状態空間時系列分析入門 第4章 (2/3)

{rstan} で 「状態空間時系列分析入門」 を再現したい。やっていること / 数式はテキストを参照。

リポジトリ: https://github.com/sinhrks/stan-statespace

必要パッケージのインストール/読み込み

library(devtools)
# devtools::install_github('hoxo-m/pforeach')
# devtools::install_github('sinhrks/ggfortify')

library(rstan)
library(pforeach)
library(ggplot2)
ggplot2::theme_set(theme_bw(base_family="HiraKakuProN-W3"))
library(ggfortify)

関数定義

# モデルが収束しているか確認
is.converged <- function(stanfit) {
  summarized <- summary(stanfit)  
  all(summarized$summary[, 'Rhat'] < 1.1)
}

# 値がだいたい近いか確認
is.almost.fitted <- function(result, expected, tolerance = 0.001) {
  if (abs(result - expected) > tolerance) {
    print(paste('Result is ', result))
    return(FALSE)
  } else {
    return(TRUE)
  }
}

データの読み込み

ukdrivers <- read.table('../data/UKdriversKSI.txt', skip = 1)
ukdrivers <- ts(ukdrivers[[1]], start = c(1969, 1), frequency = 12)
ukdrivers <- log(ukdrivers)

モデル定義

確率的レベルと確率的季節要素のあるローカルレベルモデル。

model_file <- '../models/fig04_06.stan'
cat(paste(readLines(model_file)), sep = '\n')

data {
  int<lower=1> n;
  vector[n] y;
}
parameters {
  # 確率的レベル
  vector[n] mu;
  # 確率的季節項
  vector[n] seasonal;
  # レベル撹乱項
  real<lower=0> sigma_level;
  # 季節性撹乱項
  real<lower=0> sigma_seas;
  # 観測撹乱項
  real<lower=0> sigma_irreg;
}
transformed parameters {
  vector[n] yhat;
  for(t in 1:n) {
    yhat[t] <- mu[t] + seasonal[t];
  }
}
model {
  # 式 4.1

  # frequency = 12
  for(t in 12:n) {
    seasonal[t] ~ normal(-seasonal[t-11] - seasonal[t-10] - seasonal[t-9] - seasonal[t-8] - seasonal[t-7] - seasonal[t-6] - seasonal[t-5] - seasonal[t-4] - seasonal[t-3] - seasonal[t-2] - seasonal[t-1], sigma_seas);
  }
  for(t in 2:n)
    mu[t] ~ normal(mu[t-1], sigma_level);
  for(t in 1:n)
    y[t] ~ normal(yhat[t], sigma_irreg);
}

y <- ukdrivers

standata <- within(list(), {
  y <- as.vector(y)
  n <- length(y)
})

stan_fit <- stan(file = model_file, chains = 0)

## 
## TRANSLATING MODEL 'fig04_06' FROM Stan CODE TO C++ CODE NOW.
## COMPILING THE C++ CODE FOR MODEL 'fig04_06' NOW.

fit <- pforeach(i = 1:4, .final = sflist2stanfit)({
  stan(fit = stan_fit, data = standata,
       iter = 10000, chains = 1, seed = i)
})
stopifnot(is.converged(fit))

yhat <- get_posterior_mean(fit, par = 'yhat')[, 'mean-all chains']
mu <- get_posterior_mean(fit, par = 'mu')[, 'mean-all chains']
seasonal <- get_posterior_mean(fit, par = 'seasonal')[, 'mean-all chains']

sigma_irreg <- get_posterior_mean(fit, par = 'sigma_irreg')[, 'mean-all chains']
sigma_level <- get_posterior_mean(fit, par = 'sigma_level')[, 'mean-all chains']
stopifnot(is.almost.fitted(sigma_irreg^2, 0.00351385))
stopifnot(is.almost.fitted(sigma_level^2, 0.000945723))

title <- 'Figure 4.6. Stochastic level.'
title <- '図 4.6 確率的レベル'

# 原系列
p <- autoplot(y)

# stan
mu <- ts(mu, start = start(y), frequency = frequency(y))
p <- autoplot(mu, p = p, ts.colour = 'blue')
p + ggtitle(title)

title <- 'Figure 4.7. Stochastic seasonal.'
title <- '図 4.7 確率的季節'

seasonal <- ts(seasonal, start = start(y), frequency = frequency(y))
autoplot(seasonal, ts.colour = 'blue') + ggtitle(title)

title <- 'Figure 4.8. Stochastic seasonal for the year 1969.'
title <- '図 4.8 1969年に対する確率的季節'

s1969 <- ts(seasonal[1:12], start = start(y), frequency = frequency(y))
autoplot(s1969, ts.colour = 'blue') + ggtitle(title)

title <- 'Figure 4.9. Irregular component for stochastic level and seasonal model.'
title <- '図 4.9 確率的レベルと季節モデルに対する不規則要素'

autoplot(y - yhat, ts.linetype = 'dashed') + ggtitle(title)