PREDICT 413 HW 1

2.10.1

Download some monthly Australian retail data from http://robjhyndman.com/data/retail.xlsx. These represent retail sales in various categories for different Australian states, and are stored in a MS-Excel file.

You can read the data into R with the following script:

Series ID	Description
23367261	Turnover ; New South Wales ; Supermarket and grocery stores ;
23383763	Turnover ; New South Wales ; Liquor retailing ;
23914912	Turnover ; New South Wales ; Other specialised food retailing ;
25180347	Turnover ; New South Wales ; Food retailing ;
25190644	Turnover ; New South Wales ; Furniture, floor coverings, houseware and textile goods retailing ;
25570285	Turnover ; New South Wales ; Electrical and electronic goods retailing ;
25893076	Turnover ; New South Wales ; Hardware, building and garden supplies retailing ;
26021927	Turnover ; New South Wales ; Household goods retailing ;
26096129	Turnover ; New South Wales ; Clothing retailing ;
26435934	Turnover ; New South Wales ; Footwear and other personal accessory retailing ;
26877708	Turnover ; New South Wales ; Clothing, footwear and personal accessory retailing ;
26963817	Turnover ; New South Wales ; Department stores ;
27004783	Turnover ; New South Wales ; Newspaper and book retailing ;
27307395	Turnover ; New South Wales ; Other recreational goods retailing ;
A3349401C	Turnover ; New South Wales ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349873A	Turnover ; New South Wales ; Other retailing n.e.c. ;
A3349872X	Turnover ; New South Wales ; Other retailing ;
A3349709X	Turnover ; New South Wales ; Cafes, restaurants and catering services ;
A3349792X	Turnover ; New South Wales ; Takeaway food services ;
A3349789K	Turnover ; New South Wales ; Cafes, restaurants and takeaway food services ;
A3349555V	Turnover ; New South Wales ; Total (Industry) ;
A3349565X	Turnover ; Victoria ; Supermarket and grocery stores ;
A3349414R	Turnover ; Victoria ; Liquor retailing ;
A3349799R	Turnover ; Victoria ; Other specialised food retailing ;
A3349642T	Turnover ; Victoria ; Food retailing ;
A3349413L	Turnover ; Victoria ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349564W	Turnover ; Victoria ; Electrical and electronic goods retailing ;
A3349416V	Turnover ; Victoria ; Hardware, building and garden supplies retailing ;
A3349643V	Turnover ; Victoria ; Household goods retailing ;
A3349483V	Turnover ; Victoria ; Clothing retailing ;
A3349722T	Turnover ; Victoria ; Footwear and other personal accessory retailing ;
A3349727C	Turnover ; Victoria ; Clothing, footwear and personal accessory retailing ;
A3349641R	Turnover ; Victoria ; Department stores ;
A3349639C	Turnover ; Victoria ; Newspaper and book retailing ;
A3349415T	Turnover ; Victoria ; Other recreational goods retailing ;
A3349349F	Turnover ; Victoria ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349563V	Turnover ; Victoria ; Other retailing n.e.c. ;
A3349350R	Turnover ; Victoria ; Other retailing ;
A3349640L	Turnover ; Victoria ; Cafes, restaurants and catering services ;
A3349566A	Turnover ; Victoria ; Takeaway food services ;
A3349417W	Turnover ; Victoria ; Cafes, restaurants and takeaway food services ;
A3349352V	Turnover ; Victoria ; Total (Industry) ;
A3349882C	Turnover ; Queensland ; Supermarket and grocery stores ;
A3349561R	Turnover ; Queensland ; Liquor retailing ;
A3349883F	Turnover ; Queensland ; Other specialised food retailing ;
A3349721R	Turnover ; Queensland ; Food retailing ;
