Exercises

Forecasting: Principles and Practice (3rd ed)
Chapter 3 Time series Decomposition

3.1

Consider the GDP information in global_economy. Plot the GDP per capita for each country over time. Which country has the highest GDP per capita? How has this changed over time?

top_countries_df = global_economy |>
  mutate(GDP_per_capita = GDP/Population) |>
  arrange(desc(GDP_per_capita))

# identify the top 10 countries with the highest GDP per capita to display in
# the legend of the plot
top_countries = top_countries_df |>
  distinct(Country) |>
  head(10) |>
  pull(Country)

# plot he GDP per capita for each country over time
global_economy |>
  autoplot(GDP/Population) +
  labs(title= "GDP per capita", y = "$US") +
  scale_colour_manual(values = unique(global_economy$Country), breaks = c(top_countries))

# plot the country with the highest GDP per capita over time
top_countries_year_df = top_countries_df |>
  index_by(Year) |>
  filter(GDP_per_capita == max(GDP_per_capita, na.rm = TRUE)) |>
  arrange(desc(Year))

ggplot(top_countries_year_df, aes(x = Year, y = GDP_per_capita, fill = Country)) +
  geom_col() +
  labs(title= "Max GDP per capita per Year", y = "$US")

knit_table(top_countries_df |> head(10), 'Top GDP per Capita Countries & Years')
Top GDP per Capita Countries & Years
Country Code Year GDP Growth CPI Imports Exports Population GDP_per_capita
Monaco MCO 2014 7060236168 7.1796368 NA NA NA 38132 185152.5
Monaco MCO 2008 6476490406 0.7318007 NA NA NA 35853 180640.1
Liechtenstein LIE 2014 6657170923 NA NA NA NA 37127 179308.1
Liechtenstein LIE 2013 6391735894 NA NA NA NA 36834 173528.2
Monaco MCO 2013 6553372278 9.5707988 NA NA NA 37971 172588.9
Monaco MCO 2016 6468252212 3.2138488 NA NA NA 38499 168010.9
Liechtenstein LIE 2015 6268391521 NA NA NA NA 37403 167590.6
Monaco MCO 2007 5867916781 14.4288990 NA NA NA 35111 167124.7
Liechtenstein LIE 2016 6214633651 NA NA NA NA 37666 164993.2
Monaco MCO 2015 6258178995 4.9423298 NA NA NA 38307 163369.1 
knit_table(top_countries_year_df, 'Max GDP per Capita per Year')
Max GDP per Capita per Year
Country Code Year GDP Growth CPI Imports Exports Population GDP_per_capita
Luxembourg LUX 2017 6.240446e+10 2.2979413 111.41323 193.969823 230.016430 599449 104103.037
Monaco MCO 2016 6.468252e+09 3.2138488 NA NA NA 38499 168010.915
Liechtenstein LIE 2015 6.268392e+09 NA NA NA NA 37403 167590.608
Monaco MCO 2014 7.060236e+09 7.1796368 NA NA NA 38132 185152.527
Liechtenstein LIE 2013 6.391736e+09 NA NA NA NA 36834 173528.150
Monaco MCO 2012 5.743030e+09 0.9849603 NA NA NA 37783 152000.362
Monaco MCO 2011 6.080345e+09 7.0737008 NA NA NA 37497 162155.499
Monaco MCO 2010 5.362649e+09 2.0542939 NA NA NA 37094 144569.176
Monaco MCO 2009 5.451653e+09 -11.3175072 NA NA NA 36534 149221.362
Monaco MCO 2008 6.476490e+09 0.7318007 NA NA NA 35853 180640.125
Monaco MCO 2007 5.867917e+09 14.4288990 NA NA NA 35111 167124.741
Monaco MCO 2006 4.582988e+09 5.8039365 NA NA NA 34408 133195.429
Monaco MCO 2005 4.202980e+09 1.8954147 NA NA NA 33793 124374.268
Monaco MCO 2004 4.110348e+09 2.4704857 NA NA NA 33314 123382.016
