[SOLVED] CS ETW3420 Principles of Forecasting and Applications

ETW3420 Principles of Forecasting and Applications

Principles of Forecasting and Applications

Copyright By Assignmentchef assignmentchef

Topic 7 Exercises

Question 1

Daily electricity demand for Victoria, Australia, during 2014 is contained in elecdaily. The

data is for 365 days. The data for the first 20 days can be obtained as follows.

daily20 <- head(elecdaily,20)(a) Plot the data and find the regression model for Demand with temperature as anexplanatory variable. Why is there a positive relationship?#Time series plotautoplot(daily20, facets=TRUE)#Scatter plot using ggplot(); rmbr the need to make data a dataframe objectdaily20 %>%

as.data.frame() %>%

ggplot(aes(x = Temperature, y = Demand)) +

geom_point() +

geom_smooth(method = lm, se = FALSE)

#Fit linear regression model

(fit <- tslm(Demand ~ Temperature, data = daily20))(b) Produce a residual plot. Is the model adequate? Are there any outliers or influentialobservations?checkresiduals(fit)(c) Use the model to forecast the electricity demand that you would expect for the nextday if the maximum temperature was 15 and compare it with the forecast if the withmaximum temperature was 35. Do you believe these forecasts? To forecast electricity demand, we use the forecast() function as usual. However, in this question, we are given specific values of the x-variable (i.e. temperature)for which the forecasts are to be generated. Hence, we need to specify these x-values as a data frame object in the newdata argumentwithin the forecast() function.fc <- forecast(fit, newdata = data.frame(Temperature = c(15, 35)))(d) Give prediction intervals for your forecasts.(e) Plot Demand vs Temperature for all of the available data in elecdaily. What doesthis say about your model?elecdaily %>%

as.data.frame() %>%

ggplot(aes(x = Temperature, y = Demand)) +

geom_point() +

geom_smooth(method = lm, se = FALSE)

Question 2

The data set fancy concerns the monthly sales figures of a shop which opened in January

1987 and sells gifts, souvenirs, and novelties. The shop is situated on the wharf at a beach

resort town in Queensland, Australia. The sales volume varies with the seasonal population

of tourists. There is a large influx of visitors to the town at Christmas and for the local

surfing festival, held every March since 1988. Over time, the shop has expanded its premises,

range of products, and staff.

(a) Produce a time plot of the data and describe the patterns in the graph. Identify any

unusual or unexpected fluctuations in the time series.

autoplot(fancy) +

xlab(Year) + ylab(Sales)

(b) Explain why it is necessary to take logarithms of these data before fitting a model.

# Taking logarithms of the data

autoplot(log(fancy)) + ylab(log Sales)

(c) Use R to fit a regression model to the logarithms of these sales data with a linear trend,

seasonal dummies and a surfing festival dummy variable. (Note that in R, the logical

arguments of TRUE and FALSE take the values of 1 and 0 respectively).

#Execute the following codes and observe what output is generated

help(cycle)

cycle(fancy)

cycle(fancy) == 3

#Create festival dummy:

festival <- cycle(fancy) == 3festival[3] <- FALSE #i.e. FALSE for March 1987#Print festival dummy# Fit linear model to logged data (by specifying lambda=0)fit <- tslm(fancy ~ trend + season + festival, lambda = 0)# Check fitted valuesautoplot(fancy) + ylab(“Sales”) +autolayer(fitted(fit), series=”Fitted”)(d) Plot the residuals against time and against the fitted values. Do these plots reveal anyproblems with the model?#Plot residuals against timeautoplot(residuals(fit))#Plot residuals against fitted values#Create dataframe first for subsequent ggplot()temp.df <- data.frame(Residuals = residuals(fit), Fitted = fitted(fit))ggplot(data = temp.df, aes(x = Fitted, y = Residuals)) +geom_point()(e) What do the values of the coefficients tell you about each variable?coefficients(fit)(f) What does the Breusch-Godfrey test tell you about your model?checkresiduals(fit)(g) Regardless of your answers to the above questions, use your regression model to predictthe monthly sales for 1994, 1995, and 1996. Produce prediction intervals for each ofyour forecasts.#Create dataframe of festival for predictionfuture.festival <- rep(FALSE, 36)future.festival[c(3, 15, 27)] <- TRUE#Produce forecastforecast(h=36, newdata=data.frame(festival = future.festival)) %>%

(h) Transform your predictions and intervals to obtain predictions and intervals for the raw

(i) How could you improve these predictions by modifying the model?

Question 3

The data set huron gives the level of Lake Huron in feet from 1875 to 1972.

(a) Plot the data and comment on its features.

autoplot(huron)

(b) Fit a linear regression and compare this to a piecewise linear trend model with a knot

at 1915. (Refer to Slide 39 of Topic 7 lecture notes).

#Linear Regression

(fit.lin <- tslm(huron ~ trend))#Execute the following codes to observe what outputs they generatet <-time(huron)help(pmax)pmax(t-1915,0)#Piecewise linear trend modelt <- time(huron)tb <- ts(pmax(t-1915, 0))(fit.pw <- tslm(huron ~ t + tb))(c) Generate forecasts from these two models for the period up to 1980 and comment on#Forecast length#Forecasts from linear modelfcasts.lin <- forecast(fit.lin, h=h)#Forecasts from piecewise linear model#Create the new variables and their corresponding values for forecast periodt.new <- t[length(t)] + seq(h)tb.new <- tb[length(tb)] + seq(h)#Combine new variables as data framenewdata <- cbind(t=t.new,tb=tb.new) %>% as.data.frame()

#Produce the forecast

fcasts.pw <- forecast(fit.pw, newdata = newdata)#Plot data with predictions and out-of-sample forecastsautoplot(huron) +autolayer(fitted(fit.lin), series = “Linear”) +autolayer(fitted(fit.pw), series = “Piecewise”) +autolayer(fcasts.lin, series = “Linear”) +autolayer(fcasts.pw, series=”Piecewise”)Question 4 (Optional, depending on time)The gasoline series consists of weekly data for supplies of US finished motor gasoline product,from 2 February 1991 to 20 January 2017. The units are in thousand barrels per day.Consider only the data to the end of 2004.(a) Fit a harmonic regression with trend to the data. Select the appropriate number ofFourier terms to include by minimizing the AICc or CV value.gas <- window(gasoline, end=2004.99)cv <- numeric(25)for(k in seq(25))fit <- tslm(gas ~ trend + fourier(gas, K=k))cv[k] <- CV(fit)[‘CV’]k <- which.min(cv)fit <- tslm(gas ~ trend + fourier(gas, K=k))(b) Check the residuals of the final model using the checkresiduals() function.checkresiduals(fit)(c) To forecast using harmonic regression, you will need to generate the future values ofthe Fourier terms. This can be done as follows.fc <- forecast(fit, newdata=fourier(x, K, h))where `fit` is the fitted model using `tslm`, `K` is the number of Fourier termsused in creating `fit`, and `h` is the forecast horizon required.Lets now forecast the next year of data:fc <- forecast(fit, newdata=data.frame(fourier(gas, k, 52)))(d) Plot the forecasts along with the actual data for 2005. What do you find?autoplot(fc) +autolayer(window(gasoline, start=2005, end=2005.99), series=”2005 data”)The forecasts look pretty good for the next year.Question 1Question 2Question 3Question 4 (Optional, depending on time) CS: assignmentchef QQ: 1823890830 Email: [email protected]