This is complementary vignette to the paper Helske (2022) with more detailed examples and a
short comparison with naive `Stan`

implementation for
regression model with time-varying coefficients. The varying coefficient
regression models are extension to basic linear regression models where
instead of constant but unknown regression coefficients, the underlying
coefficients are assumed to vary over time according to random walk. In
their simplest form these models can be used to model regression models
with additional time series component, and they also allow robust
modelling of phenomenas where the effect size of the predictor variables
can vary during the period of the study, for example due to interactions
with unobserved counfounders. The `R`

(R Core Team
2020) package `walker`

provides an efficient
method for fully Bayesian inference of such models, where the main
computations are performed using the Markov chain Monte Carlo (MCMC)
algorithms provided by `Stan`

Stan
Development Team (2016a). This also allows the use of many
general diagnostic and graphical tools provided by several
`Stan`

related `R`

packages such as
`ShinyStan`

(Stan Development Team 2017). the linear
model with time-varying coefficients is defined as

A linear model with time-varying coefficients defined as \[ \begin{aligned} y_t &= x'_t \beta_t + \epsilon_t, \quad t = 1,\ldots, n\\ \beta_{t+1} &= \beta_t + \eta_t, \end{aligned} \] where \(y_t\) is the observation at time \(t\), \(x_t\) contains the corresponding predictor variables, \(\beta_t\) is a \(k\) dimensional vector of regression coefficients at time \(t\), \(\epsilon_t \sim N(0, \sigma^2_{\epsilon})\), and \(\eta_t \sim N(0, D)\), with \(D\) being \(k \times k\) diagonal matrix with diagonal elements \(\sigma^2_{i,\eta}\), \(i=1,\ldots,k\). Denote the unknown parameters of the model by \(\beta = (\beta'_1, \ldots, \beta'_n)'\) and \(\sigma = (\sigma_{\epsilon}, \sigma_{1, \eta}, \ldots, \sigma_{k, \eta})\). We define priors for first \(\beta_1\) as \(N(\mu_{\beta_1}, \sigma_{\beta_1})\), and for \(\sigma_i \sim N(\mu_{\sigma_i}, \sigma_{\sigma_i})\), \(i=1,\ldots,k+1\), truncated to positive values.

Although in principle writing this model as above in
`Stan`

is straightforward, standard implementation can lead
to some computational problems (see the illustration in next Section).
An alternative solution used by `walker`

is based on the
marginalization of the regression coefficients \(\beta\) during the MCMC sampling by using
the Kalman filter. This provides fast and accurate inference of marginal
posterior \(p(\sigma | y)\). Then, the
corresponding joint posterior \(p(\sigma,
\beta | y) = p(\beta | \sigma, y)p(\sigma | y)\) can be obtained
by simulating the regression coefficients given marginal posterior of
standard deviations. This sampling can be performed for example by
simulation smoothing algorithm by Durbin and
Koopman (2002).
Note that we have opted to sample the \(\beta\) parameters given \(\sigma\)’s, but it is also possible to
obtain somewhat more accurate summary statistics such as mean and
variance of these parameters by using the standard Kalman smoother for
computation of \(\textrm{E}(\beta| \sigma,
y)\) and \(\textrm{Var}(\beta| \sigma,
y)\), and using the law of total expectation.

In this vignette we first introduce the basic use using simple linear regression model with first order random walk, and then discuss extensions to second order random walk, as well as how to deal with non-Gaussian data.

Let us consider a observations \(y\)
of length \(n=100\), generated by
random walk (i.e. time varying intercept) and two predictors. This is
rather small problem, but it was chosen in order to make possible
comparisons with the “naive” `Stan`

implementation. For
larger problems (in terms of number of observations and especially
number of predictors) it is very difficult to get naive implementation
to work properly, as even after tweaking several parameters of the
underlying MCMC sampler, one typically ends up with divergent
transitions or low BMFI index, meaning that the results are not to be
trusted.