A3349478A	Turnover ; Queensland ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349637X	Turnover ; Queensland ; Electrical and electronic goods retailing ;
A3349479C	Turnover ; Queensland ; Hardware, building and garden supplies retailing ;
A3349797K	Turnover ; Queensland ; Household goods retailing ;
A3349477X	Turnover ; Queensland ; Clothing retailing ;
A3349719C	Turnover ; Queensland ; Footwear and other personal accessory retailing ;
A3349884J	Turnover ; Queensland ; Clothing, footwear and personal accessory retailing ;
A3349562T	Turnover ; Queensland ; Department stores ;
A3349348C	Turnover ; Queensland ; Newspaper and book retailing ;
A3349480L	Turnover ; Queensland ; Other recreational goods retailing ;
A3349476W	Turnover ; Queensland ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349881A	Turnover ; Queensland ; Other retailing n.e.c. ;
A3349410F	Turnover ; Queensland ; Other retailing ;
A3349481R	Turnover ; Queensland ; Cafes, restaurants and catering services ;
A3349718A	Turnover ; Queensland ; Takeaway food services ;
A3349411J	Turnover ; Queensland ; Cafes, restaurants and takeaway food services ;
A3349638A	Turnover ; Queensland ; Total (Industry) ;
A3349654A	Turnover ; South Australia ; Supermarket and grocery stores ;
A3349499L	Turnover ; South Australia ; Liquor retailing ;
A3349902A	Turnover ; South Australia ; Other specialised food retailing ;
A3349432V	Turnover ; South Australia ; Food retailing ;
A3349656F	Turnover ; South Australia ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349361W	Turnover ; South Australia ; Electrical and electronic goods retailing ;
A3349501L	Turnover ; South Australia ; Hardware, building and garden supplies retailing ;
A3349503T	Turnover ; South Australia ; Household goods retailing ;
A3349360V	Turnover ; South Australia ; Clothing retailing ;
A3349903C	Turnover ; South Australia ; Footwear and other personal accessory retailing ;
A3349905J	Turnover ; South Australia ; Clothing, footwear and personal accessory retailing ;
A3349658K	Turnover ; South Australia ; Department stores ;
A3349575C	Turnover ; South Australia ; Newspaper and book retailing ;
A3349428C	Turnover ; South Australia ; Other recreational goods retailing ;
A3349500K	Turnover ; South Australia ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349577J	Turnover ; South Australia ; Other retailing n.e.c. ;
A3349433W	Turnover ; South Australia ; Other retailing ;
A3349576F	Turnover ; South Australia ; Cafes, restaurants and catering services ;
A3349574A	Turnover ; South Australia ; Takeaway food services ;
A3349816F	Turnover ; South Australia ; Cafes, restaurants and takeaway food services ;
A3349815C	Turnover ; South Australia ; Total (Industry) ;
A3349744F	Turnover ; Western Australia ; Supermarket and grocery stores ;
A3349823C	Turnover ; Western Australia ; Liquor retailing ;
A3349508C	Turnover ; Western Australia ; Other specialised food retailing ;
A3349742A	Turnover ; Western Australia ; Food retailing ;
A3349661X	Turnover ; Western Australia ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349660W	Turnover ; Western Australia ; Electrical and electronic goods retailing ;
A3349909T	Turnover ; Western Australia ; Hardware, building and garden supplies retailing ;
A3349824F	Turnover ; Western Australia ; Household goods retailing ;
A3349507A	Turnover ; Western Australia ; Clothing retailing ;
A3349580W	Turnover ; Western Australia ; Footwear and other personal accessory retailing ;
A3349825J	Turnover ; Western Australia ; Clothing, footwear and personal accessory retailing ;
