Time series analysis for age of death of succissive kings in England
Data science master
2018.11.06
Age of Death of Successive Kings of England.Starting with William the Conqueror
Source: McNeill, “Interactive Data Analysis”
(1)Plot the data
kings<-read.table("kings.dat.txt",skip = 3)
tk<-ts(kings)
plot.ts(tk,xlab="Time",ylab="years",main="Age of death of succissive kings in England")
(2)Test for the stationary(plotting auto-corelation function)
acf(tk,lag=30)
(3)Plot the partial auto-coretaliton function
pacf(tk,lag=30)
(4)Model Identification
Tentative models:One significance spike in ACF and one significance spike in PACF
(5)Tentative models
ACF for MA and PAFC for AR
- MA(1) model
- AR(1) model
- ARMA(1,1) model
(6)Parameter estimation
- For MA(1) model:
fit.ma1<-arima(tk,order=c(0,0,1))
fit.ma1##
## Call:
## arima(x = tk, order = c(0, 0, 1))
##
## Coefficients:
## ma1 intercept
## 0.3308 55.3263
## s.e. 0.1278 3.1159
##
## sigma^2 estimated as 233: log likelihood = -174.12, aic = 354.24- For AR(1) model:
fit.ar1<-arima(tk,order=c(1,0,0))
fit.ar1##
## Call:
## arima(x = tk, order = c(1, 0, 0))
##
## Coefficients:
## ar1 intercept
## 0.3921 55.3666
## s.e. 0.1392 3.7503
##
## sigma^2 estimated as 224.9: log likelihood = -173.41, aic = 352.82- For ARIMA(1,1) model:
fit.ar1.ma1<-arima(tk,order=c(1,0,1))
fit.ar1.ma1##
## Call:
## arima(x = tk, order = c(1, 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## 0.8341 -0.5740 56.0862
## s.e. 0.1696 0.2527 5.4798
##
## sigma^2 estimated as 216.1: log likelihood = -172.63, aic = 353.25(7)Selecting the best model
Best modl is the minimum AIC model.(i.e. AR(1) model)
(8)Testing adequacy of the model(Residual checking)
res.fit.ar1<-residuals(fit.ar1)
res.fit.ar1## Time Series:
## Start = 1
## End = 42
## Frequency = 1
## [1] 4.26228680 -14.18354185 16.48282801 -9.92851767 2.73785219
## [6] -13.61498424 -0.12503259 11.73785219 8.85576113 -17.32065708
## [11] 14.48282801 -25.14423887 0.01208264 -18.08572960 2.01208264
## [16] -11.87000841 -36.73289318 -3.75298989 5.61994324 1.56143397
## [21] -39.61498424 3.07059190 18.48282801 -1.71279648 -8.79140245
## [26] 6.52213099 29.20859755 -12.37916634 12.77715517 -9.32065708
## [31] -20.65428722 2.40422204 14.12999159 17.07148233 17.15008829
## [36] 1.58153068 11.07148233 19.50292471 2.58153068 9.67934292
## [41] 15.89506412 -7.84991171- (8.1)Residual histogram
hist(res.fit.ar1,col="light blue")
- (8.2)Residual plot
plot.ts(res.fit.ar1,ylab="Residuals")
points(res.fit.ar1,pch=16)
abline(0,0,col="red",lwd=2)
- (8.3)Auto-corretlation function for residual plot
acf(res.fit.ar1)
- (8.3) Partial auto-corelation function for residual plot
pacf(res.fit.ar1)
(9)Diagnostick checking
- Box-Pierce test:
Box.test(res.fit.ar1,type="Box-Pierce")##
## Box-Pierce test
##
## data: res.fit.ar1
## X-squared = 0.031555, df = 1, p-value = 0.859- Ljung-Box test:
Box.test(res.fit.ar1,type="Ljung-Box")##
## Box-Ljung test
##
## data: res.fit.ar1
## X-squared = 0.033864, df = 1, p-value = 0.854