_{14}\\ w_{15}\\ \end{bmatrix}_t \] \[ \text{Where }w \sim MVN \begin{pmatrix} \text{0,}\begin{bmatrix} R \end{bmatrix} \end{pmatrix} \]

\[ \text{Where }w \sim MVN \begin{pmatrix} \text{0,}\begin{bmatrix} R \end{bmatrix} \end{pmatrix} \] Hypothesis 4.1: Q matrix is “diagonal and equal” meaning the two hidden, underlying states will have equal variance but will not be correlated to each other.

Hypothesis 4.1

The Q matrix for the variance of process errors is “diagonal and equal” meaning each state (x) model has the same process error but they are not correlated to each other.

Code

U_mat4 <- matrix(c("South_group","Yakima"),2,1)
#make Z matrix correspond to 4 hidden states
Z_mat4 <- matrix(c(rep(c(1,0),3),
                  rep(c(1,0),5),
                  rep(c(1,0),3),
                  rep(c(0,1),4)),15,2, byrow=TRUE)

mod.list4.1 <- list(
  U = U_mat4,
  R = "diagonal and equal",
  Q = "diagonal and equal",
  Z = Z_mat4
)
m4.1 <- MARSS(dat, model = mod.list4.1)

Success! abstol and log-log tests passed at 130 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 130 iterations. 
Log-likelihood: -539.7939 
AIC: 1117.588   AICc: 1118.832   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                   -0.8890
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                          -1.1568
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer   0.3160
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer              -0.4616
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                 -0.0255
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                   0.2078
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                  -0.8628
A.Steelhead (Middle Columbia River DPS) Touchet River - summer                              -1.2022
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                              0.3662
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          -0.5185
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                -0.0226
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                            -0.8420
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer                -1.6434
R.diag                                                                                       0.2567
U.South_group                                                                               -0.0167
U.Yakima                                                                                     0.0440
Q.diag                                                                                       0.1153
x0.X1                                                                                        7.7289
x0.X2                                                                                        4.7154
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

On first look, this AICc does ok. Let’s look at our plots.

Code

autoplot(m4.1)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

HUGE CIs on hidden state of Yakima group where data is missing. Let’s makes sense because with no correlation, the Yakima group isn’t being informed by anything.

Yakima continues its positive drift, collective group shows negative drift

QQ plots for Yakima are clearly not normal, but this is likely because of so much missing data.

Let’s look at estimates:

Code

print(fit4.1_smooth<-tsSmooth(m4.1))

    .rownames  t .estimate       .se
1          X1  1  7.712297 0.2143823
2          X1  2  7.600601 0.2278075
3          X1  3  7.673852 0.2095918
4          X1  4  7.749386 0.2073309
5          X1  5  7.634119 0.2056141
6          X1  6  7.873875 0.1915571
7          X1  7  8.087480 0.2025566
8          X1  8  8.375908 0.1766445
9          X1  9  7.949571 0.1653731
10         X1 10  7.296163 0.1647936
11         X1 11  7.577644 0.1648552
12         X1 12  7.750212 0.1666618
13         X1 13  7.680975 0.1993935
14         X1 14  7.507111 0.1666884
15         X1 15  7.276062 0.1653717
16         X1 16  7.354580 0.1748097
17         X1 17  7.603477 0.1748302
18         X1 18  7.570314 0.1657402
19         X1 19  7.537757 0.1737707
20         X1 20  7.313922 0.1567451
21         X1 21  6.550545 0.1560083
22         X1 22  7.076916 0.1559780
23         X1 23  7.148219 0.1559768
24         X1 24  7.157472 0.1559763
25         X1 25  7.092364 0.1559650
26         X1 26  7.239600 0.1556899
27         X1 27  7.750147 0.1488156
28         X1 28  8.017426 0.1483750
29         X1 29  8.093887 0.1423875
30         X1 30  7.836010 0.1422058
31         X1 31  7.126864 0.1420543
32         X1 32  6.985240 0.1369105
33         X1 33  6.844887 0.1418969
34         X1 34  7.245269 0.1366543
35         X1 35  6.820459 0.1318227
36         X1 36  6.814317 0.1317161
37         X1 37  6.578377 0.1317137
38         X1 38  6.712012 0.1317137
39         X1 39  6.746773 0.1317137
40         X1 40  6.805910 0.1317137
41         X1 41  7.007192 0.1317137
42         X1 42  7.395275 0.1317137
43         X1 43  7.658534 0.1317137
44         X1 44  8.012211 0.1317137
45         X1 45  7.651067 0.1317137
46         X1 46  7.208946 0.1317137
47         X1 47  6.959559 0.1317137
48         X1 48  6.807582 0.1317137
49         X1 49  7.131001 0.1317137
50         X1 50  7.129599 0.1317137
51         X1 51  7.495275 0.1317137
52         X1 52  7.603977 0.1317137
53         X1 53  7.715080 0.1317137
54         X1 54  7.708201 0.1317137
55         X1 55  7.505100 0.1317137
56         X1 56  7.511069 0.1317137
57         X1 57  7.471765 0.1317137
58         X1 58  7.028010 0.1317147
59         X1 59  6.496067 0.1317603
60         X1 60  6.603709 0.1338499
61         X1 61  6.575777 0.2086323
62         X1 62  6.645130 0.2771233
63         X1 63  6.653042 0.3228770
64         X1 64  6.660953 0.2608395
65         X2  1  4.766553 0.3330594
66         X2  2  4.817707 0.4616513
67         X2  3  4.868861 0.5536967
68         X2  4  4.920014 0.6255422
69         X2  5  4.971168 0.6835868
70         X2  6  5.022322 0.7311249
71         X2  7  5.073476 0.7701046
72         X2  8  5.124629 0.8017752
73         X2  9  5.175783 0.8269768
74         X2 10  5.226937 0.8462875
75         X2 11  5.278091 0.8601043
76         X2 12  5.329244 0.8686893
77         X2 13  5.380398 0.8721970
78         X2 14  5.431552 0.8706887
79         X2 15  5.482705 0.8641383
80         X2 16  5.533859 0.8524294
81         X2 17  5.585013 0.8353453
82         X2 18  5.636167 0.8125468
83         X2 19  5.687320 0.7835354
84         X2 20  5.738474 0.7475882
85         X2 21  5.789628 0.7036428
86         X2 22  5.840782 0.6500793
87         X2 23  5.891935 0.5842584
88         X2 24  5.943089 0.5013755
89         X2 25  5.994243 0.3907196
90         X2 26  6.045396 0.2126290
91         X2 27  6.400762 0.1910445
92         X2 28  6.520126 0.1891863
93         X2 29  6.582926 0.1890349
94         X2 30  6.528614 0.1890226
95         X2 31  6.004172 0.1890216
96         X2 32  5.675394 0.1890215
97         X2 33  5.784901 0.1890215
98         X2 34  6.143324 0.1890215
99         X2 35  5.831863 0.1890215
100        X2 36  5.474207 0.1890215
101        X2 37  5.570093 0.1890215
102        X2 38  5.471317 0.1890215
103        X2 39  5.779878 0.1890215
104        X2 40  5.884468 0.1890215
105        X2 41  5.970907 0.1890215
106        X2 42  6.323477 0.1890215
107        X2 43  6.829211 0.1890215
108        X2 44  7.086948 0.1890215
109        X2 45  6.837592 0.1890215
110        X2 46  6.891836 0.1890215
111        X2 47  6.926740 0.1890215
112        X2 48  6.618598 0.1890215
113        X2 49  6.520460 0.1890215
114        X2 50  6.937462 0.1890215
115        X2 51  7.163273 0.1890215
116        X2 52  7.495525 0.1890215
117        X2 53  7.565650 0.1890216
118        X2 54  7.531363 0.1890226
119        X2 55  7.323783 0.1890358
120        X2 56  7.238950 0.1891984
121        X2 57  7.318024 0.1911914
122        X2 58  7.265110 0.2142508
123        X2 59  7.309085 0.4014966
124        X2 60  7.353059 0.5258285
125        X2 61  7.397033 0.6259327
126        X2 62  7.441008 0.7121010
127        X2 63  7.484982 0.7889132
128        X2 64  7.528957 0.8588831