A3349434X	Turnover ; Western Australia ; Department stores ;
A3349822A	Turnover ; Western Australia ; Newspaper and book retailing ;
A3349821X	Turnover ; Western Australia ; Other recreational goods retailing ;
A3349581X	Turnover ; Western Australia ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349908R	Turnover ; Western Australia ; Other retailing n.e.c. ;
A3349743C	Turnover ; Western Australia ; Other retailing ;
A3349910A	Turnover ; Western Australia ; Cafes, restaurants and catering services ;
A3349435A	Turnover ; Western Australia ; Takeaway food services ;
A3349365F	Turnover ; Western Australia ; Cafes, restaurants and takeaway food services ;
A3349746K	Turnover ; Western Australia ; Total (Industry) ;
A3349370X	Turnover ; Tasmania ; Supermarket and grocery stores ;
A3349754K	Turnover ; Tasmania ; Liquor retailing ;
A3349670A	Turnover ; Tasmania ; Other specialised food retailing ;
A3349764R	Turnover ; Tasmania ; Food retailing ;
A3349916R	Turnover ; Tasmania ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349589T	Turnover ; Tasmania ; Electrical and electronic goods retailing ;
A3349590A	Turnover ; Tasmania ; Hardware, building and garden supplies retailing ;
A3349765T	Turnover ; Tasmania ; Household goods retailing ;
A3349371A	Turnover ; Tasmania ; Clothing retailing ;
A3349588R	Turnover ; Tasmania ; Footwear and other personal accessory retailing ;
A3349763L	Turnover ; Tasmania ; Clothing, footwear and personal accessory retailing ;
A3349372C	Turnover ; Tasmania ; Department stores ;
A3349442X	Turnover ; Tasmania ; Newspaper and book retailing ;
A3349591C	Turnover ; Tasmania ; Other recreational goods retailing ;
A3349671C	Turnover ; Tasmania ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349669T	Turnover ; Tasmania ; Other retailing n.e.c. ;
A3349521W	Turnover ; Tasmania ; Other retailing ;
A3349443A	Turnover ; Tasmania ; Cafes, restaurants and catering services ;
A3349835L	Turnover ; Tasmania ; Takeaway food services ;
A3349520V	Turnover ; Tasmania ; Cafes, restaurants and takeaway food services ;
A3349841J	Turnover ; Tasmania ; Total (Industry) ;
A3349925T	Turnover ; Northern Territory ; Supermarket and grocery stores ;
A3349450X	Turnover ; Northern Territory ; Liquor retailing ;
A3349679W	Turnover ; Northern Territory ; Other specialised food retailing ;
A3349527K	Turnover ; Northern Territory ; Food retailing ;
A3349526J	Turnover ; Northern Territory ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349598V	Turnover ; Northern Territory ; Electrical and electronic goods retailing ;
A3349766V	Turnover ; Northern Territory ; Hardware, building and garden supplies retailing ;
A3349600V	Turnover ; Northern Territory ; Household goods retailing ;
A3349680F	Turnover ; Northern Territory ; Clothing retailing ;
A3349378T	Turnover ; Northern Territory ; Footwear and other personal accessory retailing ;
A3349767W	Turnover ; Northern Territory ; Clothing, footwear and personal accessory retailing ;
A3349451A	Turnover ; Northern Territory ; Department stores ;
A3349924R	Turnover ; Northern Territory ; Newspaper and book retailing ;
A3349843L	Turnover ; Northern Territory ; Other recreational goods retailing ;
A3349844R	Turnover ; Northern Territory ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349376L	Turnover ; Northern Territory ; Other retailing n.e.c. ;
A3349599W	Turnover ; Northern Territory ; Other retailing ;
A3349377R	Turnover ; Northern Territory ; Cafes, restaurants and catering services ;