First we simulate the coefficients and the predictors:

```
set.seed(1)
n <- 100
beta1 <- cumsum(c(0.5, rnorm(n - 1, 0, sd = 0.05)))
beta2 <- cumsum(c(-1, rnorm(n - 1, 0, sd = 0.15)))
x1 <- rnorm(n, mean = 2)
x2 <- cos(1:n)
rw <- cumsum(rnorm(n, 0, 0.5))
ts.plot(cbind(rw, beta1 * x1, beta2 * x2), col = 1:3)
```

```
signal <- rw + beta1 * x1 + beta2 * x2
y <- rnorm(n, signal, 0.5)
ts.plot(cbind(signal, y), col = 1:2)
```

Then we can call function `walker`

. The model is defined
as a formula like in `lm`

, and we can give several arguments
which are passed to `sampling`

method of `rstan`

,
such as number of iteration `iter`

and number of chains
`chains`

(default values for these are 2000 and 4). In
addition to these, we use arguments `beta`

and
`sigma`

, which define the Gaussian and Gamma prior
distributions for \(\beta_1\) and \(\sigma\) respectively. These arguments
should be vectors of length two defining the parameters of the prior
distributions: mean and sd for the coefficients, and shape and rate for
standard deviation parameters).

```
set.seed(1)
fit <- walker(y ~ -1 + rw1(~ x1 + x2, beta = c(0, 10), sigma = c(2, 10)),
refresh = 0, chains = 1, sigma_y = c(2, 1))
```

We sometimes get a few (typically one) warning message about
numerical problems, as the sampling algorithm warms up, but this is
nothing to be concerned with (if more errors occur, then a Github issue
for `walker`

package is more than welcome).

The output of `walker`

is `walker_fit`

object,
which is essentially a list with `stanfit`

from
`Stan`

’s `sampling`

function, and the original
observations `y`

and the covariate matrix `xreg`

.
Thus we can use all the available tools for postprocessing
`stanfit`

objects:

`print(fit$stanfit, pars = c("sigma_y", "sigma_rw1"))`

```
## Inference for Stan model: walker_lm.
## 1 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=1000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## sigma_y 0.59 0 0.11 0.38 0.52 0.59 0.66 0.82 863 1
## sigma_rw1[1] 0.47 0 0.11 0.27 0.40 0.47 0.54 0.70 836 1
## sigma_rw1[2] 0.08 0 0.04 0.02 0.06 0.08 0.10 0.16 747 1
## sigma_rw1[3] 0.26 0 0.07 0.14 0.21 0.26 0.31 0.43 873 1
##
## Samples were drawn using NUTS(diag_e) at Mon Sep 11 15:04:38 2023.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).
```

```
library(bayesplot)
mcmc_areas(as.matrix(fit$stanfit), regex_pars = c("sigma_y", "sigma_rw1"))
```

Let’s check how well our estimates of \(\beta\) coincide with the true values (the solid lines correspond to true values):

```
betas <- summary(fit$stanfit, "beta_rw")$summary[, "mean"]
ts.plot(cbind(rw, beta1, beta2, matrix(betas, ncol = 3)),
col = rep(1:3, 2), lty = rep(1:2, each = 3))
```

There is also simpler (and prettier) `ggplot2`

based
plotting function for coefficients:

`plot_coefs(fit, scales = "free") + ggplot2::theme_bw()`

The `walker`

function also returns samples from the
posterior predictive distribution \(p(y^{\textrm{rep}} | y) = \int p(y^{\textrm{rep}}
| \beta, \sigma, y) p(\beta, \sigma | y)
\textrm{d}\beta\textrm{d}\sigma\). This can be used to used for
example in assessment of model fit to the data. By comparing the
replication series (mean and 95% quantiles in black) and the original
observations (in red) we see that very good overlap, which is not that
surprising given that we know the correct model:

`pp_check(fit)`

It is also possible to perform leave-future-out cross validation
(LFO-CV) for the `walker`

models. While the exact LFO-CV can
be computationally demanding, `walker`

also supports
approximate PSIS-LFO-CV (Bürkner, Gabry, and Vehtari 2020) via
function `lfo`

.

We can also produce “counterfactual” predictions using our estimated
model. For example, we can ask what if `x1`

variable was
constant 0 from time point 60 onward:

`library(ggplot2)`

`## Warning: package 'ggplot2' was built under R version 4.3.1`

`library(dplyr)`

```
##
## Attaching package: 'dplyr'
```

```
## The following objects are masked from 'package:stats':
##
## filter, lag
```

```
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
```

```
fitted <- fitted(fit) # estimates given our actual observed data
newdata <- data.frame(x1 = c(x1[1:59], rep(0, 41)), x2 = x2)
pred_x1_1 <- predict_counterfactual(fit, newdata, type = "mean")
cbind(as.data.frame(rbind(fitted, pred_x1_1)),
type = rep(c("observed", "counterfactual"), each = n), time = 1:n) %>%
ggplot(aes(x = time, y = mean)) +
geom_ribbon(aes(ymin = `2.5%`, ymax = `97.5%`, fill = type), alpha = 0.2) +
geom_line(aes(colour = type)) + theme_bw()
```

