README.Rmd

---
output:
  github_document
---


```{r setup, include=FALSE}
knitr::opts_chunk$set(
  fig.path = "Figures/",
  comment = "#>",
  collapse = TRUE
  )
library(knitr)



```
This repository  provides code in R reproducing examples of the states space models presented in book ["An Introduction to State Space Time Series Analysis"](http://www.ssfpack.com/CKbook.html) by Jacques J.F. Commandeur and Siem Jan Koopman. 
 
![](Figures/CKbook.png)

The repository uses extensively the [KFAS package](https://cran.r-project.org/web/packages/KFAS/index.html) of Jouni Helske which includes computationally efficient functions for Kalman filtering, smoothing, forecasting, and simulation of multivariate exponential family state space models. Additionally, some own functions has been created to facilitate the calculation and presentation of diagnostics.

The code is provided in file "SSM.R" and split into sections corresponding to the following parts of the book:

* Introduction
* Chapter 2 	The Local Level Model
* Chapter 3 	The Local Linear Trend Model
* Chapter 4 	The Local Level Model with Seasonal
* Chapter 5 	The Local Level Model with Explanatory Variable
* Chapter 6 	The Local Level Model with Intervention Variable
* Chapter 7 	The UK Seat Belt and Inflation Models
* Chapter 8 	General Treatment of Univariate State Space Models
* Chapter 9 	Multivariate Time Series Analysis 