The corrplot is as expected.

Code

Q4.1 <- coef(m4.1, type = "matrix")$Q
corrmat4.1 <- diag(1/sqrt(diag(Q4.1))) %*% Q4.1 %*% diag(1/sqrt(diag(Q4.1)))
corrplot(corrmat4.1)

Code

#As expected for this Q call

The confidence intervals on the underlying state and the fitted values have really big balloon shapes to them where data is missing from each system. The Yakima system, which was assumed to have its own underlying state in this hypothesis, has much large confidence intervals than all the other systems in the fitted values plot. The ACF plots do show that about half of the systems disply autocorrelation in their residuals. The Residuals normality test for the underlying states, X1 (all but Yakima) and X2 (Yakima), show that the residuals for the X2 are not normally distributed as they vary from the qqline to a great degree. The corrplot is relatively uninformative as we forced the Q matrix to be diagonal and equal.

Hypothesis 4.2

Q matrix is “diagonal and unequal”, meaning the two hidden, underlying states will have the same variance but will not be allowed to be correlated to one another.

Code

mod.list4.2 <- list(
  U = U_mat4,
  R = "diagonal and equal",
  Q = "diagonal and unequal",
  Z = Z_mat4
)
m4.2 <- MARSS(dat, model = mod.list4.2)

Success! abstol and log-log tests passed at 130 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 130 iterations. 
Log-likelihood: -539.7371 
AIC: 1119.474   AICc: 1120.851   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                   -0.8890
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                          -1.1570
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer   0.3160
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer              -0.4616
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                 -0.0257
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                   0.2077
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                  -0.8629
A.Steelhead (Middle Columbia River DPS) Touchet River - summer                              -1.2023
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                              0.3662
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          -0.5185
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                -0.0225
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                            -0.8420
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer                -1.6433
R.diag                                                                                       0.2568
U.South_group                                                                               -0.0167
U.Yakima                                                                                     0.0438
Q.(X1,X1)                                                                                    0.1205
Q.(X2,X2)                                                                                    0.1020
x0.X1                                                                                        7.7297
x0.X2                                                                                        4.7277
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This AICc isn’t looking very promising.

Code

autoplot(m4.2)

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model looks pretty similarly to the last model, which is to say not great. There continues to be very large CIs on hidden state of Yakima group where data is missing, again because there is not data or any correlation from the other underlying state to inform it. The CIs for fitted values show the same pattern qqplots show more variation in the Yakima group.

Let’s look at the estimates:

Code

print(fit4.2_smooth<-tsSmooth(m4.2))

