Fitting multivariate Bayesian time series models

11 Feb 2021

Overview of today’s material

Using STAN for MAP estimation
Multivariate time series models
DFA models
Writing our own Stan code

MAP (maximum a posteriori) estimation

Point estimate of unknown parameters
Optimization algorithms: BFGS, Newton, etc
Estimates similar to maximum likelihood, but also incorporate prior
Quick for checking models, etc

MAP estimation in Stan

We left off with a Bayesian DLM with random walk in intercept

bayes_fit = fit_stan(y = SalmonSurvCUI$logit.s, 
               model_name="dlm-intercept")

MAP estimation in Stan

Switching this to MAP estimation

map_fit = fit_stan(y = SalmonSurvCUI$logit.s, 
               model_name="dlm-intercept", map_estimation=TRUE)

MAP estimation in Stan

print(map_fit)

## $par
##            x0       beta[1]    pro_dev[1]    pro_dev[2]    pro_dev[3] 
## -3.460000e+00 -1.503290e+00  1.046740e-01 -1.943951e-01  4.112205e-01 
##    pro_dev[4]    pro_dev[5]    pro_dev[6]    pro_dev[7]    pro_dev[8] 
## -4.336506e-01  1.943952e-01 -4.336507e-01 -1.943952e-01 -4.710349e-01 
##    pro_dev[9]   pro_dev[10]   pro_dev[11]   pro_dev[12]   pro_dev[13] 
## -6.205692e-01  7.850578e-01  8.074876e-01 -1.659836e+00  4.186974e-01 
##   pro_dev[14]   pro_dev[15]   pro_dev[16]   pro_dev[17]   pro_dev[18] 
##  4.261744e-01 -2.542092e-01 -1.719652e-01  1.345815e-01  5.607551e-01 
##   pro_dev[19]   pro_dev[20]   pro_dev[21]   pro_dev[22]   pro_dev[23] 
## -2.317783e-01 -6.729076e-02  3.514065e-01 -8.074876e-01  8.972066e-02 
##   pro_dev[24]   pro_dev[25]   pro_dev[26]   pro_dev[27]   pro_dev[28] 
##  1.271049e-01 -1.869187e-01  6.729066e-01 -7.476740e-01 -6.729063e-01 
##   pro_dev[29]   pro_dev[30]   pro_dev[31]   pro_dev[32]   pro_dev[33] 
##  6.729059e-02  8.972085e-01 -5.233691e-02 -5.308487e-01  1.375720e+00 
##   pro_dev[34]   pro_dev[35]   pro_dev[36]   pro_dev[37]   pro_dev[38] 
## -8.859995e-08  4.261745e-01 -1.046745e-01 -2.841160e-01 -4.486054e-02 
##   pro_dev[39]   pro_dev[40]   pro_dev[41] sigma_process     sigma_obs 
## -8.897319e-01  2.542093e-01 -5.607554e-01  1.337481e+00  5.537342e-07 
##       pred[1]       pred[2]       pred[3]       pred[4]       pred[5] 
## -3.460000e+00 -3.320000e+00 -3.580000e+00 -3.030000e+00 -3.610000e+00 
##       pred[6]       pred[7]       pred[8]       pred[9]      pred[10] 
## -3.350000e+00 -3.930000e+00 -4.190000e+00 -4.820000e+00 -5.650000e+00 
##      pred[11]      pred[12]      pred[13]      pred[14]      pred[15] 
## -4.600000e+00 -3.520000e+00 -5.740000e+00 -5.180000e+00 -4.610000e+00 
##      pred[16]      pred[17]      pred[18]      pred[19]      pred[20] 
## -4.950000e+00 -5.180000e+00 -5.000000e+00 -4.250000e+00 -4.560000e+00 
##      pred[21]      pred[22]      pred[23]      pred[24]      pred[25] 
## -4.650000e+00 -4.180000e+00 -5.260000e+00 -5.140000e+00 -4.970000e+00 
##      pred[26]      pred[27]      pred[28]      pred[29]      pred[30] 
## -5.220000e+00 -4.320000e+00 -5.320000e+00 -6.220000e+00 -6.130000e+00 
##      pred[31]      pred[32]      pred[33]      pred[34]      pred[35] 
## -4.930000e+00 -5.000000e+00 -5.710000e+00 -3.870000e+00 -3.870000e+00 
##      pred[36]      pred[37]      pred[38]      pred[39]      pred[40] 
## -3.300000e+00 -3.440000e+00 -3.820000e+00 -3.880000e+00 -5.070000e+00 
##      pred[41]      pred[42]  intercept[1]  intercept[2]  intercept[3] 
## -4.730000e+00 -5.480000e+00 -3.460000e+00 -3.320000e+00 -3.580000e+00 
##  intercept[4]  intercept[5]  intercept[6]  intercept[7]  intercept[8] 
## -3.030000e+00 -3.610000e+00 -3.350000e+00 -3.930000e+00 -4.190000e+00 
##  intercept[9] intercept[10] intercept[11] intercept[12] intercept[13] 
## -4.820000e+00 -5.650000e+00 -4.600000e+00 -3.520000e+00 -5.740000e+00 
## intercept[14] intercept[15] intercept[16] intercept[17] intercept[18] 
## -5.180000e+00 -4.610000e+00 -4.950000e+00 -5.180000e+00 -5.000000e+00 
## intercept[19] intercept[20] intercept[21] intercept[22] intercept[23] 
## -4.250000e+00 -4.560000e+00 -4.650000e+00 -4.180000e+00 -5.260000e+00 
## intercept[24] intercept[25] intercept[26] intercept[27] intercept[28] 
## -5.140000e+00 -4.970000e+00 -5.220000e+00 -4.320000e+00 -5.320000e+00 
## intercept[29] intercept[30] intercept[31] intercept[32] intercept[33] 
## -6.220000e+00 -6.130000e+00 -4.930000e+00 -5.000000e+00 -5.710000e+00 
## intercept[34] intercept[35] intercept[36] intercept[37] intercept[38] 
## -3.870000e+00 -3.870000e+00 -3.300000e+00 -3.440000e+00 -3.820000e+00 
## intercept[39] intercept[40] intercept[41] intercept[42]    log_lik[1] 
## -3.880000e+00 -5.070000e+00 -4.730000e+00 -5.480000e+00  1.326919e+01 
##    log_lik[2]    log_lik[3]    log_lik[4]    log_lik[5]    log_lik[6] 
##  1.348558e+01  1.348143e+01  1.343850e+01  1.346689e+01  1.348198e+01 
##    log_lik[7]    log_lik[8]    log_lik[9]   log_lik[10]   log_lik[11] 
##  1.340442e+01  1.338355e+01  1.341164e+01  1.348663e+01  1.337676e+01 
##   log_lik[12]   log_lik[13]   log_lik[14]   log_lik[15]   log_lik[16] 
##  1.348383e+01  1.341941e+01  1.342276e+01  1.346445e+01  1.348422e+01 
##   log_lik[17]   log_lik[18]   log_lik[19]   log_lik[20]   log_lik[21] 
##  1.335915e+01  1.348345e+01  1.326469e+01  1.317618e+01  1.334588e+01 
##   log_lik[22]   log_lik[23]   log_lik[24]   log_lik[25]   log_lik[26] 
##  1.348756e+01  1.340756e+01  1.348373e+01  1.326981e+01  1.348561e+01 
##   log_lik[27]   log_lik[28]   log_lik[29]   log_lik[30]   log_lik[31] 
##  1.346174e+01  1.348731e+01  1.337548e+01  1.343007e+01  1.348366e+01 
##   log_lik[32]   log_lik[33]   log_lik[34]   log_lik[35]   log_lik[36] 
##  1.333669e+01  1.348752e+01  1.342055e+01  1.331926e+01  1.342071e+01 
##   log_lik[37]   log_lik[38]   log_lik[39]   log_lik[40]   log_lik[41] 
##  1.348268e+01  1.348710e+01  1.346313e+01  1.346517e+01  1.347789e+01 
##   log_lik[42] 
##  1.346525e+01 
## 
## $value
## [1] 595.6098
## 
## $return_code
## [1] 0
## 
## $theta_tilde
##         x0  beta[1] pro_dev[1] pro_dev[2] pro_dev[3] pro_dev[4] pro_dev[5]
## [1,] -3.46 -1.50329   0.104674 -0.1943951  0.4112205 -0.4336506  0.1943952
##      pro_dev[6] pro_dev[7] pro_dev[8] pro_dev[9] pro_dev[10] pro_dev[11]
## [1,] -0.4336507 -0.1943952 -0.4710349 -0.6205692   0.7850578   0.8074876
##      pro_dev[12] pro_dev[13] pro_dev[14] pro_dev[15] pro_dev[16] pro_dev[17]
## [1,]   -1.659836   0.4186974   0.4261744  -0.2542092  -0.1719652   0.1345815
##      pro_dev[18] pro_dev[19] pro_dev[20] pro_dev[21] pro_dev[22] pro_dev[23]
## [1,]   0.5607551  -0.2317783 -0.06729076   0.3514065  -0.8074876  0.08972066
##      pro_dev[24] pro_dev[25] pro_dev[26] pro_dev[27] pro_dev[28] pro_dev[29]
## [1,]   0.1271049  -0.1869187   0.6729066   -0.747674  -0.6729063  0.06729059
##      pro_dev[30] pro_dev[31] pro_dev[32] pro_dev[33]   pro_dev[34] pro_dev[35]
## [1,]   0.8972085 -0.05233691  -0.5308487     1.37572 -8.859995e-08   