In R Studio, each section of the code can be executed with keys CTRL+ALT+T, after placing a cursor in that section. Please make sure to execute the first section of the code including the own defined functions that are used by the other sections of the code.

```{r message=FALSE, warning=FALSE, echo=FALSE}
if(!(require(normtest))){install.packages('normtest')}
library(normtest)
if(!(require(KFAS))){install.packages('KFAS')}
library(KFAS)
#if(!(require(rstudioapi))){install.packages('rstudioapi')}
#library(rstudioapi)
if(!(require(knitr))){install.packages('knitr')}
library(knitr)
if(!(require(eurostat))){install.packages('eurostat')}
library(eurostat)
if(!(require(dplyr))){install.packages('dplyr')}
library(dplyr)
if(!(require(forecast))){install.packages('forecast')}
library(forecast)

#Cleaning workspace
rm(list=ls())
#Setting directory for files with data; they should be in the same directory as the files of source code
#current_path = rstudioapi::getActiveDocumentContext()$path 
#setwd(dirname(current_path))
#print(getwd())


#Function for Q-statistic
#Q-statistic is a general omnibus test that can be used to check whether 
#the combined first k autocorrelations significantly deviate from 0,
#meaning the null hypothesis of independence must be rejected
#We assume that the residuals are independent if the test statistic does not exceed the critical value.
qStatistic <- function(predResid, k, w) {
  #Standardised residuals as predResid should be submitted into this function!
  #k - first k autocorrelations to be used in test; w - number of the disturbance variances
  #(see Commandeur and Koopman, p.90-96)
  value <- Box.test(predResid, lag = k, type = "Ljung")$statistic #Q-statistic based on the statistic calulated in the Ljung-Box test for lags from 1 to k
  criticalValue <- qchisq(0.95, k-w+1) #Critical value corresponding to the upper 5% in the chi-square-distribution with k-w+1 degrees of freedom
  list(#List of values provided by the function
    k = k, #First k autocorrelations
    value = unname(value), #Value of the test statistic
    criticalValue = criticalValue #Critical value
  )
}

#Function for r-statistic
#r-statistic checks independence of one-step-ahead prediction residuals 
#It provides values of the autocorrelations at lags 1 and l, together with the 95% confidence limits. 
#We assume that the residuals are independent if the values of the autocorrelations 
#do not exceed the critical values for the 95% confidence limits 
#(see Commandeur and Koopman, p.90-96)
rStatistic <- function(predResid, d, l) {
  #Standardised residuals as predResid should be submitted into this function!
  #d - diffuse initial value of the state, l - autocorrelation at lag l to be provided by the function
  n <- (length(predResid)-d) #Length of the series after subtracting d, the number of diffuse initial elements of the state
  acfValues <- acf(predResid[-(1:d)], plot = FALSE)$acf[-1]# List of the values of the autocorrelations for the series without the first d values
  criticalValue <- 2 / sqrt(n) # +/- critical value for the 95% confidence limits
  list( #List of values provided by the function
    l = c(1,l), #lags 1 and l 
    value1 = acfValues[1], #Value of the autocorrelation at lag 1
    value2 = acfValues[l], #Value of the autocorrelation at lag l
    criticalValue = criticalValue # +/- critical value for 95% confidence limits
  )
}

#Function for H-statistic
#H-statistic checks homoscedasticity of one-step-ahead prediction residuals 
#This is done by testing the null hypothesis of the equal variances of the residuals 
#in the first third part of the series and the last third part of the series.
#The ratio between these two variances is tested against an F-distribution with (h,h) degrees of freedom 
#applying the usual 5% rule for rejection of the null hypothesis of equal variances, for a two-tailed test.  
#We must find critical values corresponding to the upper and lower 2.5% in the two tails of the F-distribution;
#If, however, the tested statistic is larger than or equal to 1, it is enough to check 
#whether it is lower than the critical value #corresponding to the upper 2.5% in the F-distribution; 
#On the other hand, if the statistic is lower than 1 we have to test 
#if its reciprocal value (1/ratio) is lower than the above-mentioned critical value.
#We assume that the residuals are homoscedastic if the test statistic does not exceed the critical value.
#(see Commandeur and Koopman, p.90-96)
hStatistic <- function(predResid, d) {
  #Standardised residuals as predResid should be submitted into this function!
  #d - number of diffuse initial values in the state,
  n <- length(predResid) # Number of observations/residuals
  h <- round((n-d)/3, digits = 0) #One third of the series: nearest integer to (n-d)/3; also degrees of freedom for the test
  ratio <- sum(predResid[(n-h+1):n]^2) / sum(predResid[(d+1):(d+h)]^2) #Ratio between the variance of the residuals in the last third part of the series and the variance of residuals in the first third part of the series
  value <- ifelse(ratio >= 1, ratio, 1/ratio) # Value of the test statistic; if the ratio is smaller than 1 then the reciprocal value is used for testing (1/ratio)
  criticalValue <- qf(0.975, h, h) # Critical value corresponding to the upper 2.5% in the F-distribution with (h,h) degrees of freedom