    .rownames  t .estimate       .se
1          X1  1  7.713088 0.2171542
2          X1  2  7.597869 0.2302268
3          X1  3  7.673348 0.2113186
4          X1  4  7.750450 0.2090800
5          X1  5  7.629621 0.2074018
6          X1  6  7.873388 0.1929693
7          X1  7  8.088784 0.2044279
8          X1  8  8.384151 0.1778855
9          X1  9  7.951740 0.1664611
10         X1 10  7.287923 0.1659052
11         X1 11  7.577577 0.1659624
12         X1 12  7.752995 0.1677206
13         X1 13  7.683337 0.2013568
14         X1 14  7.507536 0.1677448
15         X1 15  7.272938 0.1664597
16         X1 16  7.352788 0.1761058
17         X1 17  7.606312 0.1761249
18         X1 18  7.571590 0.1668217
19         X1 19  7.540859 0.1751015
20         X1 20  7.317274 0.1577066
21         X1 21  6.540679 0.1569962
22         X1 22  7.079167 0.1569687
23         X1 23  7.149301 0.1569676
24         X1 24  7.157923 0.1569672
25         X1 25  7.090095 0.1569570
26         X1 26  7.236472 0.1566930
27         X1 27  7.751988 0.1496947
28         X1 28  8.019884 0.1492727
29         X1 29  8.097283 0.1431839
30         X1 30  7.839286 0.1430106
31         X1 31  7.123193 0.1428661
32         X1 32  6.984256 0.1376312
33         X1 33  6.841647 0.1427164
34         X1 34  7.250119 0.1373876
35         X1 35  6.818691 0.1324899
36         X1 36  6.815270 0.1323887
37         X1 37  6.576289 0.1323866
38         X1 38  6.712362 0.1323866
39         X1 39  6.746573 0.1323866
40         X1 40  6.804876 0.1323866
41         X1 41  7.006022 0.1323866
42         X1 42  7.395977 0.1323866
43         X1 43  7.658772 0.1323866
44         X1 44  8.016926 0.1323866
45         X1 45  7.652119 0.1323866
46         X1 46  7.207774 0.1323866
47         X1 47  6.958430 0.1323866
48         X1 48  6.804689 0.1323866
49         X1 49  7.132426 0.1323866
50         X1 50  7.127746 0.1323866
51         X1 51  7.496731 0.1323866
52         X1 52  7.604357 0.1323866
53         X1 53  7.716102 0.1323866
54         X1 54  7.709484 0.1323866
55         X1 55  7.504097 0.1323866
56         X1 56  7.511643 0.1323866
57         X1 57  7.474494 0.1323866
58         X1 58  7.028453 0.1323874
59         X1 59  6.492198 0.1324282
60         X1 60  6.603952 0.1344222
61         X1 61  6.574936 0.2104532
62         X1 62  6.646393 0.2804938
63         X1 63  6.654190 0.3277061
64         X1 64  6.661987 0.2618778
65         X2  1  4.778820 0.3132035
66         X2  2  4.829931 0.4341431
67         X2  3  4.881041 0.5207221
68         X2  4  4.932151 0.5883112
69         X2  5  4.983261 0.6429278
70         X2  6  5.034371 0.6876697
71         X2  7  5.085482 0.7243690
72         X2  8  5.136592 0.7542007
73         X2  9  5.187702 0.7779552
74         X2 10  5.238812 0.7961767
75         X2 11  5.289922 0.8092390
76         X2 12  5.341033 0.8173895
77         X2 13  5.392143 0.8207746
78         X2 14  5.443253 0.8194532
79         X2 15  5.494363 0.8134025
80         X2 16  5.545473 0.8025155
81         X2 17  5.596584 0.7865914
82         X2 18  5.647694 0.7653158
83         X2 19  5.698804 0.7382262
84         X2 20  5.749914 0.7046524
85         X2 21  5.801024 0.6636110
86         X2 22  5.852135 0.6136053
87         X2 23  5.903245 0.5522053
88         X2 24  5.954355 0.4750131
89         X2 25  6.005465 0.3723326
90         X2 26  6.056575 0.2095544
91         X2 27  6.394413 0.1874200
92         X2 28  6.513676 0.1852326
93         X2 29  6.572805 0.1850284
94         X2 30  6.512455 0.1850094
95         X2 31  6.011024 0.1850076
96         X2 32  5.693510 0.1850075
97         X2 33  5.792239 0.1850075
98         X2 34  6.122720 0.1850075
99         X2 35  5.828509 0.1850075
100        X2 36  5.488243 0.1850075
101        X2 37  5.571224 0.1850075
102        X2 38  5.484049 0.1850075
103        X2 39  5.777213 0.1850075
104        X2 40  5.885964 0.1850075
105        X2 41  5.981148 0.1850075
106        X2 42  6.327910 0.1850075
107        X2 43  6.817186 0.1850075
108        X2 44  7.068274 0.1850075
109        X2 45  6.841617 0.1850075
110        X2 46  6.889785 0.1850075
111        X2 47  6.917703 0.1850075
112        X2 48  6.628168 0.1850075
113        X2 49  6.539539 0.1850075
114        X2 50  6.936611 0.1850075
115        X2 51  7.163445 0.1850075
116        X2 52  7.484730 0.1850075
117        X2 53  7.557295 0.1850077
118        X2 54  7.524409 0.1850095
119        X2 55  7.327414 0.1850300
120        X2 56  7.244774 0.1852498
121        X2 57  7.316352 0.1876029
122        X2 58  7.268717 0.2113120
123        X2 59  7.312541 0.3828910
124        X2 60  7.356366 0.4985561
125        X2 61  7.400190 0.5920396
126        X2 62  7.444015 0.6726542
127        X2 63  7.487839 0.7445914
128        X2 64  7.531664 0.8101661

Let’s look at the corrplot

Code

Q4.2 <- coef(m4.2, type = "matrix")$Q
corrmat4.2 <- diag(1/sqrt(diag(Q4.2))) %*% Q4.2 %*% diag(1/sqrt(diag(Q4.2)))
corrplot(corrmat4.2)

Code

#As expected for this Q call

This model, like the previous one, did not perform well with very large confidence intervals for X2 (Yakima), which is not surprising given that there wasn’t much informing X2. The Yakima group only has large balloon shaped confidence intervals on its fitted values while the other river systems have some structure to their confidence intervals in sections with missing data. Very similar ACF plots to the other Hypothesis Four models, showing structuring and multiple significant lags in approximately half of the plots. The variance was allowed to vary independently between the two underlying states but they were estimated to be very similar to one another

Hypothesis 4.3

The Q matrix is “equal variance and covariance”. This will result in both of the hidden, underlying states having the same variance and they will be correlated to one another. This model should perform better than the previous models.

Code

mod.list4.3 <- list(
  U = U_mat4,
  R = "diagonal and equal",
  Q = "equalvarcov",
  Z = Z_mat4
)
m4.3 <- MARSS(dat, model = mod.list4.3)

Success! abstol and log-log tests passed at 464 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 464 iterations. 
Log-likelihood: -522.7461 
AIC: 1085.492   AICc: 1086.869   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                   -0.9112
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                          -1.1767
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer   0.2939
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer              -0.4838
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                 -0.0477
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                   0.1856
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                  -0.8850
A.Steelhead (Middle Columbia River DPS) Touchet River - summer                              -1.2192
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                              0.3440
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          -0.5291
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                -0.0136
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                            -0.8330
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer                -1.6343
R.diag                                                                                       0.2571
U.South_group                                                                               -0.0167
U.Yakima                                                                                     0.0334
Q.diag                                                                                       0.1176
Q.offdiag                                                                                    0.1175
x0.X1                                                                                        7.7516
x0.X2                                                                                        4.8589
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model did in fact perform better based on AICc! Let’s look at our plots:

Code

autoplot(m4.3)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model looks much better, with tighter confidence levels that are being informed with the correlated process errors. However, there is a clear residual pattern in X2 (Yakima) which also has a wiggly QQ plot. Meanwhile X1 has a fat left tail.

