Efficient Bayesian generalized linear models with time-varying coefficients
Jouni Helske
13 October 2022
Introduction
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.
Illustration
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$stanfit, pars = c("sigma_y", "sigma_rw1"))## Inference for Stan model: walker_lm.
## 1 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=1000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## sigma_y 0.59 0 0.11 0.39 0.52 0.59 0.66 0.81 687 1.01
## sigma_rw1[1] 0.48 0 0.10 0.30 0.41 0.47 0.54 0.70 729 1.00
## sigma_rw1[2] 0.08 0 0.04 0.02 0.06 0.08 0.10 0.17 795 1.00
## sigma_rw1[3] 0.26 0 0.08 0.14 0.21 0.25 0.31 0.43 805 1.00
##
## Samples were drawn using NUTS(diag_e) at Thu Aug 29 20:11:02 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).mcmc_areas(as.matrix(fit$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 <- rstan::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()
Posterior predictive checks
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.
Counterfactual predictions
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 4.3.3library(dplyr)## Warning: package 'dplyr' was built under R version 4.3.3##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionfitted <- 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()
Out-of-sample prediction
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)
Extensions: Smoother effects and non-Gaussian models
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.
Comparison with naive approach
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$stanfit)## warmup sample
## chain:1 716.958 416.457
## chain:2 761.639 409.833get_elapsed_time(kalman_fit$stanfit)## warmup sample
## chain:1 21.213 23.804
## chain:2 20.374 24.557Naive 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).Discussion
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.
Acknowledgements
This work has been supported by the Academy of Finland research grants 284513, 312605, 311877, and 331817.