predict.mcmc_output: Predictions for State Space Models

Description

Draw samples from the posterior predictive distribution for future time points given the posterior draws of hyperparameters $θ$ and latent state $a l p h a_{n + 1}$ returned by run_mcmc. Function can also be used to draw samples from the posterior predictive distribution $p ({\tilde{y}}_{1}, \dots, {\tilde{y}}_{n} | y_{1}, \dots, y_{n})$ .

Usage

# S3 method for mcmc_output
predict(
  object,
  model,
  nsim,
  type = "response",
  future = TRUE,
  seed = sample(.Machine$integer.max, size = 1),
  ...
)

Value

A data.frame consisting of samples from the predictive posterior distribution.

Arguments

object: Results object of class mcmc_output from run_mcmc.
model: A bssm_model object. Should have same structure and class as the original model which was used in run_mcmc, in order to plug the posterior samples of the model parameters to the right places. It is also possible to input the original model for obtaining predictions for past time points. In this case, set argument future to FALSE.
nsim: Positive integer defining number of samples to draw. Should be less than or equal to sum(object$counts) i.e. the number of samples in the MCMC output. Default is to use all the samples.
type: Type of predictions. Possible choices are "mean" "response", or "state" level.
future: Default is TRUE, in which case predictions are for the future, using posterior samples of (theta, alpha_T+1) i.e. the posterior samples of hyperparameters and latest states. Otherwise it is assumed that model corresponds to the original model.
seed: Seed for the C++ RNG (positive integer). Note that this affects only the C++ side, and predict also uses R side RNG for subsampling, so for replicable results you should call set.seed before predict.
...: Ignored.

Examples

Run this code

library("graphics")
y <- log10(JohnsonJohnson)
prior <- uniform(0.01, 0, 1)
model <- bsm_lg(window(y, end = c(1974, 4)), sd_y = prior,
  sd_level = prior, sd_slope = prior, sd_seasonal = prior)

mcmc_results <- run_mcmc(model, iter = 5000)
future_model <- model
future_model$y <- ts(rep(NA, 25), 
  start = tsp(model$y)[2] + 2 * deltat(model$y), 
  frequency = frequency(model$y))
# use "state" for illustrative purposes, we could use type = "mean" directly
pred <- predict(mcmc_results, model = future_model, type = "state", 
  nsim = 1000)

library("dplyr")
sumr_fit <- as.data.frame(mcmc_results, variable = "states") |>
  group_by(time, iter) |> 
  mutate(signal = 
      value[variable == "level"] + 
      value[variable == "seasonal_1"]) |>
  group_by(time) |>
  summarise(mean = mean(signal), 
    lwr = quantile(signal, 0.025), 
    upr = quantile(signal, 0.975))

sumr_pred <- pred |> 
  group_by(time, sample) |>
  mutate(signal = 
      value[variable == "level"] + 
      value[variable == "seasonal_1"]) |>
  group_by(time) |>
  summarise(mean = mean(signal),
    lwr = quantile(signal, 0.025), 
    upr = quantile(signal, 0.975)) 
    
# If we used type = "mean", we could do
# sumr_pred <- pred |> 
#   group_by(time) |>
#   summarise(mean = mean(value),
#     lwr = quantile(value, 0.025), 
#     upr = quantile(value, 0.975)) 
    
library("ggplot2")
rbind(sumr_fit, sumr_pred) |> 
  ggplot(aes(x = time, y = mean)) + 
  geom_ribbon(aes(ymin = lwr, ymax = upr), 
   fill = "#92f0a8", alpha = 0.25) +
  geom_line(colour = "#92f0a8") +
  theme_bw() + 
  geom_point(data = data.frame(
    mean = log10(JohnsonJohnson), 
    time = time(JohnsonJohnson)))

# Posterior predictions for past observations:
yrep <- predict(mcmc_results, model = model, type = "response", 
  future = FALSE, nsim = 1000)
meanrep <- predict(mcmc_results, model = model, type = "mean", 
  future = FALSE, nsim = 1000)
  
sumr_yrep <- yrep |> 
  group_by(time) |>
  summarise(earnings = mean(value),
    lwr = quantile(value, 0.025), 
    upr = quantile(value, 0.975)) |>
  mutate(interval = "Observations")

sumr_meanrep <- meanrep |> 
  group_by(time) |>
  summarise(earnings = mean(value),
    lwr = quantile(value, 0.025), 
    upr = quantile(value, 0.975)) |>
  mutate(interval = "Mean")
    
rbind(sumr_meanrep, sumr_yrep) |> 
  mutate(interval = 
    factor(interval, levels = c("Observations", "Mean"))) |>
  ggplot(aes(x = time, y = earnings)) + 
  geom_ribbon(aes(ymin = lwr, ymax = upr, fill = interval), 
   alpha = 0.75) +
  theme_bw() + 
  geom_point(data = data.frame(
    earnings = model$y, 
    time = time(model$y)))

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