Let’s look at the estimates:

Code

print(fit4.3_smooth<-tsSmooth(m4.3))

    .rownames  t .estimate       .se
1          X1  1  7.734863 0.2156542
2          X1  2  7.621694 0.2289443
3          X1  3  7.695833 0.2104342
4          X1  4  7.771997 0.2081805
5          X1  5  7.654469 0.2064783
6          X1  6  7.895860 0.1922594
7          X1  7  8.110207 0.2034529
8          X1  8  8.401457 0.1772674
9          X1  9  7.972633 0.1659261
10         X1 10  7.314966 0.1653559
11         X1 11  7.599785 0.1654158
12         X1 12  7.773530 0.1672038
13         X1 13  7.704131 0.2003253
14         X1 14  7.529461 0.1672294
15         X1 15  7.296955 0.1659247
16         X1 16  7.376017 0.1754538
17         X1 17  7.626808 0.1754738
18         X1 18  7.593020 0.1662909
19         X1 19  7.561213 0.1744284
20         X1 20  7.337474 0.1572410
21         X1 21  6.568709 0.1565145
22         X1 22  7.100114 0.1564853
23         X1 23  7.171366 0.1564828
24         X1 24  7.182443 0.1564491
25         X1 25  7.126649 0.1556058
26         X1 26  7.325691 0.1327852
27         X1 27  7.854397 0.1280392
28         X1 28  8.050286 0.1278854
29         X1 29  8.119447 0.1240084
30         X1 30  7.934258 0.1239458
31         X1 31  7.167471 0.1238918
32         X1 32  6.945001 0.1204123
33         X1 33  6.871715 0.1238425
34         X1 34  7.365617 0.1203173
35         X1 35  6.869345 0.1170224
36         X1 36  6.728017 0.1169813
37         X1 37  6.596640 0.1169863
38         X1 38  6.627159 0.1169923
39         X1 39  6.764356 0.1169985
40         X1 40  6.804720 0.1170047
41         X1 41  6.936846 0.1170111
42         X1 42  7.322716 0.1170176
43         X1 43  7.671446 0.1170242
44         X1 44  8.028496 0.1170309
45         X1 45  7.600749 0.1170378
46         X1 46  7.289629 0.1170449
47         X1 47  7.128139 0.1170521
48         X1 48  6.864724 0.1170595
49         X1 49  7.060122 0.1170671
50         X1 50  7.194498 0.1170750
51         X1 51  7.511118 0.1170830
52         X1 52  7.688430 0.1170912
53         X1 53  7.768679 0.1170997
54         X1 54  7.744574 0.1171085
55         X1 55  7.489668 0.1171174
56         X1 56  7.462192 0.1171258
57         X1 57  7.473148 0.1171297
58         X1 58  7.075461 0.1172746
59         X1 59  6.517079 0.1318246
60         X1 60  6.622739 0.1341650
61         X1 61  6.593270 0.2095121
62         X1 62  6.664116 0.2786770
63         X1 63  6.673330 0.3250464
64         X1 64  6.682543 0.2614355
65         X2  1  4.892252 0.2157009
66         X2  2  4.829193 0.2290564
67         X2  3  4.953415 0.2106386
68         X2  4  5.079662 0.2084640
69         X2  5  5.012245 0.2068399
70         X2  6  5.303695 0.1927314
71         X2  7  5.568105 0.2039695
72         X2  8  5.909408 0.1779508
73         X2  9  5.530739 0.1667459
74         X2 10  4.923260 0.1662638
75         X2 11  5.258133 0.1664065
76         X2 12  5.481947 0.1682641
77         X2 13  5.462651 0.2012684
78         X2 14  5.338100 0.1684454
79         X2 15  5.155721 0.1672264
80         X2 16  5.284866 0.1767528
81         X2 17  5.585715 0.1768411
82         X2 18  5.602025 0.1678061
83         X2 19  5.620317 0.1759367
84         X2 20  5.446704 0.1589883
85         X2 21  4.728143 0.1583388
86         X2 22  5.309566 0.1583757
87         X2 23  5.430902 0.1584308
88         X2 24  5.492071 0.1584206
89         X2 25  5.486378 0.1574377
90         X2 26  5.735486 0.1337102
91         X2 27  6.314140 0.1289557
92         X2 28  6.559913 0.1288329
93         X2 29  6.678984 0.1251261
94         X2 30  6.543762 0.1251085
95         X2 31  5.826899 0.1250979
96         X2 32  5.654243 0.1217841
97         X2 33  5.630870 0.1251254
98         X2 34  6.174598 0.1217852
99         X2 35  5.728119 0.1186873
100        X2 36  5.636469 0.1186984
101        X2 37  5.554989 0.1187435
102        X2 38  5.635322 0.1187888
103        X2 39  5.822526 0.1188346
104        X2 40  5.912876 0.1188812
105        X2 41  6.094990 0.1189285
106        X2 42  6.530957 0.1189768
107        X2 43  6.929947 0.1190259
108        X2 44  7.337240 0.1190761
109        X2 45  6.959835 0.1191274
110        X2 46  6.699198 0.1191798
111        X2 47  6.588064 0.1192335
112        X2 48  6.374747 0.1192886
113        X2 49  6.620113 0.1193450
114        X2 50  6.804672 0.1194030
115        X2 51  7.171327 0.1194625
116        X2 52  7.398733 0.1195237
117        X2 53  7.528966 0.1195867
118        X2 54  7.554813 0.1196516
119        X2 55  7.349859 0.1197187
120        X2 56  7.372354 0.1197901
121        X2 57  7.433397 0.1198866
122        X2 58  7.085883 0.1203695
123        X2 59  6.577633 0.1368446
124        X2 60  6.733331 0.1395225
125        X2 61  6.753918 0.2130669
126        X2 62  6.874807 0.2814064
127        X2 63  6.934071 0.3274303
128        X2 64  6.993336 0.2644798

Let’s look at the corrplots

Code

Q4.3 <- coef(m4.3, type = "matrix")$Q
corrmat4.3 <- diag(1/sqrt(diag(Q4.3))) %*% Q4.3 %*% diag(1/sqrt(diag(Q4.3)))
corrplot(corrmat4.3)

The confidence intervals for the underlying states and the fitted values now fit the estimated abundance in each river well rather than being oval shaped over any missing values. There were very similar ACF plots to the other models in this model with about half of them showing sine wave patterns. The variance-covariance matrix was forced to be equal.

