INTRODUCTION TO STATISTICAL DATA ANALYSIS

Time Series Analysis

Work with a Time Series Dataset:

Install Packages and Load Libraries

library(ggplot2)
library(forecast)
library(tseries)

Set a Working Directory

setwd("~/R_TRAINING")

Import a Time Series Dataset and Visualize it

cpi_data<-read.csv("cpi_data.csv")
head(cpi_data)
      DATE  CPI
1 1/1/1967 34.8
2 2/1/1967 34.7
3 3/1/1967 34.7
4 4/1/1967 34.6
5 5/1/1967 34.6
6 6/1/1967 34.9

Extract the Data of Interest

CPI <- cpi_data$CPI

Convert to time series object

ts_CPI<-ts(CPI,start =c(1967,1), frequency = 12)
head(ts_CPI)
      Jan  Feb  Mar  Apr  May  Jun
1967 34.8 34.7 34.7 34.6 34.6 34.9
#View(CPI)

Plot time series data

plot(ts_CPI, main="Time Series of CPI", col="magenta", lwd=2)

Decompose the time series into trend, seasonality, and residuals.

plot(decompose(ts_CPI))

Check Stationarity using ADF

adf.test(ts_CPI)

    Augmented Dickey-Fuller Test

data:  ts_CPI
Dickey-Fuller = -1.7508, Lag order = 8, p-value = 0.6838
alternative hypothesis: stationary
  • if p-value< 0.05, the series is stationary
  • If p-value > 0.05, the data is non-stationary. Take the first difference

Take the first difference and test for stationarity

diff_data<-diff(ts_CPI)
plot(diff_data, main="Differenced Time Series", col="tan2")

adf.test(diff_data) 

    Augmented Dickey-Fuller Test

data:  diff_data
Dickey-Fuller = -5.1378, Lag order = 8, p-value = 0.01
alternative hypothesis: stationary

Analyze autocorrelation using ACF.

Plot the ACF

par(mfrow=c(1,2))
acf(ts_CPI, main="ACF Plot")
pacf(ts_CPI, main="PACF Plot")

par(mfrow=c(1,1))

Build and evaluate an ARIMA model for forecasting.

Fit ARIMA

model<-auto.arima(ts_CPI)
summary(model)
Series: ts_CPI 
ARIMA(1,2,1)(0,0,2)[12] 

Coefficients:
         ar1      ma1     sma1     sma2
      0.2005  -0.7875  -0.1435  -0.1493
s.e.  0.0643   0.0464   0.0390   0.0388

sigma^2 = 0.1455:  log likelihood = -305.59
AIC=621.19   AICc=621.28   BIC=643.75

Training set error measures:
                      ME      RMSE       MAE        MPE      MAPE       MASE
Training set 0.008997016 0.3797776 0.2660481 0.01059324 0.2197267 0.05301554
                     ACF1
Training set -0.003908124

Model Diagnostics

tsdiag(model)

checkresiduals(model)


    Ljung-Box test

data:  Residuals from ARIMA(1,2,1)(0,0,2)[12]
Q* = 18.379, df = 20, p-value = 0.5624

Model df: 4.   Total lags used: 24

Forecasting

forecast_values<-forecast(model, h=12)
head(forecast_values)
$method
[1] "ARIMA(1,2,1)(0,0,2)[12]"

$model
Series: ts_CPI 
ARIMA(1,2,1)(0,0,2)[12] 

Coefficients:
         ar1      ma1     sma1     sma2
      0.2005  -0.7875  -0.1435  -0.1493
s.e.  0.0643   0.0464   0.0390   0.0388

sigma^2 = 0.1455:  log likelihood = -305.59
AIC=621.19   AICc=621.28   BIC=643.75

$level
[1] 80 95

$mean
          Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
2023                                     319.3141 320.0558 320.9268 321.9420
2024 326.4583 327.3673 328.4571 329.4843                                    
          Sep      Oct      Nov      Dec
2023 322.7623 323.5906 324.5323 325.5392
2024                                    

$lower
              80%      95%
May 2023 318.8252 318.5664
Jun 2023 319.2095 318.7616
Jul 2023 319.7379 319.1085
Aug 2023 320.4088 319.5972
Sep 2023 320.8763 319.8778
Oct 2023 321.3397 320.1482
Nov 2023 321.9033 320.5115
Dec 2023 322.5180 320.9187
Jan 2024 323.0310 321.2166
Feb 2024 323.5198 321.4830
Mar 2024 324.1756 321.9091
Apr 2024 324.7552 322.2518

$upper
              80%      95%
May 2023 319.8030 320.0618
Jun 2023 320.9021 321.3501
Jul 2023 322.1158 322.7452
Aug 2023 323.4751 324.2867
Sep 2023 324.6484 325.6468
Oct 2023 325.8415 327.0331
Nov 2023 327.1614 328.5532
Dec 2023 328.5604 330.1597
Jan 2024 329.8856 331.7000
Feb 2024 331.2148 333.2515
Mar 2024 332.7386 335.0051
Apr 2024 334.2133 336.7167

Plot the Forecasted Values

plot(forecast_values, main = "ARIMA FORECAST CPI", col = "magenta")