  list( #List of values provided by the function
    h = h, #Degrees of freedom
    ratio = ratio, #Ratio between the two variances
    value = value, #Value of the test statistic
    criticalValue = criticalValue #Critical value 
  )
}

#Function for N-statistic
#H-statistic checks normality of one-step-ahead prediction residuals 
#This is done by testing the null hypothesis of normality
#We assume that the residuals are normally distributed if the test statistic does not exceed the critical value at 5% level
#(see Commandeur and Koopman, p.90-96)
nStatistic <- function(predResid, d) {
  #Standardised residuals as predResid should be submitted into this function!
  #d - number of diffuse initial values in the state
  value <- jb.norm.test(predResid[-(1:d)])$statistic #N-statistic based on the statistic calculated in the Jarque and Bera or Shenton and Bowman test;
  criticalValue <- qchisq(0.95,2) #Critical value corresponding to the upper 5% in the chi-square-distribution with 2 degrees of freedom
  list(#List of values provided by the function
    value = unname(value), #Value of the test statistic
    criticalValue = criticalValue #Critical value
  )
}

#Function to create a table with statistics
dTable <- function(qStatistic, rStatistic, hStatistic, nStatistic, title){

cat(title)
cat("\n")

diagnosticTemplateTable <- c(
  
  "-----------------------------------------------------------------------------",    
  "                    statistic    value   critical value   asumption satisfied",    
  "-----------------------------------------------------------------------------",    
  "independence           Q(%2d)   %7.3f            %5.2f        %1s",  # Q-statistic, 4 args    
  "                        r(%1d)   %7.3f           +-%4.2f        %1s", # r-statistics,      4 args    
  "                       r(%2d)   %7.3f           +-%4.2f        %1s", # r,      4 args    
  "homoscedasticity     %-3s(%2d)   %7.3f            %5.2f        %1s",  # Homo,     5 args    
  "normality                  N   %7.3f            %5.2f        %1s",    # N,        3 args    
  "-----------------------------------------------------------------------------"  
) 

cat(    sprintf(      paste(diagnosticTemplateTable, collapse = "\n"),       
                      # Q-statistic, 4 args      
                      qStatistic$k,          
                      qStatistic$value,      
                      qStatistic$criticalValue,      
                      ifelse(qStatistic$value < qStatistic$criticalValue, "+", "-"),     
                      # r-statistic, 4 args      
                      rStatistic$l[1],       
                      rStatistic$value1,      
                      rStatistic$criticalValue,      
                      ifelse(abs(rStatistic$value1) < rStatistic$criticalValue, "+", "-"),      
                      # r-statistic, 4 args   
                      rStatistic$l[2],   
                      rStatistic$value2,
                      rStatistic$criticalValue,      
                      ifelse(abs(rStatistic$value2) < rStatistic$criticalValue, "+", "-"),      
                      # H-statistic, 5 args      
                      ifelse(hStatistic$ratio > 1, "  H", "1/H"),      
                      hStatistic$h,       
                      hStatistic$value,       
                      hStatistic$criticalValue,       
                      ifelse(hStatistic$value < hStatistic$criticalValue, "+", "-"),      
                      # N, 3 args      
                      nStatistic$value,        
                      nStatistic$criticalValue,      
                      ifelse(nStatistic$value < nStatistic$criticalValue, "+", "-")    )  )  

}

#Function to find best initial values for optim ver. 1 
initValOpt <- function(w_ = w , model_ = model, updatefn_ = ownupdatefn, method = "Nelder-Mead", maxLoop = 100){
  results  <- matrix(NA, maxLoop, 2) %>% 
    data.frame() %>%
    `colnames<-`(c("Log.likelihood", "Initial.value"))
  #set.seed(123)
  cat("Loop: ")
  for (j in 1:maxLoop){
    cat(paste(j, " "))
    x <- runif(1, min = 0.00001, max = 2) %>% round(3)
    fit <- fitSSM(inits = log(rep(x, w_)), model = model_, updatefn = updatefn_, method = method)
    maxLik <- (logLik(fit$model, method = method)/n) %>% round(7)
    #results[j, ] <- c(round(maxLik, 7), x)
    results[j, ] <- c(maxLik, x)
  }     
  cat("\n")
  results %>% arrange(desc(Log.likelihood), Initial.value) %>% print()
  return(results[1,2])
}

#Function to find best initial values for optim ver. 2
initValOpt2 <- function(formula = "log(rep(x, 3))", model_ = model, updatefn_ = ownupdatefn, method = "Nelder-Mead", maxLoop = 100){
  results  <- matrix(NA, maxLoop, 2) %>% 
    data.frame() %>%
    `colnames<-`(c("Log.likelihood", "Initial.value"))
  #set.seed(123)
  cat("Loop: ")
  for (j in 1:maxLoop){
    cat(paste(j, ""))
    x <- runif(1, min = 0.00001, max = 2) %>% round(3)
    fit <- fitSSM(inits = eval(parse(text = formula)), model = model_, updatefn = updatefn_, method = method)
    maxLik <- (logLik(fit$model, method = method)/n) %>% round(7)
    results[j, ] <- c(maxLik, x)
  }     
  cat("\n")
  results %>% arrange(desc(Log.likelihood), Initial.value) %>% print()
  return(results[1,2])
}