Hypothesis 4.4

The Q matrix is “unconstrained”. Meaning the two hidden, underlying states will be allowed to vary independently of one another and correlation is allowed to vary between the two states.

Code

mod.list4.4 <- list(
  U = U_mat4,
  R = "diagonal and equal",
  Q = "unconstrained",
  Z = Z_mat4
)
m4.4 <- MARSS(dat, model = mod.list4.4)

Success! abstol and log-log tests passed at 468 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 468 iterations. 
Log-likelihood: -522.7025 
AIC: 1087.405   AICc: 1088.922   
 
                                                                                           Estimate
A.Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer                   -0.9118
A.Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter                          -1.1772
A.Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer   0.2933
A.Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer              -0.4844
A.Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer                 -0.0482
A.Steelhead (Middle Columbia River DPS) North Fork John Day River - summer                   0.1850
A.Steelhead (Middle Columbia River DPS) South Fork John Day River - summer                  -0.8856
A.Steelhead (Middle Columbia River DPS) Touchet River - summer                              -1.2198
A.Steelhead (Middle Columbia River DPS) Umatilla River - summer                              0.3434
A.Steelhead (Middle Columbia River DPS) Walla Walla River - summer                          -0.5298
A.Steelhead (Middle Columbia River DPS) Satus Creek - summer                                -0.0130
A.Steelhead (Middle Columbia River DPS) Toppenish Creek - summer                            -0.8324
A.Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer                -1.6338
R.diag                                                                                       0.2571
U.South_group                                                                               -0.0167
U.Yakima                                                                                     0.0332
Q.(1,1)                                                                                      0.1169
Q.(2,1)                                                                                      0.1187
Q.(2,2)                                                                                      0.1206
x0.X1                                                                                        7.7520
x0.X2                                                                                        4.8616
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model did just a little worse than the model with a U matrix that had equal variance and covariance. Let’s look at plots:

Code

autoplot(m4.4)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

While this model does ok in some streams, it’s missing data in places, X2 (Yakima) has a clear residual structure and the QQ plot is very wiggly. X1 (the rest) does ok, but there are some outliers and the QQ plot continues to have a fat left tail.

Let’s look at estimates:

Code

print(fit4.4_smooth<-tsSmooth(m4.4))

    .rownames  t .estimate       .se
1          X1  1  7.735328 0.2152863
2          X1  2  7.622619 0.2286223
3          X1  3  7.696466 0.2102034
4          X1  4  7.772426 0.2079469
5          X1  5  7.655625 0.2062396
6          X1  6  7.896490 0.1920702
7          X1  7  8.110598 0.2032034
8          X1  8  8.400935 0.1771009
9          X1  9  7.972909 0.1657799
10         X1 10  7.316607 0.1652066
11         X1 11  7.600355 0.1652671
12         X1 12  7.773726 0.1670614
13         X1 13  7.704384 0.2000638
14         X1 14  7.529966 0.1670873
15         X1 15  7.297925 0.1657785
16         X1 16  7.376812 0.1752802
17         X1 17  7.626995 0.1753003
18         X1 18  7.593410 0.1661455
19         X1 19  7.561365 0.1742502
20         X1 20  7.337596 0.1571115
21         X1 21  6.570566 0.1563816
22         X1 22  7.100384 0.1563520
23         X1 23  7.171796 0.1563494
24         X1 24  7.182990 0.1563145
25         X1 25  7.127678 0.1554471
26         X1 26  7.327259 0.1321084
27         X1 27  7.853412 0.1274180
28         X1 28  8.047414 0.1272675
29         X1 29  8.116314 0.1234446
30         X1 30  7.932935 0.1233835
31         X1 31  7.169347 0.1233307
32         X1 32  6.946792 0.1198966
33         X1 33  6.874801 0.1232828
34         X1 34  7.367199 0.1198036
35         X1 35  6.872387 0.1165497
36         X1 36  6.729889 0.1165094
37         X1 37  6.600488 0.1165146
38         X1 38  6.629466 0.1165207
39         X1 39  6.767352 0.1165269
40         X1 40  6.807312 0.1165332
41         X1 41  6.938011 0.1165397
42         X1 42  7.322114 0.1165463
43         X1 43  7.670394 0.1165530
44         X1 44  8.025781 0.1165598
45         X1 45  7.599116 0.1165668
46         X1 46  7.290979 0.1165739
47         X1 47  7.131233 0.1165812
48         X1 48  6.867630 0.1165887
49         X1 49  7.060457 0.1165964
50         X1 50  7.196049 0.1166043
51         X1 51  7.510535 0.1166124
52         X1 52  7.687958 0.1166208
53         X1 53  7.767433 0.1166293
54         X1 54  7.743210 0.1166381
55         X1 55  7.488829 0.1166471
56         X1 56  7.460986 0.1166556
57         X1 57  7.472388 0.1166595
58         X1 58  7.076754 0.1168084
59         X1 59  6.518256 0.1317245
60         X1 60  6.623272 0.1340868
61         X1 61  6.593993 0.2092689
62         X1 62  6.664570 0.2782300
63         X1 63  6.673772 0.3244088
64         X1 64  6.682974 0.2612937
65         X2  1  4.894785 0.2187770
66         X2  2  4.830469 0.2323901
67         X2  3  4.955669 0.2137661
68         X2  4  5.083015 0.2115504
69         X2  5  5.014542 0.2098920
70         X2  6  5.309409 0.1956031
71         X2  7  5.577095 0.2069581
72         X2  8  5.922218 0.1806002
73         X2  9  5.537583 0.1692309
74         X2 10  4.921052 0.1687343
75         X2 11  5.259482 0.1688766
76         X2 12  5.485784 0.1707675
77         X2 13  5.465523 0.2041844
78         X2 14  5.338519 0.1709458
79         X2 15  5.152979 0.1696998
80         X2 16  5.283298 0.1793513
81         X2 17  5.587631 0.1794385
82         X2 18  5.603695 0.1702816
83         X2 19  5.621322 0.1785097
84         X2 20  5.444185 0.1613299
85         X2 21  4.715168 0.1606638
86         X2 22  5.303572 0.1606986
87         X2 23  5.426298 0.1607523
88         X2 24  5.487850 0.1607390
89         X2 25  5.481842 0.1597170
90         X2 26  5.734770 0.1350962
91         X2 27  6.319381 0.1303184
92         X2 28  6.566465 0.1301950
93         X2 29  6.686474 0.1264769
94         X2 30  6.550227 0.1264583
95         X2 31  5.824445 0.1264465
96         X2 32  5.648245 0.1231190
97         X2 33  5.625114 0.1264716
98         X2 34  6.175318 0.1231175
99         X2 35  5.722479 0.1200041
100        X2 36  5.627476 0.1200138
101        X2 37  5.545992 0.1200575
102        X2 38  5.625334 0.1201014
103        X2 39  5.815514 0.1201458
104        X2 40  5.906177 0.1201909
105        X2 41  6.089028 0.1202367
106        X2 42  6.529446 0.1202834
107        X2 43  6.933626 0.1203309
108        X2 44  7.345012 0.1203795
109        X2 45  6.961933 0.1204290
110        X2 46  6.699418 0.1204797
111        X2 47  6.587544 0.1205315
112        X2 48  6.369896 0.1205846
113        X2 49  6.615853 0.1206391
114        X2 50  6.803870 0.1206949
115        X2 51  7.173498 0.1207523
116        X2 52  7.403928 0.1208113
117        X2 53  7.534736 0.1208719
118        X2 54  7.560159 0.1209343
119        X2 55  7.351743 0.1209988
120        X2 56  7.373511 0.1210676
121        X2 57  7.435267 0.1211612
122        X2 58  7.083558 0.1216480
123        X2 59  6.566338 0.1387321
124        X2 60  6.723157 0.1414692
125        X2 61  6.743552 0.2160934
126        X2 62  6.865386 0.2853517
127        X2 63  6.924871 0.3319324
128        X2 64  6.984357 0.2684488

