LAB 9

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George Mason University *

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MISC

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Industrial Engineering

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Dec 6, 2023

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docx

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LAB 9: Autocorrelation and Autoregressive Models PART I: Practice Problems with code and outputs given in the R Demonstration Videos . Refer to the R Demonstration Videos for a walk-through of Part I. You are expected to duplicate this demonstration as Part I of the lab and submit it for credit and grading. Save the Amtrak.csv file into a known folder. The dataset file is found in the left-hand menu of Blackboard in a link titled Data Sets Open RStudio. Go to File > New File > R Script Save your Untitled1.R file into the same directory that your csv is saved. Give it the name Autocorrelation.R From the Menu Bar, go to Sessions > Set Working Directory > To Source File Location; This sets the working directory as the place where both the R file and the csv file are saved . Duplicate the following code: Autocorrelation and AR models The dataset Amtrak.csv contains counts of riders on Amtrak trains in thousands Install the forecast package library(forecast) amtrak.df<-read.csv(“Amtrak.csv, header=T) amtrak.ts<-ts(amtrak.df$Ridership, start = c(1991, 1), end = c(2004, 3), freq = 12) create time series object using ts() ts() takes three arguments: start, end, and freq. With monthly data, the frequency of periods per cycle is 12 (per year). Arguments start and end are (cycle [=year], seasonal period [=month] number) Here start is Jan 1991: start = c(1991, 1); end is Mar 2004: end = c(2004,3) plot(amtrak.ts, xlab = "Time", ylab = "Ridership (in 000s)“) plot the data amtrak.full.qseason<-tslm(amtrak.ts~trend+I(trend^2)+season) amtrak.full.qseason.pred<-forecast(amtrak.full.qseason, h=1, level=0) amtrak.full.qseason.pred combining the training and validation dataset to get an improved forecast from chosen “best” model Acf(amtrak.ts, lag.max = 12, main = “ACF of Data”) Compute and display autocorrelation for different lags Acf(amtrak.full.qseason$residuals, lag.max = 12, main = “ACF of Residuals”)
Autocorrelation for the residuals amtrak.res.ar1 <- Arima(amtrak.full.qseason$residuals, order = c(1,0,0)) summary(amtrak.res.ar1) Fit an AR(1) model to residuals. Use Arima() in the forecast package to fit an ARIMA model. The ARIMA models include the AR models; order c(1,0,0) gives an AR(1) model amtrak.res.ar1.pred <-forecast(amtrak.res.ar1,h = 1, level=0) amtrak.res.ar1.pred predicted residual amtrak.res.ar1.pred$mean+amtrak.full.qseason.pred$mean improved forecast that combines the predicted residual and the previously predicted output Acf(amtrak.res.ar1$residuals, lag.max=12,main=”ACF of residuals for AR(1)) ACF on the residuals of the AR(1) model plot(amtrak.full.qseason$residuals, ylim=c(-250,250),ylab=”Residuals”) lines(amtrak.res.ar1.pred$fitted,col=”blue”) A plot of actual residuals vs. the forecasted residuals PART 2: Practice Problems with outputs given . Using R, answer the following questions and check your results with the given output provided. 1. For the Amtrak dataset, fit a cubic plus seasonal model to the entire data set and generate a graph of the ACF function on the residuals. Graph the plot for two years (lag.max=24) Solution: 2. For the model generated in the previous question (cubic plus seasonal), fit an AR(1) model to the residuals and predict the residual for April 2004. Combine the predicted residual with the predicted output of April 2004 using the cubic + seasonal model to get an improved forecast. Report that improved forecast. Solution: Apr 2004 2149.518
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