状態空間時系列分析入門 第7章 (3/3)

{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

ukinflation <- read.table('../data/UKinflation.txt', skip = 1)
ukinflation <- ts(ukinflation[[1]], start = c(1950, 1), frequency = 4)

ukpulse <- rep(0, length.out = length(ukinflation))
ukpulse[4*(1975-1950)+2] <- 1
ukpulse[4*(1979-1950)+3] <- 1
ukpulse <- ts(ukpulse, start = start(ukinflation), frequency = frequency(ukinflation))

モデル定義

model_file <- '../models/fig07_04.stan'
cat(paste(readLines(model_file)), sep = '\n')

data {
  int<lower=1> n;
  vector[n] y;
  vector[n] w;
}
parameters {
  # 確率的レベル
  vector[n] mu;
  # 確率的季節項
  vector[n] seasonal;
  # 確定的係数
  real lambda;
  # 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] + lambda * w[t];
  }
}
model {
  # 式 7.1

  # frequency = 4
  for(t in 4:n) {
    seasonal[t] ~ normal(-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] + seasonal[t], sigma_irreg);
}

y <- ukinflation
w <- ukpulse

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

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

## 
## TRANSLATING MODEL 'fig07_04' FROM Stan CODE TO C++ CODE NOW.
## COMPILING THE C++ CODE FOR MODEL 'fig07_04' NOW.

fit <- pforeach(i = 1:4, .final = sflist2stanfit)({
  stan(fit = stan_fit, data = standata, 
       iter = 2000, 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']
lambda <- get_posterior_mean(fit, par = 'lambda')[, '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']
sigma_seas <- get_posterior_mean(fit, par = 'sigma_seas')[, 'mean-all chains']

stopifnot(is.almost.fitted(sigma_irreg^2, 2.1990e-5))
stopifnot(is.almost.fitted(sigma_level^2, 1.8595e-5))
stopifnot(is.almost.fitted(sigma_seas^2, 0.0110e-5))

title <- paste('Figure 7.7.1. Local level (including pulse interventions) ',
               'for UK inflation time series data.', sep = '\n')
title <- paste('図 7.7.1 英国インフレーション時系列データに',
               '対するローカル・レベル(含むパルス干渉変数)', sep = '\n')
p <- autoplot(y)
yhat <- ts(yhat, start = start(y), frequency = frequency(y))
p <- autoplot(yhat, p = p, ts.colour = 'blue')
p + ggtitle(title)

title <- 'Figure 7.7.2. Local seasonal for UK inflation time series data.'
title <- '図 7.7.2 ローカル季節'
seasonal <- ts(seasonal, start = start(y), frequency = frequency(y))
autoplot(seasonal, ts.colour = 'blue') + ggtitle(title)

title <- 'Figure 7.7.3. Irregular for UK inflation time series data.'
title <- '図 7.7.3 不規則要素'
autoplot(y - yhat, ts.linetype = 'dashed') + ggtitle(title)