Finally we’ll look at corrplots. As hinted at by the model output there is very high correlation even though this was an unconstrained model.

Code

Q4.4 <- coef(m4.4, type = "matrix")$Q
corrmat4.4 <- diag(1/sqrt(diag(Q4.4))) %*% Q4.4 %*% diag(1/sqrt(diag(Q4.4)))
corrplot(corrmat4.4)

The confidence intervals on the underlying state and the fitted values fit the estimated abundances well in areas with missing data. QQplots for X1 (all but Yakima) had a fat left tail and X2 (Yakima) had a lot of structure in the residuals and wiggly QQ plots. The AFC plots show many of the streams have autocorrelated residuals. Even though this hypothesis allowed the Q matrix to be unconstrained, it still estimated variances and covariances that were essentially equal to the “equal variance and covariance” hypothesis.

AICc Results and Selected Model

Code

mods <- c("1","2.1","2.2","2.3","2.4","3.1","3.2", "3.3", "3.4","4.1","4.2","4.3","4.4")
aic <- c(m1$AICc, m2.1$AICc, m2.2$AICc, m2.3$AICc, m2.4$AICc,m3.1$AICc, m3.2$AICc, m3.3$AICc, m3.4$AICc, m4.1$AICc, m4.2$AICc, m4.3$AICc, m4.4$AICc)
daic <- aic-min(aic)
tab <- cbind.data.frame(mods, aic, daic)
kable(tab, col.names = c("Hypothesis", "AICc", "delta AICc"))

Hypothesis	AICc	delta AICc
1	1274.985	265.95836
2.1	1072.048	63.02114
2.2	1058.819	49.79222
2.3	1032.863	23.83661
2.4	1009.027	0.00000
3.1	1127.770	118.74285
3.2	1127.268	118.24138
3.3	1112.854	103.82729
3.4	1107.787	98.76043
4.1	1118.832	109.80504
4.2	1120.851	111.82460
4.3	1086.869	77.84256
4.4	1088.922	79.89551

The best model is Hypothesis 2.4 where it is assumed that the four main population groups form separate sub-populations. In this hypothesis we are utilizing 4 separate underlying states to model the observations from each of the main population groups. The Q matrix for the variance of process errors is “unconstrained”. Meaning that each hidden state is allowed to vary separately as is the correlation between the underlying states.

Cycling considerations for best model

Simple Cycling

First we try a simple approach as outlined in example code and assume a periodicity of about four years, as seen in some of the ACF plots.

Code

TT <- years
p <- 4 #try a period of 4

Z <- array(1, dim = c(15, 3, TT))
Z[1, 2, ] <- sin(2 * pi * (1:TT)/p)
Z[1, 3, ] <- cos(2 * pi * (1:TT)/p)

mod.list_test <- list(U = "zero", 
                      Q = "diagonal and unequal", 
                      Z = Z, 
                      A = "zero")

m <- dim(Z)[2]
m_test <- MARSS(dat, model = mod.list_test, inits = list(x0 = matrix(0,m, 1)))

Warning! Abstol convergence only. Maxit (=500) reached before log-log convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
WARNING: Abstol convergence only no log-log convergence.
 maxit (=500) reached before log-log convergence.
 The likelihood and params might not be at the ML values.
 Try setting control$maxit higher.
Log-likelihood: -850.9578 
AIC: 1715.916   AICc: 1716.095   
 
           Estimate
R.diag     0.790060
Q.(X1,X1)  0.051311
Q.(X2,X2)  0.017798
Q.(X3,X3)  0.000461
x0.X1      7.855865
x0.X2     -0.044586
x0.X3     -0.289316
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

Convergence warnings
 Warning: the  Q.(X3,X3)  parameter value has not converged.
 Warning: the  x0.X2  parameter value has not converged.
 Type MARSSinfo("convergence") for more info on this warning.

This model struggled to converge and did pretty poorly in terms of AICc.

Let’s look at some plots:

Code

plot_test<-autoplot(m_test)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

To be honest, with the U matrix equal to 0, I’m unsure what states we’re looking at. But they get worse as we go from X1, to X2, to X3 in terms of CI, residual patterns and QQ plots. Additionally, some of the models are completly missing data.

Corrplot is as expectd with the Q matrix set to diagonal and unequal.

Code

Qtest <- coef(m_test, type = "matrix")$Q
corrmat_test <- diag(1/sqrt(diag(Qtest))) %*% Qtest %*% diag(1/sqrt(diag(Qtest)))
corrplot(corrmat_test)

This model isn’t it. Let’s move onto a model based on our best performer with cycling considerations.

