19 Jan 2021
Weeks 1-3: building blocks for analysis of multivariate time-series data with observation error, structure, and missing values
Week 4: putting this all together to start analyzing ecological data sets
\[x_t = x_{t-1}+u+w_t, \,\,\, w_t \sim N(0,q)\] \[y_t = x_t + v_t, \,\,\, v_t \sim N(0,r)\] The \(x\) model is the classic “random walk” with drift.
\(y\) are the observatons.
This model is a random walk observed with (Gaussian) error.
\[x_t = x_{t-1}+u+w_t, \,\,\, w_t \sim N(0,q)\] \[y_t = x_t + v_t, \,\,\, v_t \sim N(0,r)\]
There are many textbooks on this class of model. It is used in extensively in economics and engineering.
All of these are examples:
\[x_t = x_{t-1} + u + w_t\] \[x_{t+1} = x_t + w_t\] \[x_t = b x_{t-1} + u + w_t\]
Additive random walks
\[x_t = x_{t-1}+u+w_t, \,\,\, w_t \sim N(0,q)\]
Movement, changes in gene frequency, somatic growth if growth is by fixed amounts
Why Gaussian? The average of many small perturbations, regardless of their distribution, is Gaussian.
Multiplicative random walks
\[n_t = \lambda n_{t-1}e_t, \,\,\, log(e_t) \sim N(0,q)\]
Population growth, somatic growth if growth is by percentage
Take the log and you get the linear additive model above. log-normal error distribution means that 10% increase is as likely as 10% decrease
Addition of \(b\) with \(0<b<1\) leads to process model with mean-reversion.
In the ecological literature on density-dependent processes, you may see this in non-log notation:
\[N_t = exp(u+w_t)N^b_{t-1}\]
\(N_t\) is population size.
Take the log, and we have
\[x_t = bx_{t-1}+u+w_t\] \[w_t \sim N(0,q)\] It is not required that \(w_t\) is Gaussian but that is a common assumption. Dynamics of processes with non-Gaussian errors, esp long-tailed errors, is a common extension. Autocorrelated errors could be implemented with MA process or covariates.
An random walk can show a wide-range of trajectories, even for the same parameter values. All trajectories below came from the same random walk model: \(x_t= x_{t-1} -0.02+w_t\), \(w_t \sim N(mean=0.0, var=0.01)\).
The “state” is a hidden (dynamical) variable. In this class, it will be a hidden random walk or AR(1) process.
Our data are observations of this hidden state.
Often state-space models include inputs (explanatory variables) and the state or the data may be multivariate.
The model you are seeing today is a simple univariate state-space model with no inputs.
state: \(x_t = x_{t-1} + u + w_t\)
observation: \(y_t = x_t + v_t\)
Yearly, usually, population or subpopulation counts, possibly with missing values.
The data are observations of a hidden ‘true’ population size. The data are observations of that hidden state and have observation error.
This is a survey photograph for Steller sea lions in the Gulf of Alaska. There IS some number of sea lions in our population in year \(t\), but we don’t know that number precisely. It is “hidden”.
Sightability varies due to factors that may not be fully understood or measureable
Environmental factors (tides, temperature, etc.)
Population factors (age structure, sex ratio, etc.)
Species interactions (prey distribution, prey density, predator distribution or density, etc.)
Sampling variability–due to how you actually count animals–is just one component of observation variance
Suppose we have the following data (say, population density logged)
The model of the hidden state in this case is \(x_t = \alpha + \beta t\). The observation model is \(y_t=x_t+v_t\). All variability = non-process or observation variability.
The model of the hidden state in this case is \(x_t = \alpha + x_{t-1}+w_t\). The observation model is \(y_t=x_t\). All variability = process variability. 
Autoregressive state-space models fit a random walk AR(1) through the data. The variabilty in the data contains both process and non-process (observation) variability.
Observation or “non-process” error is the difference between the hidden state (blue line) and the observation (X).
Process error is the difference between the expected \(x_t\) given data up to time \(t-1\) (x in the plot) and the actual \(x\) at time \(t\).
One use of univariate state-space models is “count-based” population viability analysis (chap 7 HWS2014)
## Kalman filter and smoother
The Kalman filter and smoother is an algorithm for computing the expected value of the \(x_t\) from the data and the model parameters.
\[x_t = x_{t-1}+u+w_t, \,\,\, w_t \sim N(0,q)\]
\[y_t = x_t + v_t, \,\,\, v_t \sim N(0,r)\]
Innovations residuals =
data at time \(t\) minus model predictions given data up to \(t-1\)
\[\hat{y_t} = E[Y_{t}|y_{t-1}]\]
residuals(fit)
Standard diagnostics
MARSS packageWe will be using the MARSS package to fit univariate and multivariate state-space models.
The main function is MARSS():
fit <- MARSS(data, model=list())
data are a vector or a matrix with time going along the columns.
model list is a list with the structure of all the parameters.
\[x_t = \mathbf{B}x_{t-1} + \mathbf{U} + w_t, \,\,\, w_t \sim N(0,\mathbf{Q})\] \[y_t = \mathbf{Z}x_{t} + \mathbf{A} + v_t, \,\,\, v_t \sim N(0,\mathbf{R})\]
The MARSS model list follows this notation one-to-one.
\[x_t = x_{t-1} + u + w_t, \,\,\, w_t \sim N(0,q)\] \[y_t = x_{t} + v_t, \,\,\, v_t \sim N(0,r)\]
Write as where everything bold is a matrix.
\[x_t = \mathbf{B}x_{t-1} + \mathbf{U} + w_t, \,\,\, w_t \sim N(0,\mathbf{Q})\] \[y_t = \mathbf{Z}x_{t} + \mathbf{A} + v_t, \,\,\, v_t \sim N(0,\mathbf{R})\]
mod.list <- list(
U = matrix("u"),
x0 = matrix("x0"),
B = matrix(1),
Q = matrix("q"),
Z = matrix(1),
A = matrix(0),
R = matrix("r"),
tinitx = 0
)