walker_rw1: Comparison of naive and state space implementation of RW1...

In walker: Bayesian Generalized Linear Models with Time-Varying Coefficients

Comparison of naive and state space implementation of RW1 regression model

Description

This function is the first iteration of the function walker, which supports only time-varying model where all coefficients ~ rw1. This is kept as part of the package in order to compare "naive" and state space versions of the model in the vignette, but there is little reason to use it for other purposes.

Usage

walker_rw1(
  formula,
  data,
  beta,
  sigma,
  init,
  chains,
  naive = FALSE,
  return_x_reg = FALSE,
  ...
)

Arguments

formula	An object of class formula. See lm for details.
data	An optional data.frame or object coercible to such, as in lm.
beta	A matrix with k rows and 2 columns, where first columns defines the prior means of the Gaussian priors of the corresponding k regression coefficients, and the second column defines the the standard deviations of those prior distributions.
sigma	A matrix with k + 1 rows and two colums with similar structure as beta, with first row corresponding to the prior of the standard deviation of the observation level noise, and rest of the rows define the priors for the standard deviations of random walk noise terms. The prior distributions for all sigmas are Gaussians truncated to positive real axis. For non-Gaussian models, this should contain only k rows. For second order random walk model, these priors correspond to the slope level standard deviations.
init	Initial value specification, see sampling.
chains	Number of Markov chains. Default is 4.
naive	Only used for walker function. If TRUE, use "standard" approach which samples the joint posterior p(beta, sigma | y). If FALSE (the default), use marginalisation approach where we sample the marginal posterior p(sigma | y) and generate the samples of p(beta | sigma, y) using state space modelling techniques (namely simulation smoother by Durbin and Koopman (2002)). Both methods give asymptotically identical results, but the latter approach is computationally much more efficient.
return_x_reg	If TRUE, does not perform sampling, but instead returns the matrix of predictors after processing the formula.
...	Additional arguments to sampling.

Examples

## Not run: 
## Comparing the approaches, note that with such a small data 
## the differences aren't huge, but try same with n = 500 and/or more terms...
set.seed(123)
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, 1)
x2 <- 0.25 * cos(1:n)
ts.plot(cbind(beta1 * x1, beta2 *x2), col = 1:2)
u <- cumsum(rnorm(n))
y <- rnorm(n, u + beta1 * x1 + beta2 * x2)
ts.plot(y)
lines(u + beta1 * x1 + beta2 * x2, col = 2)
kalman_walker <- walker_rw1(y ~ -1 + 
  rw1(~ x1 + x2, beta = c(0, 2), sigma = c(0, 2)), 
  sigma_y = c(0, 2), iter = 2000, chains = 1)
print(kalman_walker$stanfit, pars = c("sigma_y", "sigma_rw1"))
betas <- extract(kalman_walker$stanfit, "beta")[[1]]
ts.plot(cbind(u, beta1, beta2, apply(betas, 2, colMeans)), 
  col = 1:3, lty = rep(2:1, each = 3))
sum(get_elapsed_time(kalman_walker$stanfit))
naive_walker <- walker_rw1(y ~ x1 + x2, iter = 2000, chains = 1, 
  beta = cbind(0, rep(2, 3)), sigma = cbind(0, rep(2, 4)), 
  naive = TRUE)
print(naive_walker$stanfit, pars = c("sigma_y", "sigma_b"))
sum(get_elapsed_time(naive_walker$stanfit))

## Larger problem, this takes some time with naive approach

set.seed(123)
n <- 500
beta1 <- cumsum(c(1.5, rnorm(n - 1, 0, sd = 0.05)))
beta2 <- cumsum(c(-1, rnorm(n - 1, 0, sd = 0.5)))
beta3 <- cumsum(c(-1.5, rnorm(n - 1, 0, sd = 0.15)))
beta4 <- 2
x1 <- rnorm(n, 1)
x2 <- 0.25 * cos(1:n)
x3 <- runif(n, 1, 3)
ts.plot(cbind(beta1 * x1, beta2 * x2, beta3 * x3), col = 1:3)
a <- cumsum(rnorm(n))
signal <- a + beta1 * x1 + beta2 * x2 + beta3 * x3
y <- rnorm(n, signal)
ts.plot(y)
lines(signal, col = 2)
kalman_walker <- walker_rw1(y ~ x1 + x2 + x3, iter = 2000, chains = 1,
  beta = cbind(0, rep(2, 4)), sigma = cbind(0, rep(2, 5)))
print(kalman_walker$stanfit, pars = c("sigma_y", "sigma_b"))
betas <- extract(kalman_walker$stanfit, "beta")[[1]]
ts.plot(cbind(u, beta1, beta2, beta3, apply(betas, 2, colMeans)), 
  col = 1:4, lty = rep(2:1, each = 4))
sum(get_elapsed_time(kalman_walker$stanfit))
# need to increase adapt_delta in order to get rid of divergences
# and max_treedepth to get rid of related warnings
# and still we end up with low BFMI warning after hours of computation
naive_walker <- walker_rw1(y ~ x1 + x2 + x3, iter = 2000, chains = 1, 
  beta = cbind(0, rep(2, 4)), sigma = cbind(0, rep(2, 5)),
  naive = TRUE, control = list(adapt_delta = 0.9, max_treedepth = 15)) 
print(naive_walker$stanfit, pars = c("sigma_y", "sigma_b"))
sum(get_elapsed_time(naive_walker$stanfit))

## End(Not run)

walker documentation built on Sept. 11, 2023, 5:10 p.m.