It is also possible to obtain actual forecasts given new covariates \(x^{new}\):

```
new_data <- data.frame(x1 = rnorm(10, mean = 2), x2 = cos((n + 1):(n + 10)))
pred <- predict(fit, new_data)
plot_predict(pred)
```

When modelling regression coefficients as a simple random walk, the
posterior estimates of these coefficients can have large short-term
variation which might not be realistic in practice. One way of imposing
more smoothness for the estimates is to switch from random walk
coefficients to integrated second order random walk coefficients: \[
\beta_{t+1} = \beta_t + \nu_t,\\
\nu_{t+1} = \nu_t + \eta_t.
\] This is essentially local linear trend model (Harvey
1989) with restriction that there is no noise on the \(\beta\) level. This model can be estimated
by switching `rw1`

function inside of the walker formula to
`rw2`

, with almost identical interface, but now \(\sigma\) correspond to the standard
deviations of the slope terms \(\nu\).
The Gaussian prior for \(\nu_1\) most
also be defined. Using RW2 model, the coefficient estimates of our
example model are clearly smoother:

```
fit_rw2 <-walker(y ~ -1 +
rw2(~ x1 + x2, beta = c(0, 10), sigma = c(2, 0.001), nu = c(0, 10)),
refresh = 0, init = 0, chains = 1, sigma_y = c(2, 0.001))
plot_coefs(fit_rw2, scales = "free") + ggplot2::theme_bw()
```

So far we have focused on simple Gaussian linear regression. The
above treatment cannot be directly extended to non-Gaussian models such
as Poisson regression, as the marginal log-likelihood is intractable.
However, it is possible to use relatively accurate Gaussian
approximations, and the resulting approximate posterior can then be
weighted using importance sampling type correction (Vihola,
Helske, and Franks 2020), leading to asymptotically exact
inference (if necessary). Function `walker_glm`

extends the
the package to handle Poisson and binomial observations using the
aforementioned methodology. Further extension to negative binomial
distribution is planned in future.

We now compare the efficiency of the “naive” implementation and the
state space approach. For this, we use function `walker_rw1`

,
which supports only basic random walk models (here priors for standard
deviations are defined as truncated normal distributions). We can
perform the same analysis with naive implementation by setting the
argument `naive`

to `TRUE`

. Note that in this case
we need to adjust the default sampling parameters (argument
`control`

) to avoid (most of the) problems in sampling.

```
set.seed(1) # set seed to simulate same initial values for both methods
naive_fit <- walker_rw1(y ~ x1 + x2, refresh = 0,
chains = 2, cores = 2, iter = 1e4,
beta = cbind(0, rep(5, 3)), sigma = cbind(0, rep(2, 4)),
naive = TRUE,
control = list(adapt_delta = 0.999, max_treedepth = 12))
set.seed(1)
kalman_fit <- walker_rw1(y ~ x1 + x2, refresh = 0,
chains = 2, cores = 2, iter = 1e4,
beta = cbind(0, rep(5, 3)), sigma = cbind(0, rep(2, 4)),
naive = FALSE)
```

(In order to keep CRAN checks fast, the results here are precomputed.)

With naive implementation we get some warnings and much higher computation time:

`check_hmc_diagnostics(naive_fit$stanfit)`

```
##
## Divergences:
```

```
## 14 of 10000 iterations ended with a divergence (0.14%).
## Try increasing 'adapt_delta' to remove the divergences.
```

```
##
## Tree depth:
```

`## 0 of 10000 iterations saturated the maximum tree depth of 12.`

```
##
## Energy:
```

`## E-BFMI indicated no pathological behavior.`

`check_hmc_diagnostics(kalman_fit$stanfit)`

```
##
## Divergences:
```

`## 0 of 10000 iterations ended with a divergence.`

```
##
## Tree depth:
```

`## 0 of 10000 iterations saturated the maximum tree depth of 10.`

```
##
## Energy:
```

`## E-BFMI indicated no pathological behavior.`

`get_elapsed_time(naive_fit$stanfit)`

```
## warmup sample
## chain:1 716.958 416.457
## chain:2 761.639 409.833
```

`get_elapsed_time(kalman_fit$stanfit)`

```
## warmup sample
## chain:1 21.213 23.804
## chain:2 20.374 24.557
```

Naive implementation produces also much smaller effective sample sizes:

