Learning Log

On Tuesday we learned about time series data and how to build and make predictions with the model. The two topics we discussed were polynomial trend and detecting autocorrelation. We start by graphing a linar model and add a power everytime to see if we can get an ideal model. After the cubic power, we can use the poly function to save time in terms of writing our commands. Autocorrelation is a problem because it violates our assumptions that the residuals are random and independent. Autocorrelation affects our prediction result and tend to over estimate our p-values.We use Durbin Watson test to test our hypothesis when it comes to autocorrelation.

library(lmtest)
library(faraway)
data(airpass)
attach(airpass)
head(airpass)
plot(pass~year,type="l")#we see a linear trend with this time series
mod1<-lm(pass~year)#create the linear model to see the four plots
summary(mod1)
plot(mod1)

Since there seems to be a curvature with the residual plot which suggests overprediction, we will try a quadratic function.

mod2<-lm(pass~year+I(year^2))#p-values are small so this graph is acceptable
summary(mod2)
plot(mod2)

Our residual plot looks random which is what we are looking for. Once we have a model that’s best, we can use it for point prediction and test for autocorrelation.

coef(mod1)%*%c(1,62)

         [,1]
[1,] 502.1735

coef(mod2)%*%c(1,62,62^2)

         [,1]
[1,] 538.9268

This example shows that if we use the two different models to do point prediction, we get different answers. This holds because our model are different and we would expect the second model (quadratic) to have a larger value.

dwtest(mod2)

    Durbin-Watson test

data:  mod2
DW = 0.56939, p-value < 2.2e-16
alternative hypothesis: true autocorrelation is greater than 0

dwtest(mod2,alternative = "less")

    Durbin-Watson test

data:  mod2
DW = 0.56939, p-value = 1
alternative hypothesis: true autocorrelation is less than 0

dwtest(mod2,alternative = "two.sided")

    Durbin-Watson test

data:  mod2
DW = 0.56939, p-value < 2.2e-16
alternative hypothesis: true autocorrelation is not 0

The null hypothesis: there is no auto correlation. There are three alternative hypothesis which covers all possible combinations. The test will state the alternative hypothesis. The default is that the true autocorrelation is greater than 0. No matter which way you state the alternative hypothesis, you should get the same result. We know that there is a positive correlation.