A3349779F	Turnover ; Northern Territory ; Takeaway food services ;
A3349379V	Turnover ; Northern Territory ; Cafes, restaurants and takeaway food services ;
A3349842K	Turnover ; Northern Territory ; Total (Industry) ;
A3349532C	Turnover ; Australian Capital Territory ; Supermarket and grocery stores ;
A3349931L	Turnover ; Australian Capital Territory ; Liquor retailing ;
A3349605F	Turnover ; Australian Capital Territory ; Other specialised food retailing ;
A3349688X	Turnover ; Australian Capital Territory ; Food retailing ;
A3349456L	Turnover ; Australian Capital Territory ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349774V	Turnover ; Australian Capital Territory ; Electrical and electronic goods retailing ;
A3349848X	Turnover ; Australian Capital Territory ; Hardware, building and garden supplies retailing ;
A3349457R	Turnover ; Australian Capital Territory ; Household goods retailing ;
A3349851L	Turnover ; Australian Capital Territory ; Clothing retailing ;
A3349604C	Turnover ; Australian Capital Territory ; Footwear and other personal accessory retailing ;
A3349608L	Turnover ; Australian Capital Territory ; Clothing, footwear and personal accessory retailing ;
A3349609R	Turnover ; Australian Capital Territory ; Department stores ;
A3349773T	Turnover ; Australian Capital Territory ; Newspaper and book retailing ;
A3349852R	Turnover ; Australian Capital Territory ; Other recreational goods retailing ;
A3349775W	Turnover ; Australian Capital Territory ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349776X	Turnover ; Australian Capital Territory ; Other retailing n.e.c. ;
A3349607K	Turnover ; Australian Capital Territory ; Other retailing ;
A3349849A	Turnover ; Australian Capital Territory ; Cafes, restaurants and catering services ;
A3349850K	Turnover ; Australian Capital Territory ; Takeaway food services ;
A3349606J	Turnover ; Australian Capital Territory ; Cafes, restaurants and takeaway food services ;
A3349932R	Turnover ; Australian Capital Territory ; Total (Industry) ;
A3349862V	Turnover ; Total (State) ; Supermarket and grocery stores ;
A3349462J	Turnover ; Total (State) ; Liquor retailing ;
A3349463K	Turnover ; Total (State) ; Other specialised food retailing ;
A3349334R	Turnover ; Total (State) ; Food retailing ;
A3349863W	Turnover ; Total (State) ; Furniture, floor coverings, houseware and textile goods retailing ;
A3349781T	Turnover ; Total (State) ; Electrical and electronic goods retailing ;
A3349861T	Turnover ; Total (State) ; Hardware, building and garden supplies retailing ;
A3349626T	Turnover ; Total (State) ; Household goods retailing ;
A3349617R	Turnover ; Total (State) ; Clothing retailing ;
A3349546T	Turnover ; Total (State) ; Footwear and other personal accessory retailing ;
A3349787F	Turnover ; Total (State) ; Clothing, footwear and personal accessory retailing ;
A3349333L	Turnover ; Total (State) ; Department stores ;
A3349860R	Turnover ; Total (State) ; Newspaper and book retailing ;
A3349464L	Turnover ; Total (State) ; Other recreational goods retailing ;
A3349389X	Turnover ; Total (State) ; Pharmaceutical, cosmetic and toiletry goods retailing ;
A3349461F	Turnover ; Total (State) ; Other retailing n.e.c. ;
A3349788J	Turnover ; Total (State) ; Other retailing ;
A3349547V	Turnover ; Total (State) ; Cafes, restaurants and catering services ;
A3349388W	Turnover ; Total (State) ; Takeaway food services ;
A3349870V	Turnover ; Total (State) ; Cafes, restaurants and takeaway food services ;
A3349396W	Turnover ; Total (State) ; Total (Industry) ;