0.4261745
##      pro_dev[36] pro_dev[37] pro_dev[38] pro_dev[39] pro_dev[40] pro_dev[41]
## [1,]  -0.1046745   -0.284116 -0.04486054  -0.8897319   0.2542093  -0.5607554
##      sigma_process    sigma_obs pred[1] pred[2] pred[3] pred[4] pred[5] pred[6]
## [1,]      1.337481 5.537342e-07   -3.46   -3.32   -3.58   -3.03   -3.61   -3.35
##      pred[7] pred[8] pred[9] pred[10] pred[11] pred[12] pred[13] pred[14]
## [1,]   -3.93   -4.19   -4.82    -5.65     -4.6    -3.52    -5.74    -5.18
##      pred[15] pred[16] pred[17] pred[18] pred[19] pred[20] pred[21] pred[22]
## [1,]    -4.61    -4.95    -5.18       -5    -4.25    -4.56    -4.65    -4.18
##      pred[23] pred[24] pred[25] pred[26] pred[27] pred[28] pred[29] pred[30]
## [1,]    -5.26    -5.14    -4.97    -5.22    -4.32    -5.32    -6.22    -6.13
##      pred[31] pred[32] pred[33] pred[34] pred[35] pred[36] pred[37] pred[38]
## [1,]    -4.93       -5    -5.71    -3.87    -3.87     -3.3    -3.44    -3.82
##      pred[39] pred[40] pred[41] pred[42] intercept[1] intercept[2] intercept[3]
## [1,]    -3.88    -5.07    -4.73    -5.48        -3.46        -3.32        -3.58
##      intercept[4] intercept[5] intercept[6] intercept[7] intercept[8]
## [1,]        -3.03        -3.61        -3.35        -3.93        -4.19
##      intercept[9] intercept[10] intercept[11] intercept[12] intercept[13]
## [1,]        -4.82         -5.65          -4.6         -3.52         -5.74
##      intercept[14] intercept[15] intercept[16] intercept[17] intercept[18]
## [1,]         -5.18         -4.61         -4.95         -5.18            -5
##      intercept[19] intercept[20] intercept[21] intercept[22] intercept[23]
## [1,]         -4.25         -4.56         -4.65         -4.18         -5.26
##      intercept[24] intercept[25] intercept[26] intercept[27] intercept[28]
## [1,]         -5.14         -4.97         -5.22         -4.32         -5.32
##      intercept[29] intercept[30] intercept[31] intercept[32] intercept[33]
## [1,]         -6.22         -6.13         -4.93            -5         -5.71
##      intercept[34] intercept[35] intercept[36] intercept[37] intercept[38]
## [1,]         -3.87         -3.87          -3.3         -3.44         -3.82
##      intercept[39] intercept[40] intercept[41] intercept[42] log_lik[1]
## [1,]         -3.88         -5.07         -4.73         -5.48   13.26919
##      log_lik[2] log_lik[3] log_lik[4] log_lik[5] log_lik[6] log_lik[7]
## [1,]   13.48558   13.48143    13.4385   13.46689   13.48198   13.40442
##      log_lik[8] log_lik[9] log_lik[10] log_lik[11] log_lik[12] log_lik[13]
## [1,]   13.38355   13.41164    13.48663    13.37676    13.48383    13.41941
##      log_lik[14] log_lik[15] log_lik[16] log_lik[17] log_lik[18] log_lik[19]
## [1,]    13.42276    13.46445    13.48422    13.35915    13.48345    13.26469
##      log_lik[20] log_lik[21] log_lik[22] log_lik[23] log_lik[24] log_lik[25]
## [1,]    13.17618    13.34588    13.48756    13.40756    13.48373    13.26981
##      log_lik[26] log_lik[27] log_lik[28] log_lik[29] log_lik[30] log_lik[31]
## [1,]    13.48561    13.46174    13.48731    13.37548    13.43007    13.48366
##      log_lik[32] log_lik[33] log_lik[34] log_lik[35] log_lik[36] log_lik[37]
## [1,]    13.33669    13.48752    13.42055    13.31926    13.42071    13.48268
##      log_lik[38] log_lik[39] log_lik[40] log_lik[41] log_lik[42]
## [1,]     13.4871    13.46313    13.46517    13.47789    13.46525