Hypthothesis 2.4 with Cycling

For this section, we’re going to explore 2 cycling options, 4 years and 9 years, as these are period where salmon are generally known to cycle (I think….need a source).

Four Years

We’ll use the U and Z matrices from Hypothesis 2:

Code

U_cyl <- matrix(c("Cascades","JohnDay","Walla","Yakima"),4,1)

Z_cyl <- matrix(c(rep(c(1,0,0,0),3),
                  rep(c(0,1,0,0),5),
                  rep(c(0,0,1,0),3),
                  rep(c(0,0,0,1),4)),15,4, byrow=TRUE)

And we’ll set up a co-variate matrix to allow for some cycling and set up our model list with Q unconstrained, and D unconstrained.

Code

d_cyl <- matrix(0,2,TT)

d_cyl[1,] <- sin(2 * pi * (1:TT)/p)
d_cyl[2,] <- cos(2 * pi * (1:TT)/p)


 
mod.list <- list(U = U_cyl, 
                 Q = "unconstrained",
                 Z = Z_cyl, 
                 A = "zero",
                 D="unconstrained",
                 d = d_cyl) 

m <- dim(Z_cyl)[2]
m_cyl_4 <- MARSS(dat, model = mod.list, inits = list(x0 = matrix(0, m, 1)))

Success! abstol and log-log tests passed at 456 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 456 iterations. 
Log-likelihood: -744.479 
AIC: 1586.958   AICc: 1595.392   
 
                                                                                               Estimate
R.diag                                                                                          0.53728
U.Cascades                                                                                     -0.02389
U.JohnDay                                                                                      -0.01849
U.Walla                                                                                        -0.02611
U.Yakima                                                                                        0.02795
Q.(1,1)                                                                                         0.06525
Q.(2,1)                                                                                         0.07433
Q.(3,1)                                                                                         0.04350
Q.(4,1)                                                                                         0.06408
Q.(2,2)                                                                                         0.13865
Q.(3,2)                                                                                         0.07049
Q.(4,2)                                                                                         0.10751
Q.(3,3)                                                                                         0.03716
Q.(4,3)                                                                                         0.05612
Q.(4,4)                                                                                         0.08504
x0.X1                                                                                           6.91912
x0.X2                                                                                           7.57238
x0.X3                                                                                           7.77344
x0.X4                                                                                           4.38479
D.(Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer,1)                   0.16957
D.(Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer,1)                   0.08829
D.(Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter,1)                          0.17364
D.(Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer,1)  0.02798
D.(Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer,1)              0.04088
D.(Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer,1)                 0.00434
D.(Steelhead (Middle Columbia River DPS) North Fork John Day River - summer,1)                 -0.06285
D.(Steelhead (Middle Columbia River DPS) South Fork John Day River - summer,1)                 -0.03264
D.(Steelhead (Middle Columbia River DPS) Touchet River - summer,1)                              0.03742
D.(Steelhead (Middle Columbia River DPS) Umatilla River - summer,1)                            -0.00726
D.(Steelhead (Middle Columbia River DPS) Walla Walla River - summer,1)                         -0.05636
D.(Steelhead (Middle Columbia River DPS) Naches River - summer,1)                              -0.10493
D.(Steelhead (Middle Columbia River DPS) Satus Creek - summer,1)                               -0.05741
D.(Steelhead (Middle Columbia River DPS) Toppenish Creek - summer,1)                           -0.18220
D.(Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer,1)               -0.07321
D.(Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer,2)                   0.11903
D.(Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer,2)                   0.05723
D.(Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter,2)                          0.14154
D.(Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer,2)  0.00397
D.(Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer,2)              0.01291
D.(Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer,2)                 0.10192
D.(Steelhead (Middle Columbia River DPS) North Fork John Day River - summer,2)                 -0.03743
D.(Steelhead (Middle Columbia River DPS) South Fork John Day River - summer,2)                  0.09649
D.(Steelhead (Middle Columbia River DPS) Touchet River - summer,2)                              0.05861
D.(Steelhead (Middle Columbia River DPS) Umatilla River - summer,2)                            -0.04049
D.(Steelhead (Middle Columbia River DPS) Walla Walla River - summer,2)                          0.06629
D.(Steelhead (Middle Columbia River DPS) Naches River - summer,2)                              -0.04680
D.(Steelhead (Middle Columbia River DPS) Satus Creek - summer,2)                               -0.04567
D.(Steelhead (Middle Columbia River DPS) Toppenish Creek - summer,2)                           -0.11343
D.(Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer,2)               -0.02867
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

Wow, this model AICc is BAD.

Code

autoplot(m_cyl_4)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

Well, this model does very poorly. The model is missing data, there are residual patterns in all four states, the QQ plots aren’t all terrible but not totally normal and there is some temporal correlation in the ACFs.

Let’s look at the corrplot:

Code

Q_4 <- coef(m_cyl_4, type = "matrix")$Q
corrmat_4 <- diag(1/sqrt(diag(Q_4))) %*% Q_4 %*% diag(1/sqrt(diag(Q_4)))
corrplot(corrmat_4)

The unconstrained Q matrix shows that there is a fair amount of correlation between states. This model overall is MUCH WORSE than no cycling.

Will different cycling assumptions perform any better?

Nine Years

And we’ll set up a co-variate matrix and change the p to 9.

Code

d_cyl <- matrix(0,2,TT)

p<-9

d_cyl[1,] <- sin(2 * pi * (1:TT)/p)
d_cyl[2,] <- cos(2 * pi * (1:TT)/p)


 
mod.list <- list(U = U_cyl, 
                 Q = "unconstrained",
                 Z = Z_cyl, 
                 A = "zero",
                 D="unconstrained",
                 d = d_cyl) 

m <- dim(Z_cyl)[2]
m_cyl_9 <- MARSS(dat, model = mod.list, inits = list(x0 = matrix(0, m, 1)))

Success! abstol and log-log tests passed at 456 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

MARSS fit is
Estimation method: kem 
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 456 iterations. 
Log-likelihood: -740.5612 
AIC: 1579.122   AICc: 1587.556   
 