Select one of the time series as follows (but replace the column name with your own chosen column):

myts

       Jan   Feb   Mar   Apr   May   Jun   Jul   Aug   Sep   Oct   Nov   Dec
1982                    17.3  18.1  18.1  18.9  19.0  18.4  20.9  22.4  29.7
1983  22.9  20.8  23.5  21.7  21.4  20.8  21.3  22.6  22.4  22.9  24.0  33.5
1984  22.2  20.8  21.9  20.4  20.6  20.4  19.3  20.4  20.5  21.4  23.5  29.6
1985  23.8  21.6  23.2  22.3  22.4  20.8  20.7  21.8  21.9  23.4  24.9  32.4
1986  25.8  24.7  26.9  23.5  24.2  22.4  22.5  23.9  23.6  24.7  26.9  34.2
1987  28.1  25.9  26.4  25.5  25.4  24.0  26.4  26.4  27.7  28.6  28.5  38.7
1988  31.0  27.9  29.1  28.0  26.2  25.8  28.5  28.2  31.0  29.7  34.5  51.3
1989  22.4  23.4  25.1  22.4  22.5  22.4  22.5  23.5  23.1  22.8  27.1  44.5
1990  22.7  23.9  26.4  25.8  25.1  24.0  23.6  25.3  24.5  26.9  29.3  45.7
1991  24.3  22.3  25.1  22.5  25.0  22.8  23.4  26.4  24.2  26.7  28.9  42.6
1992  27.3  26.4  29.3  29.0  27.0  26.1  28.5  27.0  28.7  30.8  29.7  51.8
1993  28.3  26.5  28.5  28.1  26.6  23.1  25.1  25.2  26.5  27.8  28.1  47.8
1994  24.8  25.5  30.9  27.1  27.5  27.5  25.7  25.0  26.9  28.4  28.8  46.8
1995  26.4  25.2  28.7  26.5  27.9  28.8  27.7  28.8  30.1  28.3  32.8  52.5
1996  27.6  28.5  30.9  30.2  30.5  30.2  33.5  35.4  36.5  39.7  41.6  64.3
1997  39.4  37.2  41.4  39.4  42.1  39.8  42.1  44.5  43.5  46.8  51.5  82.8
1998  46.0  43.9  47.1  46.5  49.7  44.5  52.7  52.6  53.8  51.6  60.0  93.8
1999  56.7  54.1  59.3  54.2  56.3  53.8  56.7  56.3  60.6  65.7  69.1 110.0
2000  54.8  56.7  60.6  56.5  54.5  62.8  57.6  62.1  60.6  67.3  75.7 114.9
2001  65.3  62.0  70.1  67.0  69.6  65.9  62.5  62.8  64.1  71.1  77.0 119.1
2002  68.7  60.9  70.4  60.1  63.6  58.6  66.6  70.3  69.9  74.0  80.6 120.1
2003  75.5  68.8  72.7  75.2  74.1  73.0  77.4  76.3  78.2  80.8  87.8 135.6
2004  85.7  81.5  87.3  83.5  81.2  78.5  79.2  79.3  80.3  90.4  91.3 144.9
2005  76.4  75.7  87.5  86.9  82.8  82.6  92.1  94.9  94.7  85.9  95.3 139.7
2006  91.0  84.5  96.8  95.4  89.9  92.2  92.2  96.9  98.5  94.3  98.6 159.9
2007  94.0  93.0 102.7  90.5  88.6  87.5  89.7  90.3  97.1 103.5 109.1 154.3
2008 104.4  88.6  98.0  91.2  92.1  91.0  92.2  99.9 102.8 121.3 135.9 204.4
2009 134.9 123.0 128.1 116.6 122.6 115.1 122.1 121.6 129.1 143.0 145.6 198.8
2010 134.8 117.1 138.3 125.6 124.5 121.0 132.4 135.3 143.5 161.2 167.8 232.1
2011 160.7 135.0 149.5 151.1 136.8 129.6 148.0 149.1 155.3 161.3 172.6 260.8
2012 176.6 159.2 171.1 159.2 153.8 148.8 144.6 149.6 149.6 157.0 179.3 268.7
2013 168.3 152.4 180.2 145.7 145.3 139.2 146.6 150.5 149.1 163.2 176.5 273.8

Explore your chosen retail time series using the following functions:

autoplot, ggseasonplot, ggsubseriesplot, gglagplot, ggAcf

Can you spot any seasonality, cyclicity and trend? What do you learn about the series?

ap <- autoplot(myts)+
  theme_yaz()
seas <- ggseasonplot(myts)+
  theme_yaz()
subs <- ggsubseriesplot(myts)+
  theme_yaz()
lag <- gglagplot(myts)+
  theme_yaz()
acf <- ggAcf(myts)+
  theme_yaz()
pacf <- ggPacf(myts)+
  theme_yaz()
library(gridExtra)
grid.arrange(ap, subs, seas, lag, acf, pacf, nrow = 3)

Liquor sales exhibit annual seasonality where sales spike during December (around the holidays) and are otherwise flat. The lag plot, partial autocorrelation plot, and subseries plot each show strong correlations with 12 month seasons. That said, there is also a general upward trend over time since 1992.