Monaco MCO 2003 3.588989e+09 1.0875362 NA NA NA 32933 108978.489
Monaco MCO 2002 2.905973e+09 1.0264955 NA NA NA 32629 89061.051
Monaco MCO 2001 2.671401e+09 2.1877308 NA NA NA 32360 82552.567
Monaco MCO 2000 2.647884e+09 3.9102306 NA NA NA 32082 82534.874
Monaco MCO 1999 2.906009e+09 3.3005427 NA NA NA 31800 91383.940
Monaco MCO 1998 2.934579e+09 3.5032655 NA NA NA 31523 93093.260
Monaco MCO 1997 2.840182e+09 2.2372502 NA NA NA 31251 90882.923
Monaco MCO 1996 3.137849e+09 1.1107282 NA NA NA 30967 101328.795
Monaco MCO 1995 3.130271e+09 2.1171526 NA NA NA 30691 101993.122
Monaco MCO 1994 2.720298e+09 2.2156235 NA NA NA 30427 89404.075
Monaco MCO 1993 2.574440e+09 -0.9139168 NA NA NA 30138 85421.727
Monaco MCO 1992 2.737067e+09 1.3666397 NA NA NA 29863 91654.121
Monaco MCO 1991 2.480498e+09 1.0151285 NA NA NA 29624 83732.701
Monaco MCO 1990 2.481316e+09 2.6434542 NA NA NA 29439 84286.696
Monaco MCO 1989 2.010117e+09 4.1636593 NA NA NA 29312 68576.585
Monaco MCO 1988 2.000675e+09 4.5986015 NA NA NA 29235 68434.228
Monaco MCO 1987 1.839096e+09 2.4845742 NA NA NA 29172 63043.178
Monaco MCO 1986 1.515210e+09 2.4523647 NA NA NA 29041 52174.842
Monaco MCO 1985 1.082851e+09 1.7086997 NA NA NA 28835 37553.358
Monaco MCO 1984 1.037315e+09 1.4845156 NA NA NA 28512 36381.697
Monaco MCO 1983 1.092552e+09 1.1949733 NA NA NA 28095 38887.766
Monaco MCO 1982 1.143229e+09 2.4323895 NA NA NA 27624 41385.356
Monaco MCO 1981 1.205166e+09 0.9223784 NA NA NA 27164 44366.295
Monaco MCO 1980 1.378131e+09 1.6854574 NA NA NA 26745 51528.547
Monaco MCO 1979 1.209898e+09 3.5342049 NA NA NA 26395 45838.162
Monaco MCO 1978 1.000536e+09 3.9535513 NA NA NA 26087 38353.806
United Arab Emirates ARE 1977 2.487178e+10 21.4393302 NA NA NA 748117 33245.836
United Arab Emirates ARE 1976 1.921302e+10 16.5268565 NA NA NA 646943 29698.169
Monaco MCO 1975 7.119230e+08 -0.9726375 NA NA NA 25197 28254.276
Monaco MCO 1974 5.639397e+08 4.4746111 NA NA NA 24835 22707.456
Monaco MCO 1973 5.235528e+08 6.5537575 NA NA NA 24439 21422.841
Monaco MCO 1972 4.024603e+08 4.6483096 NA NA NA 24051 16733.622
Monaco MCO 1971 3.276515e+08 5.2280600 NA NA NA 23720 13813.301
Monaco MCO 1970 2.930739e+08 NA NA NA NA 23484 12479.725
United States USA 1969 1.019900e+12 3.1000000 16.82293 4.951466 5.088734 202677000 5032.145
United States USA 1968 9.425000e+11 4.8000000 15.95160 4.944297 5.082228 200706000 4695.923
United States USA 1967 8.617000e+11 2.5000000 15.29809 4.630382 5.048161 198712000 4336.427
Kuwait KWT 1966 2.391487e+09 12.3155611 NA 24.355972 65.690866 524856 4556.463
Kuwait KWT 1965 2.097452e+09 NA NA 23.097463 67.690254 473554 4429.171
United States USA 1964 6.858000e+11 5.8000000 14.22421 4.097404 5.103529 191889000 3573.941
United States USA 1963 6.386000e+11 4.4000000 14.04459 4.087065 4.870028 189242000 3374.515
United States USA 1962 6.051000e+11 6.1000000 13.87261 4.131549 4.809122 186538000 3243.843
United States USA 1961 5.633000e+11 2.3000000 13.70828 4.029824 4.899698 183691000 3066.563
United States USA 1960 5.433000e+11 NA 13.56306 4.196576 4.969630 180671000 3007.123