MAP estimation in Stan

Check that model converged

map_fit$return_code

## [1] 0

MAP value when converged

map_fit$value

## [1] 595.6098

MAP estimation in Stan

grep or other string matching functions needed

grep("pred",names(map_fit$par))

##  [1] 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
## [26] 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

pred = map_fit$par[grep("pred",names(map_fit$par))]

MAP estimation in Stan

grep or other string matching functions needed

grep("pred",names(map_fit$par))

##  [1] 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
## [26] 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

pred = map_fit$par[grep("pred",names(map_fit$par))]

MAP estimation in Stan

grep or other string matching functions needed

grep("pred",names(map_fit$par))

##  [1] 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
## [26] 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

pred = map_fit$par[grep("pred",names(map_fit$par))]

MAP estimation in Stan

include SEs of estimates, via Hessian

map_fit = fit_stan(y = SalmonSurvCUI$logit.s, 
     model_name="dlm-intercept", 
     map_estimation=TRUE, 
     hessian=TRUE)

or change algorithm (“LBFGS” default) to “BFGS” or “Newton”

map_fit = fit_stan(y = SalmonSurvCUI$logit.s, 
     model_name="dlm-intercept", 
     map_estimation=TRUE, 
     hessian=TRUE, 
     algorithm="BFGS")

MAP estimation in Stan

Posterior means, 95% CIs

MAP estimation in Stan

MAP estimates = posterior

Multivariate time series models

Multivariate state space models
Stan package for fitting models similar to MARSS

devtools::install_github("atsa-es/tvvarss")

Multivariate time series models

Most models in tvvarss() are exactly the same as MARSS
Takes in data matrix y
family argument defaults to Gaussian, but can be many others

tvvarss::tvvarss(y = y, family = "poisson",...)

Multivariate time series models

Like MARSS, variances can be shared (default) or unique by species, or time series

tvvarss::tvvarss(y = y, shared_r = R, shared_q = Q,...)