2.10.2

Repeat for the following series:

bicoal, chicken, dole, usdeaths, bricksq, lynx, ibmclose

Use the help files to find out what the series are.

bicoal

The bicoal data is not seasonal (to the point where ggseasonalplot threw an error). There may be some cyclicality around a four year lag, but it’s weak if at all existent.

library(fma)
a <- autoplot(bicoal)+ theme_yaz()
# b <- ggseasonplot(bicoal)+theme_yaz()
c <- ggsubseriesplot(bicoal)+theme_yaz()
d <- ggPacf(bicoal)+theme_yaz()
grid.arrange(a,c,d, nrow = 2)

chicken

The chicken data is not seasonal (again, to the point where ggseasonalplot threw an error). The overall trend is negative between 1950 and 2000.

a <- autoplot(chicken)+ theme_yaz()
# b <- ggseasonplot(chicken)+theme_yaz()
c <- ggsubseriesplot(chicken)+theme_yaz()
d <- ggPacf(chicken)+theme_yaz()
grid.arrange(a,c,d, nrow = 2)

dole

The dole data has a general positive trend indicating either an increase in population or an increase in the generosity of Austrailia’s welfare state (or both). There appears to have been a shift in seasonality in recent years. Before the spike in enrollment, there were no seasonal trends - welfare receipt was similar year-round. In recent years, seasonality has shifted towards earlier in the year. This may reflect the seasonality of job availability in retail.

a <- autoplot(dole)+ theme_yaz()
b <- ggseasonplot(dole)+theme_yaz()+theme(legend.position = 'none')
c <- ggsubseriesplot(dole)+theme_yaz()
d <- ggPacf(dole)+theme_yaz()
grid.arrange(a,b,c,d, nrow = 2)

usdeaths

Data on accidental deaths in the US in the 1970s indicates a seasonal trend in the summers with a peak in July of every year. On a brighter note, the sheer number of deaths appears to have a slightly negative trend, which should be viewed positively in light of the fact that the US population is continuously growing.

a <- autoplot(usdeaths)+ theme_yaz()
b <- ggseasonplot(usdeaths)+theme_yaz()
c <- ggsubseriesplot(usdeaths)+theme_yaz()
d <- ggPacf(usdeaths)+theme_yaz()
grid.arrange(a,b,c,d, nrow = 2)

bricksq

The bricksq data has a seasonal trend with peaks in Q2 and Q3. This doesn’t make a ton of sense since those quarters are winter in Austrailia and I would assume brick production wouldl heavily correlate with building. Perhaps building season is in winter in Austrailia due to teh summer heat?

a <- autoplot(bricksq)+ theme_yaz()
b <- ggseasonplot(bricksq)+theme_yaz()+theme(legend.position = 'none')
c <- ggsubseriesplot(bricksq)+theme_yaz()
d <- ggPacf(bricksq)+theme_yaz()
grid.arrange(a,b,c,d, nrow = 2)

lynx

Lynx population growth may be cyclical but it’s not seasonal. The trend in population size is also relatively flat.

a <- autoplot(lynx)+ theme_yaz()
# b <- ggseasonplot(lynx)+theme_yaz()
c <- ggsubseriesplot(lynx)+theme_yaz()
d <- ggPacf(lynx)+theme_yaz()
grid.arrange(a,c,d, nrow = 2)

lynx ibmclose

Lastly, IBM closing prices are trending downward in this time series with no clear seasonal trend. There may be some cyclicality, but that is unclear based on this data.

a <- autoplot(ibmclose)+ theme_yaz()
# b <- ggseasonplot(ibmclose)+theme_yaz()
c <- ggsubseriesplot(ibmclose)+theme_yaz()
d <- ggPacf(ibmclose)+theme_yaz()
grid.arrange(a,c,d, nrow = 2)

2.10.3

The arrivals data set comprises quarterly international arrivals (in thousands) to Australia from Japan, New Zealand, UK and the US. Use autoplot and ggseasonplot and compare the differences between the arrivals from these four countries. Can you identify any unusual observations?