Monaco had the highest GDP per capita in 2014 at $185,152.5. In the 1960s, the United States had the highest GDP per capita other than 1965 and 1966 in which it was Kuwait. From 1970 to 2017, Monaco maintained the highest GDP per capita per year for almost every year. The United Arab Emirates surpassed Monaco in 1976 and 1977 and Liechtenstein took the leader status in more recent years including 2013, 2015, and 2017.

3.2

For each of the following series, make a graph of the data. If transforming seems appropriate, do so and describe the effect.

United States GDP from global_economy

global_economy |>
  filter(Country == 'United States') |>
  autoplot(GDP/Population) +
  labs(title= "United States GDP per Capita", y = "$US")

Transformed to GDP per capita to account for population changes.

Slaughter of Victorian “Bulls, bullocks and steers” in aus_livestock

aus_livestock |>
  filter(
    Animal == 'Bulls, bullocks and steers',
    State == 'Victoria'
    ) |>
  mutate(Year = year(Month)) |>
  group_by(Year) |>
  mutate(Avg_Count = mean(Count)) |>
  distinct(Year, Avg_Count) |>
  as_tsibble(index=Year) |>
  autoplot(Avg_Count)  +
  labs(title= "Average Slaughter of Victorian “Bulls, bullocks and steers” per year", y = "Average Count")

Applied a calendar adjustment to account for variation between the months due to the different numbers of days in each month by taking the average number of animals slaughtered per year.

Victorian Electricity Demand from vic_elec

vic_elec |>
  mutate(Date = date(Time)) |>
  group_by(Date) |>
  mutate(Avg_Demand = mean(Demand)) |>
  distinct(Date, Avg_Demand) |>
  as_tsibble(index=Date) |>
  autoplot(Avg_Demand)  +
  labs(title= "Average Victorian Electricity Demand per day", y = "Average Demand (MWh)")

Applied a calendar adjustment to account for variation between the half-hourly data due to large variation in demand throughout a typical day by taking the average demand per day.

vic_elec |>
  mutate(Month = yearmonth(Time)) |>
  group_by(Month) |>
  mutate(Avg_Demand = mean(Demand)) |>
  distinct(Month, Avg_Demand) |>
  as_tsibble(index=Month) |>
  autoplot(Avg_Demand)  +
  labs(title= "Average Victorian Electricity Demand per Month", y = "Average Demand (MWh)")

Due to there still being large variation at the daily level, another calendar adjustment was applied by taking the average demand per month to further simplify the seasonal patterns.

Gas production from aus_production

lambda <-aus_production |>
  features(Gas, features = guerrero) |>
  pull(lambda_guerrero)

aus_production |>
  autoplot(box_cox(Gas, lambda)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed gas production with $\\lambda$ = ",
         round(lambda,2))))

Due to the data showing variation that increases/decreases with the level of the series, the data was transformed using \(\lambda\) = 0.11 to standardize the seasonal variation or make it about the same across the whole series.

3.3

Why is a Box-Cox transformation unhelpful for the canadian_gas data?

canadian_gas |>
  autoplot(Volume) +
  labs(title='Canadian gas production')

lambda <-canadian_gas |>
  features(Volume, features = guerrero) |>
  pull(lambda_guerrero)

canadian_gas |>
  autoplot(box_cox(Volume, lambda)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed Canadian gas production with $\\lambda$ = ",
         round(lambda,2))))

A Box-Cox transformation is helpful if the data shows variation that increases or decreases with the level of the series. The canadian_gas data does not have this pattern and portrays inconsistent variation with seasonal variability being larger between 1975-1990 when compared to either end of the series. Thus, the Box-Cox transformation is unhelpful for the canadian_gas data. This is further supported by seeing that when applying the Box-Cox transformation, the seasonal variation is not consistent across the series.