`print(naive_fit$stanfit, pars = c("sigma_y", "sigma_b"))`

```
## Inference for Stan model: rw1_model_naive.
## 2 chains, each with iter=10000; warmup=5000; thin=1;
## post-warmup draws per chain=5000, total post-warmup draws=10000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## sigma_y 0.51 0.01 0.13 0.23 0.43 0.52 0.60 0.75 315 1
## sigma_b[1] 0.59 0.01 0.13 0.36 0.50 0.58 0.67 0.85 579 1
## sigma_b[2] 0.08 0.00 0.04 0.01 0.05 0.07 0.10 0.17 1335 1
## sigma_b[3] 0.31 0.00 0.10 0.15 0.24 0.30 0.37 0.54 2162 1
##
## Samples were drawn using NUTS(diag_e) at Thu Jan 27 16:03:57 2022.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).
```

`print(kalman_fit$stanfit, pars = c("sigma_y", "sigma_b"))`

```
## Inference for Stan model: rw1_model.
## 2 chains, each with iter=10000; warmup=5000; thin=1;
## post-warmup draws per chain=5000, total post-warmup draws=10000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## sigma_y 0.49 0 0.14 0.14 0.41 0.50 0.58 0.74 2417 1
## sigma_b[1] 0.60 0 0.13 0.36 0.51 0.59 0.68 0.88 3837 1
## sigma_b[2] 0.08 0 0.04 0.01 0.05 0.08 0.10 0.17 5404 1
## sigma_b[3] 0.31 0 0.10 0.15 0.24 0.30 0.37 0.55 6033 1
##
## Samples were drawn using NUTS(diag_e) at Thu Jan 27 16:04:48 2022.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).
```

In this vignette we illustrated the benefits of marginalisation in
the context of time-varying regression models. The underlying idea is
not new; this approach is typical in classic Metropolis-type algorithms
for linear-Gaussian state space models where the marginal likelihood
\(p(y | \theta)\) (where \(\theta\) denotes the hyperparameters
i.e. not the latents states such as \(\beta\)’s in current context) is used in
the computation of the acceptance probability. Here we rely on readily
available Hamiltonian Monte Carlo based `Stan`

software, thus
allowing us to enjoy the benefits of diverse tools of the
`Stan`

community.

The original motivation behind the `walker`

was to test
the efficiency of importance sampling type weighting method of Vihola, Helske, and Franks (2020) in a dynamic generalized linear
model setting within the HMC context. Although some of our other
preliminary results have suggested that naive combination of the HMC
with IS weighting can provide rather limited computational gains, in
case of GLMs with time-varying coefficients so called global
approximation technique (Vihola, Helske, and Franks 2020)
provides fast and robust alternative to standard HMC approach of
`Stan`

.

This work has been supported by the Academy of Finland research grants 284513, 312605, 311877, and 331817.

Bürkner, Paul-Christian, Jonah Gabry, and Aki Vehtari. 2020.
“Approximate Leave-Future-Out Cross-Validation for Bayesian Time
Series Models.” *Journal of Statistical Computation and
Simulation* 90 (14): 2499–2523. https://doi.org/10.1080/00949655.2020.1783262.

Durbin, J., and S. J. Koopman. 2002. “A Simple and Efficient
Simulation Smoother for State Space Time Series Analysis.”
*Biometrika* 89: 603–15.

Harvey, A. C. 1989. *Forecasting, Structural Time Series Models and
the Kalman Filter*. Cambridge University Press.

Helske, Jouni. 2022. “Efficient Bayesian Generalized Linear Models
with Time-Varying Coefficients: The Walker Package in r.”
*SoftwareX* 18: 101016. https://doi.org/10.1016/j.softx.2022.101016.

R Core Team. 2020. *R: A Language and Environment for Statistical
Computing*. Vienna, Austria: R Foundation for Statistical Computing.
https://www.R-project.org/.

Stan Development Team. 2016a. *RStan: The R
Interface to Stan*. https://mc-stan.org/.

———. 2016b. *The Stan C++ Library*. https://mc-stan.org/.

———. 2017. *Shinystan: Interactive Visual and Numerical Diagnostics
and Posterior Analysis for Bayesian Models*. https://mc-stan.org/.

Vihola, Matti, Jouni Helske, and Jordan Franks. 2020. “Importance
Sampling Type Estimators Based on Approximate Marginal Markov Chain
Monte Carlo.” *Scandinavian Journal of Statistics*. https://doi.org/10.1111/sjos.12492.