I couldn’t find the arrivals data set so I used the woolyrnq data set instead. Generally woollen yarn peaks in either Q2 or Q3, but that is not really consistent from year to year. It looks like Q3 is the peak for about half of the years and Q2 is the peak for another half.

grid.arrange(
  autoplot(woolyrnq)+theme_yaz(),
  ggseasonplot(woolyrnq)+theme_yaz(),
  nrow = 1
)

2.10.4

1 = B 2 = C 3 = D 4 = A

Chapter 5 Questions

5.10.1

Electricity consumption was recorded for a small town on 12 consecutive days. The following maximum temperatures (degrees Celsius) and consumption (megawatt-hours) were recorded for each day. TODO: change the econsumption to a ts of 12 concecutive days - change the lm to tslm below

• Plot the data and find the regression model for Mwh with temperature as an explanatory variable. Why is there a negative relationship?

library(GGally)
ggpairs(econsumption)+
  theme_yaz()

• Produce a residual plot. Is the model adequate? Are there any outliers or influential observations? The residuals violate the assumption that they are normally distributed with mean zero. There is one outlier with a residual over 2.

lm.fit <- tslm(Mwh ~ temp, data = ts(econsumption))
library(gridExtra)
grid.arrange(
  data.frame(Mwh = econsumption$Mwh, error = residuals(lm.fit))%>%
    ggplot(aes(Mwh, error))+
      geom_point(color = yaz_cols[1])+
      theme_yaz()+
      geom_hline(yintercept =  0, linetype = 'dashed'),
  data.frame(error = residuals(lm.fit))%>%
    ggplot(aes(x = ' ', y = error))+
      geom_boxplot(fill = yaz_cols[1])+
      labs(y = 'error', x = element_blank())+
      theme_yaz()
  , nrow = 1, widths = c(3,1)
  )

• Use the model to predict the electricity consumption that you would expect for the next day if the maximum temperature was 10∘ and compare it with the forecast if the with maximum temperature was 35∘. Do you believe these predictions? It makes sense that the 35 degree prediction would be lower than the 10 degree prediction has energy consumption tends to be lower in cooler temperatures. That said, the prediction intervals are very wide!

forecast(lm.fit, newdata=data.frame(temp=c(10,35)))

   Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
13       18.74795 17.27010 20.22579 16.34824 21.14766
14       15.11902 13.50469 16.73335 12.49768 17.74035

autoplot(ts(econsumption$Mwh)) +
  forecast::autolayer(forecast(lm.fit, newdata=data.frame(temp=c(10))), PI=TRUE, series="10 Degrees") +
  forecast::autolayer(forecast(lm.fit, newdata=data.frame(temp=c(35))), PI=TRUE, series="35 Degrees")+
  labs(title = 'Consumption Forecast', y = 'Electricity Consumption',
       x = element_blank())+
  theme_yaz()+
  scale_color_manual(name = 'Possible Temperature', values = yaz_cols[c(3,4)])

• Give prediction intervals for your forecasts.

5.10.2

Data set olympic contains the winning times (in seconds) for the men’s 400 meters final in each Olympic Games from 1896 to 2012.

olympic%>%head()

• Plot the winning time against the year. Describe the main features of the scatterplot. There is a downward trend in 400M times over the years. Runners are getting faster!

ggplot(olympic, aes(Year, time))+
  geom_point()+
  theme_yaz()+
  labs(y = 'Seconds',
       title = 'Men\'s 400M Time (seconds): Olympic Games from 1896 to 2012')

• Fit a regression line to the data. Obviously the winning times have been decreasing, but at what average rate per year? The average rate of change for 400M times is -.07 seconds per year.

olympic.fit <- tslm(time~Year, data = ts(olympic))
summary(olympic.fit)

Call:
tslm(formula = time ~ Year, data = ts(olympic))

Residuals:
    Min      1Q  Median      3Q     Max 
-1.5215 -0.7037 -0.1642  0.4952  3.7141 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) 196.079876  14.177031   13.83 5.09e-12 ***
Year         -0.076790   0.007278  -10.55 7.51e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.089 on 21 degrees of freedom
Multiple R-squared:  0.8413,    Adjusted R-squared:  0.8337 
F-statistic: 111.3 on 1 and 21 DF,  p-value: 7.515e-10

• Plot the residuals against the year. What does this indicate about the suitability of the fitted line? For the most part, the regression line fits the data appropriately, but the first year is an influential point.

