Overview of mvdlm package

Overview

For examples, we’re going to use one of the same datasets that are widely documented in the MARSS manual. This data consists of three columns, * year, representing the year of the observation * logit.s, representing logit-transformed survival of Columbia River salmon * CUI.apr, representing coastal upwelling index values in April

library(mvdlm)
library(MARSS)
library(ggplot2)
library(broom.mixed)
library(loo)

## This is loo version 2.8.0

## - Online documentation and vignettes at mc-stan.org/loo

## - As of v2.0.0 loo defaults to 1 core but we recommend using as many as possible. Use the 'cores' argument or set options(mc.cores = NUM_CORES) for an entire session.

data(SalmonSurvCUI)
head(SalmonSurvCUI)

##   year logit.s CUI.apr
## 1 1964   -3.46      57
## 2 1965   -3.32       5
## 3 1966   -3.58      43
## 4 1967   -3.03      11
## 5 1968   -3.61      47
## 6 1969   -3.35     -21

g1 <- ggplot(SalmonSurvCUI, aes(year, logit.s)) + 
  geom_point() + 
  geom_line()
g1

g2 <- ggplot(SalmonSurvCUI, aes(year, CUI.apr)) + 
  geom_point() + 
  geom_line()
g2

Model 1: time varying intercept and slope

For a first model, we’ll fit the same model used in the MARSS manual, with a time varying intercept and time varying coefficient on CUI. We can specify these time varying effects using the time_varying argument. Note an intercept is not included, like with lm, glm, and similar packages this the intercept is implicitly included. Note that alternatively you could specify this formula as time_varying = logit.s ~ 1 + CUI.apr, where we add the extra 1 to signify the intercept.

Like the MARSS example, we also standardize the covariates,

SalmonSurvCUI$CUI.apr = scale(SalmonSurvCUI$CUI.apr)

fit <- fit_dlm(time_varying = logit.s ~ CUI.apr,
        data = SalmonSurvCUI, 
        chains=1,
        iter=4000)

## 
## SAMPLING FOR MODEL 'dlm' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 4.3e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.43 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:    1 / 4000 [  0%]  (Warmup)
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## Chain 1: 
## Chain 1:  Elapsed Time: 7.403 seconds (Warm-up)
## Chain 1:                15.725 seconds (Sampling)
## Chain 1:                23.128 seconds (Total)
## Chain 1:

## Warning: There were 18 divergent transitions after warmup. See
## https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.

## Warning: There were 1 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
## https://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded

## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## https://mc-stan.org/misc/warnings.html#bfmi-low

## Warning: Examine the pairs() plot to diagnose sampling problems

## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess

fit <- fit_dlm(time_varying = logit.s ~ CUI.apr,
        data = SalmonSurvCUI[-36,], 
        chains=1,
        iter=1000)

## 
## SAMPLING FOR MODEL 'dlm' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 4e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.4 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:   1 / 1000 [  0%]  (Warmup)
## Chain 1: Iteration: 100 / 1000 [ 10%]  (Warmup)
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## Chain 1: Iteration: 1000 / 1000 [100%]  (Sampling)
## Chain 1: 
## Chain 1:  Elapsed Time: 2.565 seconds (Warm-up)
## Chain 1:                2.87 seconds (Sampling)
## Chain 1:                5.435 seconds (Total)
## Chain 1:

## Warning: There were 11 divergent transitions after warmup. See
## https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.

## Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
## https://mc-stan.org/misc/warnings.html#bfmi-low

## Warning: Examine the pairs() plot to diagnose sampling problems

## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess

## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess

With only 1000 iterations and 1 chain, we might not expect the model to converge (see additional guidance via Stan developers here: https://mc-stan.org/misc/warnings.html). A couple things to consider:

Did you get any warnings about divergent transitions after warmup? If so, try increasing the iterations / burn-in period, and number of chains. It’s also worth increasing the value of adapt_delta. (this will slow the sampling down a little). If you’re still getting these warnings, the model may be mis-specified or data not informative.

fit <- fit_dlm(time_varying = logit.s ~ CUI.apr,
        data = SalmonSurvCUI, 
        chains=1,
        iter=1000,
        control = list(adapt_delta=0.99))

Did you get a warning about the maximum tree depth? If so, you can increase the maximum tree (max_treedepth) depth with the following

fit <- fit_dlm(time_varying = logit.s ~ CUI.apr,
        data = SalmonSurvCUI,
        chains=1,
        iter=1000,
        control = list(max_treedepth=15))

We can extract tidied versions of any of the parameters with

broom.mixed::tidy(fit$fit)

## # A tibble: 169 × 3
##    term    estimate std.error
##    <chr>      <dbl>     <dbl>
##  1 eta[1]     -3.48     0.354
##  2 eta[2]     -3.34     0.322
##  3 eta[3]     -3.53     0.303
##  4 eta[4]     -3.27     0.296
##  5 eta[5]     -3.62     0.317
##  6 eta[6]     -3.49     0.330
##  7 eta[7]     -3.98     0.279
##  8 eta[8]     -4.23     0.252
##  9 eta[9]     -4.65     0.283
## 10 eta[10]    -5.23     0.416
## # ℹ 159 more rows

We also have a helper function for extracting and visualizing trends for time varying parameters.

dlm_trends(fit)

## $plot

## 
## $b_varying
## # A tibble: 84 × 5
##    term            estimate std.error par          time
##    <chr>              <dbl>     <dbl> <chr>       <int>
##  1 b_varying[1,1]     -3.34     0.501 (Intercept)     1
##  2 b_varying[2,1]     -3.34     0.326 (Intercept)     2
##  3 b_varying[3,1]     -3.34     0.392 (Intercept)     3
##  4 b_varying[4,1]     -3.24     0.322 (Intercept)     4
##  5 b_varying[5,1]     -3.43     0.351 (Intercept)     5
##  6 b_varying[6,1]     -3.61     0.290 (Intercept)     6
##  7 b_varying[7,1]     -3.91     0.295 (Intercept)     7
##  8 b_varying[8,1]     -4.26     0.243 (Intercept)     8
##  9 b_varying[9,1]     -4.68     0.270 (Intercept)     9
## 10 b_varying[10,1]    -4.94     0.371 (Intercept)    10
## # ℹ 74 more rows

This function returns a plot and the values used to make the plot (b_varying)

Model 2: time varying intercept and constant slope

As a second example, we could fit a model with a constant effect of CUI, but a time varying slope. The fit_dlm function has two formula parameters, formula for static effects, and time_varying for time varying ones. This model is specified as

fit <- fit_dlm(time_varying = logit.s ~ 1,
               formula = logit.s ~ CUI.apr,
        data = SalmonSurvCUI,
        chains=1,
        iter=1000)

Model 3: constant intercept and time varying slope

We could do the same with a model that had a time varying effect of CUI, but a time constant slope.

fit <- fit_dlm(time_varying = logit.s ~ CUI.apr,
               formula = logit.s ~ 1,
        data = SalmonSurvCUI,
        chains=1,
        iter=1000)

Comparing models

All three of the above formulations of the DLM can be compared with the loo package, and statistics such as LOOIC can be easily calculated. For example,

library(loo)
loo::loo(fit$fit)