```

Below, the code of the stochastic level and slope model of chapter 3 is shown as an example.

Loading data on UK drivers killed or seriously injured (KSI):
```{r message=FALSE, warning=FALSE}
dataUKdriversKSI <- log(read.table("UKdriversKSI.txt")) %>% 
  ts(start = 1969, frequency = 12)
head(dataUKdriversKSI, 24)
tail(dataUKdriversKSI, 24)
```

Defining the model using function `SSModel()` of the KFAS package:
```{r message=FALSE, warning=FALSE}
model <- SSModel(dataUKdriversKSI ~ SSMtrend(degree = 2, 
         Q = list(matrix(NA), matrix(NA))),  H = matrix(NA))
ownupdatefn <- function(pars, model){
  model$H[,,1] <- exp(pars[1])
  diag(model$Q[,,1]) <- exp(pars[2:3])
  model
}
(model)
```

Providing the number of diffuse initial values in the state:
```{r message=FALSE, warning=FALSE}
d <- q <- 2 
```

Defining the number of estimated hyperparameters (two state disturbance variances + irregular disturbance variance):
```{r message=FALSE, warning=FALSE}
w <- 3
```

Providing the autocorrelation lag l for r-statistic (ACF function):
```{r message=FALSE, warning=FALSE}
l <- 12
```
Defining the first k autocorrelations to be used in Q-statistic:
```{r message=FALSE, warning=FALSE}
k <- 15
```

Providing the number of observations:
```{r message=FALSE, warning=FALSE}
n <- 192
```

Fitting the model using function `fitSSM()` and extracting the output using function `KFS()` of the KFAS package:
```{r message=FALSE, warning=FALSE}
fit <- fitSSM(model, inits = log(c(0.001, 0001, 0001)), method = "BFGS")
outKFS <- KFS(fit$model, smoothing = c("state", "mean", "disturbance"))
```

Extracting the maximum likelihood using function `logLik()` of the KFAS package: 
```{r message=FALSE, warning=FALSE}
(maxLik <- logLik(fit$model)/n)
```

Calculating the Akaike information criterion (AIC):
```{r message=FALSE, warning=FALSE}
(AIC <- (-2*logLik(fit$model)+2*(w+q))/n)
```

Extracting the maximum likelihood estimate of the irregular variance:
```{r message=FALSE, warning=FALSE}
(H <- fit$model$H)
```

Extracting the maximum likelihood estimate of the state disturbance variances for level and slope:
```{r message=FALSE, warning=FALSE}
(Q <- fit$model$Q)
```

Extracting the initial values of the smoothed estimates of states using function `coef()` of the KFAS package:
```{r message=FALSE, warning=FALSE}
smoothEstStat <- coef(outKFS)
(initSmoothEstStat <- smoothEstStat[1,])
```

Extracting the values for trend (stochastic level + slope) using function `signal()` of the KFAS package:
```{r message=FALSE, warning=FALSE}
trend <-signal(outKFS, states = "trend")$signal
head(trend, 24)
tail(trend, 24)
```

Showing Figure 3.1. of the book for trend of stochastic linear trend model:
```{r message=FALSE, warning=FALSE}
plot(dataUKdriversKSI , xlab = "", ylab = "", lty = 1)
lines(trend, lty = 3)
title(main = "Figure 3.1. Trend of stochastic linear trend model", cex.main = 0.8)
legend("topright",leg = c("log UK drivers KSI", "stochastic level and slope"), 
       cex = 0.5, lty = c(1, 3), horiz = T)
```

Showing Figure 3.2. of the book for slope of stochastic linear trend model:
```{r message=FALSE, warning=FALSE}
plot(smoothEstStat[, "slope"], xlab = "", ylab = "", lty = 1)
title(main = "Figure 3.2. Slope of stochastic linear trend model", 
      cex.main = 0.8)
legend("topleft",leg = "stochastic slope", 
       cex = 0.5, lty = 1, horiz = T)

```

Extracting auxiliary irregular residuals (non-standardised) using function `residuals()` of the KFAS package:
```{r message=FALSE, warning=FALSE}
irregResid <- residuals(outKFS, "pearson") 
head(irregResid, 24)
tail(irregResid, 24)
```


Showing Figure 3.3. of the book for irregular component of stochastic trend model:
```{r message=FALSE, warning=FALSE}
plot(irregResid  , xlab = "", ylab = "", lty = 2)
abline(h = 0, lty = 1)
title(main = "Figure 3.3. Irregular component of stochastic trend model", cex.main = 0.8)
legend("topright",leg = "irregular",cex = 0.5, lty = 2, horiz = T)
```

Extracting one-step-ahead prediction residuals (standardised) using function `rstandard()` of the KFAS package and calculating diagnostic for these residuals using own defined functions `qStatistic()`, `rStatistic()`, `hStatistic()` and `nStatistic()`:
```{r message=FALSE, warning=FALSE}
predResid <- rstandard(outKFS) 
qStat <- qStatistic(predResid, k, w)
rStat <- rStatistic(predResid, d, l)
hStat <- hStatistic(predResid, d)
nStat <- nStatistic(predResid, d)
```

Showing Table 3.2 of the book for diagnostic tests for the local linear trend model applied to the log of the UK drivers KSI using own defined function `dTable()`:
```{r message=FALSE, warning=FALSE}
title = "Table 3.2 Diagnostic tests for the local linear trend model applied to \n
the log of the UK drivers KSI"
dTable(qStat, rStat, hStat, nStat, title)
```