Multivariate time series models

Optional ‘process’ argument acts as Z in MARSS and maps time series to latent state processes

tvvarss::tvvarss(y = y, shared_r = R, shared_q = Q,...)

Multivariate time series models

Perhaps most exciting feature is time-varying interactions (AR coefficients)
Time-varying vector autoregressive models
AR coefficients behave like a DLM

tvvarss::tvvarss(y = y, dynamicB = TRUE,...)

Time-varying parameters are data hungry! We’ll dive into this more later in the quarter

Bayesian DFA models

DFA models from atsar bundled with other DFA code we’ve developed

devtools::install_github("fate-ewi/bayesdfa")

Bayesian DFA models

DFA poses interesting identifiability challenges
Bayesian models generally involve > 1 MCMC chain
Direction of DFA loadings / trends don’t have meaning

Bayesian DFA models

Solution 1: priors for identifiability
Solution 2: post-hoc ‘chain flipping’ before convergence tests run

Bayesian DFA models

Fitting DFA models will give very similar answers to using MARSS

data("harborSealWA")

Bayesian DFA models

We’ll extract predictions from the best model,

fit = bayesdfa::fit_dfa(y = t(harborSealWA[,-1]), num_trends = 1)

And as an stanfit object, we can extract summaries of states x

pars = extract(fit$model)

Bayesian DFA models

bayesdfa::plot_fitted(fit) + theme_bw()

Bayesian DFA models

DFA extension 1: long format data (optional), replicate observations

Bayesian DFA models

DFA extension 2: temporal extremes

\[x_{t}=x_{t-1}+\delta_{t-1}\] \[\delta_{t-1} \sim Normal(0,q)\]

But do these deviations have to be normal? NO!

\[\delta_{t-1} \sim Student-t(\nu,0,q)\]

Bayesian DFA models

Extremes example: coho salmon body size in coastal fisheries

Bayesian DFA models

Fitting the model

coho = readRDS("coho_mean_length.rds")
# rename variables
coho=dplyr::rename(coho, obs=mean_length,time=recovery_year,ts=fishery)
fit = fit_dfa(y = coho, data_shape="long",estimate_nu = TRUE)

Bayesian DFA models

Let’s look at df parameter, \(\nu\)
Mean ~ 19.5, median ~ 16.3

pars = extract(fit$model)

Bayesian DFA models

\(\nu\) parameter ranges from 2 (heavy tails) to normal (30)

(source: Wikipedia)

Bayesian DFA models

DFA extension 2: increased flexibility in trends
Sampling might be ‘chunky’

Bayesian DFA models

AR models (conventional DFA) may not be well suited
Alternative approach: use Gaussian Process models to model smooth trends
Feddern et al. 2021 (Global Change Biology)

Bayesian DFA models

What’s a Gaussian Process model?
Instead of modeling process \[x_{t} = x_{t-1}...\]
We model the covariance of the \(\textbf{x}\)
But there are a lot of \(\textbf{x}\)!!

Bayesian DFA models

For model with \(T\) timesteps
\(\Sigma\) has \(T*(T-1)/2\) parameters
GP models implement covariance functions, e.g. exponential, Gaussian, Matern
Model covariance as function of space or distance in time

Bayesian DFA models

Example: Gaussian covariance (aka ‘squared exponential’)

\[\Sigma_{t,t+3} = \sigma^2exp(\frac{(t-(t+3))^2}{2\tau^2})\]

\(\sigma^2\) controls variability
\(\tau\) controls how quickly covariance decays between points

Bayesian DFA models

Implementation

fit_dfa(y= y, num_trends = 2, ..., trend_model = c("gp"))

Bayesian DFA models

DFA extension 3: alternate constraints on loadings matrix
Conventional constraints on loadings matrix

##             [,1]       [,2]       [,3]
## [1,] -0.56047565  0.0000000  0.0000000
## [2,] -0.23017749  0.4609162  0.0000000
## [3,]  1.55870831 -1.2650612  0.4007715
## [4,]  0.07050839 -0.6868529  0.1106827
## [5,]  0.12928774 -0.4456620 -0.5558411