                                                                                                Estimate
R.diag                                                                                          0.537851
U.Cascades                                                                                     -0.024773
U.JohnDay                                                                                      -0.019922
U.Walla                                                                                        -0.027059
U.Yakima                                                                                        0.027994
Q.(1,1)                                                                                         0.057938
Q.(2,1)                                                                                         0.057843
Q.(3,1)                                                                                         0.035569
Q.(4,1)                                                                                         0.051534
Q.(2,2)                                                                                         0.105069
Q.(3,2)                                                                                         0.054112
Q.(4,2)                                                                                         0.082292
Q.(3,3)                                                                                         0.029194
Q.(4,3)                                                                                         0.043774
Q.(4,4)                                                                                         0.065984
x0.X1                                                                                           6.923487
x0.X2                                                                                           7.606757
x0.X3                                                                                           7.795524
x0.X4                                                                                           4.362978
D.(Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer,1)                  -0.009839
D.(Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer,1)                   0.000495
D.(Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter,1)                         -0.099255
D.(Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer,1) -0.115589
D.(Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer,1)             -0.077972
D.(Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer,1)                -0.114415
D.(Steelhead (Middle Columbia River DPS) North Fork John Day River - summer,1)                  0.013860
D.(Steelhead (Middle Columbia River DPS) South Fork John Day River - summer,1)                 -0.175876
D.(Steelhead (Middle Columbia River DPS) Touchet River - summer,1)                              0.122691
D.(Steelhead (Middle Columbia River DPS) Umatilla River - summer,1)                             0.067741
D.(Steelhead (Middle Columbia River DPS) Walla Walla River - summer,1)                         -0.147210
D.(Steelhead (Middle Columbia River DPS) Naches River - summer,1)                               0.055801
D.(Steelhead (Middle Columbia River DPS) Satus Creek - summer,1)                               -0.040935
D.(Steelhead (Middle Columbia River DPS) Toppenish Creek - summer,1)                           -0.193744
D.(Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer,1)               -0.030299
D.(Steelhead (Middle Columbia River DPS) Deschutes River Eastside - summer,2)                   0.005567
D.(Steelhead (Middle Columbia River DPS) Deschutes River Westside - summer,2)                   0.050042
D.(Steelhead (Middle Columbia River DPS) Fifteenmile Creek - winter,2)                          0.324118
D.(Steelhead (Middle Columbia River DPS) John Day River Lower Mainstem Tributaries - summer,2)  0.227233
D.(Steelhead (Middle Columbia River DPS) John Day River Upper Mainstem - summer,2)              0.255739
D.(Steelhead (Middle Columbia River DPS) Middle Fork John Day River - summer,2)                 0.257472
D.(Steelhead (Middle Columbia River DPS) North Fork John Day River - summer,2)                  0.297389
D.(Steelhead (Middle Columbia River DPS) South Fork John Day River - summer,2)                  0.296448
D.(Steelhead (Middle Columbia River DPS) Touchet River - summer,2)                              0.260476
D.(Steelhead (Middle Columbia River DPS) Umatilla River - summer,2)                            -0.033824
D.(Steelhead (Middle Columbia River DPS) Walla Walla River - summer,2)                          0.247593
D.(Steelhead (Middle Columbia River DPS) Naches River - summer,2)                               0.252340
D.(Steelhead (Middle Columbia River DPS) Satus Creek - summer,2)                                0.237560
D.(Steelhead (Middle Columbia River DPS) Toppenish Creek - summer,2)                            0.238715
D.(Steelhead (Middle Columbia River DPS) Yakima River Upper Mainstem - summer,2)                0.140014
Initial states (x0) defined at t=0

Standard errors have not been calculated. 
Use MARSSparamCIs to compute CIs and bias estimates.

This model converged, but still has a bad AIC. this model AICc is BAD.

Code

autoplot(m_cyl_9)

MARSSresiduals.tt1 reported warnings. See msg element of returned residuals object.
MARSSresiduals.tT reported warnings. See msg element or attribute of returned residuals object.

plot.type = xtT

Hit <Return> to see next plot (q to exit):

plot.type = fitted.ytT

Hit <Return> to see next plot (q to exit):

plot.type = model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = std.model.resids.ytT

Hit <Return> to see next plot (q to exit):

plot.type = std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.model.resids.ytt1

Hit <Return> to see next plot (q to exit):

plot.type = qqplot.std.state.resids.xtT

Hit <Return> to see next plot (q to exit):

plot.type = acf.std.model.resids.ytt1

Finished plots.

This model performs similarly to the last model. Cycling my 9 years doesn’t seem to have improved anything.

Let’s look at the corrplot: They are pretty similar to the last model.

Code

Q_9 <- coef(m_cyl_9, type = "matrix")$Q
corrmat_9 <- diag(1/sqrt(diag(Q_9))) %*% Q_9 %*% diag(1/sqrt(diag(Q_9)))
corrplot(corrmat_9)

AICc Results for Cycling

Code

mods_cyl <- c("2.4","m_test","m_cyl_4", "m_cyl_9")
aic_cyl <- c(m2.4$AICc, m_test$AICc, m_cyl_4$AICc, m_cyl_9$AICc)
daic_cyl <- aic_cyl-min(aic_cyl)
tab2 <- cbind.data.frame(mods_cyl, aic_cyl, daic_cyl)
kable(tab2, col.names = c("Hypothesis", "AICc", "delta AICc"))

Hypothesis	AICc	delta AICc
2.4	1009.027	0.0000
m_test	1716.095	707.0687
m_cyl_4	1595.392	586.3650
m_cyl_9	1587.556	578.5295

The cycling assumptions tested in this excersize clearly worsened model fits.

Discussion

Ultimately the most informative model for streams with missing data was the model tested in hypothesis 2.4, which assumed four underlying states, one for each of the main distinct population centers (DPC), the Cascades, John Day, Walla Walla, and Yakima tributaries, where the Q matrix was unconstrained allowing for correlation in the process errors.

Based on initial results, cycling only worsened fits, but only one method and two periods, 4 and 9 were tests, so perhaps with more exploraiton cycling considerations would have improved model fits.

Ultimately, the model that assumed four states performed the best, and from this we can interpret that while salmon generally return to their native streams, there is correlation in the systems, and allowing the models to explore that correlation in process error freely resulting in the best model fits and lowest confidence intervals.

Description of each team member’s contributions

Dylan: Hypothesis conceptualization, code for hypothesis 2 and 4, matrix display code, and AICc comparison methods. Madison: Hypothesis 1 and 3, cycling code, and Rmarkdown formatting.

Both Dylan and Madi helped to write the report.