grid.arrange(
  ggplot(data.frame(year = olympic$Year, error = olympic.fit$residuals),
         aes(year, error))+
        geom_point(color = yaz_cols[1])+
        theme_yaz()+
        geom_hline(yintercept =  0, linetype = 'dashed'),
    data.frame(error = residuals(olympic.fit))%>%
      ggplot(aes(x = ' ', y = error))+
        geom_boxplot(fill = yaz_cols[1])+
        labs(y = 'error', x = element_blank())+
        theme_yaz()
    , nrow = 1, widths = c(3,1)
)

• Predict the winning time for the men’s 400 meters final in the 2000, 2004, 2008 and 2012 Olympics. Give a prediction interval for each of your forecasts. What assumptions have you made in these calculations? The assumptions in this model are the same as the assumptions for OLS regression in a non-time series context.

forecast(olympic.fit, newdata=data.frame(Year = years))

   Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
24       42.49977 40.94361 44.05593 40.05401 44.94554
25       42.19261 40.62355 43.76167 39.72657 44.65866
26       41.88545 40.30266 43.46825 39.39782 44.37308
27       41.57829 39.98095 43.17563 39.06780 44.08879

• Find out the actual winning times for these Olympics (see www.databaseolympics.com). How good were your forecasts and prediction intervals? The predictions were OK, although I think the influential point from 1896 led to the model being a bit overly optimistic. All four actual times fell within the 80% confidence interval, but all were off in the same direction.

new_years <- data.frame(Year= c(24, 25, 26, 27),
                        time = c(44, 43.75 ,43.94 , 43.03))
autoplot(ts(olympic$time, frequency = 1))+
  geom_point()+
  ylim(0,55)+
  autolayer(forecast(olympic.fit, newdata= data.frame(Year = years)), PI=TRUE)+
  labs(y = 'Olympic Times', x = 'Olympic Games Since 1896',
       title = 'Prediction Quality Demonstration')+
  geom_point(data = new_years, aes(Year, time))+
  theme_yaz()

5.10.3

Type easter(ausbeer) and interpret what you see. Easter always occurs in the first or second quarter of the year. The date is close enough to the border of the quarter that occasionally the holiday falls partially in Q1 and partially in Q2 (as happened in 1956).

library(fpp)
easter(ausbeer)%>%
  head(20)

     Qtr1 Qtr2 Qtr3 Qtr4
1956 0.67 0.33 0.00 0.00
1957 0.00 1.00 0.00 0.00
1958 0.00 1.00 0.00 0.00
1959 1.00 0.00 0.00 0.00
1960 0.00 1.00 0.00 0.00

Chapter 6 Questions

6.8.1

Show that a 3×5 MA is equivalent to a 7-term weighted moving average with weights of 0.067, 0.133, 0.200, 0.200, 0.200, 0.133, and 0.067.

6.8.2

The plastics data set consists of the monthly sales (in thousands) of product A for a plastics manufacturer for five years. * Plot the time series of sales of product A. Can you identify seasonal fluctuations and/or a trend-cycle? There is a general upward trend over the course of the years covered by this data. Additionally, there does appear to be a seasonal trend where most plastics are sold around the fall and early winter, which may coincide with holiday gift purchasing.

• Use a classical multiplicative decomposition to calculate the trend-cycle and seasonal indices.

• Do the results support the graphical interpretation from part (a)? Yes!
• Compute and plot the seasonally adjusted data.

• Change one observation to be an outlier (e.g., add 500 to one observation), and recompute the seasonally adjusted data. What is the effect of the outlier? I changed the fifth value to an outlier by adding 1000 to it. The seasonally adjusted data is inflated for eariler values but that effect is eliminated further along the time series.

• Does it make any difference if the outlier is near the end rather than in the middle of the time series? After going back and adding another outlier in the middle of the dataset, the inflation pattern (the bump in the trend line) appears to behave pretty similarly.

6.8.3

Recall your retail time series data (from Exercise 1 in Section ??). Decompose the series using X11. Does it reveal any outliers, or unusual features that you had not noticed previously? The x11 decomposition enveils a few outliers in the 1990s and mid 2000s that were not easy to see in the initial plotting of the data! Additionally, we can see that seasonal effects start to increase in teh 1990s leading to a more general updard trend going forward.