Bayesian DFA models

Alternative: compositional DFA model

##           [,1]       [,2]       [,3]
## [1,] 0.6871757 0.24516978 0.06765452
## [2,] 0.1491745 0.06014499 0.79068047
## [3,] 0.3885419 0.59306319 0.01839489
## [4,] 0.2919682 0.30545457 0.40257721
## [5,] 0.3760087 0.33262301 0.29136825

Bayesian DFA models

Under compositional DFA model, time series are true mixtures of underlying trends

\[\textbf{Y} = \textbf{Zx}\]

fit_dfa(y = y, num_trends = 2, ..., z_model = "proportion")

When might this be useful?
Stable isotope data, environmental monitoring, pollutant data, etc

Writing our own Stan scripts

Stan scripts always start with a data block
Data needs to be typed

data {
  int<lower=0> N;
  int<lower=0> K;
  real y[N];
  int P;
  int y_int[N];
  matrix[N, K] x;
}

Stan manual

Writing our own Stan scripts

transformed data block is optional

transformed data {
  int zeros[N];
  for(i in 1:N) {
    zeros[i] = 0;
  }
}

Writing our own Stan scripts

parameters block containts any parameters

parameters {
  real x0;
  vector[K] beta0;
  vector[K] pro_dev[N-1];
  real<lower=0> sigma_process[K];
  real<lower=0> sigma_obs;
}

Writing our own Stan scripts

parameters block containts any parameters

parameters {
  real x0;
  vector[K] beta0;
  vector[K] pro_dev[N-1];
  real<lower=0> sigma_process[K];
  real<lower=0> sigma_obs;
}

Writing our own Stan scripts

transformed parameters block contains derived quantities
examples: predicted states for state space time series model

transformed parameters {
  vector[N] pred;
  pred[1] = x0;
  for(i in 2:N) {
    pred[i] = phi*pred[i-1] + sigma_process*pro_dev[i-1];
  }
}

Writing our own Stan scripts

model block contains priors on parameters
likelihood (or data model)

model {
  x0 ~ normal(0,10);
  phi ~ normal(0,1);
  sigma_process ~ student_t(3,0,2);
  sigma_obs ~ student_t(3,0,2);
  pro_dev ~ std_normal();
  for(i in 1:N) {
    y[i] ~ normal(pred[i], sigma_obs);
  }
}

Writing our own Stan scripts

generated data block (optional)
includes quantities useful for model selection, prediction, forecasts, etc

generated quantities {
  vector[N] log_lik;
  for (n in 1:N) log_lik[n] = normal_lpdf(y[n] | pred[n], sigma_obs);
}

Writing our own Stan scripts

Ok – let’s do this in practice
Problem # 1: let’s take our state space random walk model and modify it to better handle extreme events with the Student - t distribution
Mathematically

\[\delta_{t-1} \sim Normal(0,q)\] becomes

\[\delta_{t-1} \sim Student-t(\nu,0,q)\]

Writing our own Stan scripts

You could do this in RStudio (File -> New File -> Stan File)
Scripts for the atsar package linked below

https://github.com/atsa-es/atsar/tree/master/inst/stan

We’re just going to modify this code (ss_ar.stan) rather than start from scratch

Writing our own Stan scripts

2 things need to change
Change process error deviations from Normal to Student-t distribution
Add \(\nu\) as a parameter, with constraints (> 2)

Writing our own Stan scripts

parameters {
real<lower=2> nu;
...
}

Writing our own Stan scripts

model{
...
nu ~ student_t(3, 2, 3);
pro_dev ~ student_t(nu, 0, 1);
//pro_dev ~ std_normal();
...
}

Writing our own Stan scripts

Done! Now we can fit the model

y = gme$Close
N = length(y)
n_pos = length(which(!is.na(y)))
pos_indx = c(which(!is.na(y)),0,0)

fit = stan(file = "ss_ar_t.stan", 
  data = list(y = y, N = N, n_pos = n_pos, pos_indx = pos_indx))