3.4

What Box-Cox transformation would you select for your retail data (from Exercise 7 in Section 2.10)?

set.seed(87654321)
myseries <- aus_retail |>
  filter(`Series ID` == sample(aus_retail$`Series ID`,1))

myseries |>
  autoplot(Turnover) +
  labs(y=' $Million AUD', title='Australian retail trade turnover')

lambda <- myseries |>
  features(Turnover, features = guerrero) |>
  pull(lambda_guerrero)

myseries |>
  autoplot(box_cox(Turnover, lambda)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed Australian retail trade turnover with $\\lambda$ = ",
         round(lambda,2))))

myseries |>
  autoplot(box_cox(Turnover, 0)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed Australian retail trade turnover with $\\lambda$ = ",
         0)))

The guerrero feature ouput an optimal lambda value of -0.08. Due to this being negative and producing very similar results as a lamda value or 0 or applying a log transformation, I would select. \(\lambda\) = 0.

3.5

For the following series, find an appropriate Box-Cox transformation in order to stabilize the variance. Tobacco from aus_production, Economy class passengers between Melbourne and Sydney from ansett, and Pedestrian counts at Southern Cross Station from pedestrian.

Tobacco from aus_production

aus_production |>
  autoplot(Tobacco) +
  labs(title='Australia Tobacco production')
## Warning: Removed 24 rows containing missing values or values outside the scale range
## (`geom_line()`).

lambda <- aus_production |>
  features(Tobacco, features = guerrero) |>
  pull(lambda_guerrero)

aus_production |>
  autoplot(box_cox(Tobacco, lambda)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed Australia Tobacco production with $\\lambda$ = ",
         round(lambda,2))))
## Warning: Removed 24 rows containing missing values or values outside the scale range
## (`geom_line()`).

Economy class passengers between Melbourne and Sydney from ansett

ansett |>
  filter(
    Class == 'Economy',
    Airports == "MEL-SYD"
    ) |>
  autoplot(Passengers) +
  labs(title='Economy passengers between Melbourne and Sydney')

lambda <- ansett |>
  filter(
    Class =='Economy',
    Airports == "MEL-SYD"
    ) |>
  features(Passengers, features = guerrero) |>
  pull(lambda_guerrero)

ansett |>
  filter(
    Class == 'Economy',
    Airports == "MEL-SYD"
    ) |>
  autoplot(box_cox(Passengers, lambda)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed Economy passengers between MEL-SYD with $\\lambda$ = ",
         round(lambda,2))))

Pedestrian counts at Southern Cross Station from pedestrian

# adjusted to be average passenger count per week
pedestrian_df = pedestrian |>
  filter(Sensor == "Southern Cross Station") |>
  mutate(Week = yearweek(Date)) |>
  group_by(Week) |>
  mutate(Avg_Count = mean(Count)) |>
  distinct(Week, Avg_Count) |>
  as_tsibble(index=Week)


pedestrian_df |>
  autoplot(Avg_Count) +
  labs(title='Average Pedestrianat Southern Cross Station per Week')

lambda <- pedestrian_df |>
  features(Avg_Count, features = guerrero) |>
  pull(lambda_guerrero)

pedestrian_df |>
  autoplot(box_cox(Avg_Count, lambda)) +
  labs(y = "",
       title = latex2exp::TeX(paste0(
         "Transformed Average Pedestrian per Week with $\\lambda$ = ",
         round(lambda,2))))

3.7

Consider the last five years of the Gas data from aus_production.

gas <- tail(aus_production, 5*4) |> select(Gas)

a.

Plot the time series. Can you identify seasonal fluctuations and/or a trend-cycle?

gas |>
  autoplot(Gas)

The series appears to have an overall increasing trend with a strong seasonality pattern. Gas production increases significantly in the second and third quarters of the year and declines during the first and fourth quarters.

b.

Use classical_decomposition with type=multiplicative to calculate the trend-cycle and seasonal indices.

gas_decomp = gas |>
  model(classical_decomposition(Gas, type = "multiplicative")) |>
  components()

gas_decomp |>
  autoplot() +
  labs(title = "Classical multiplicative decomposition of last five years of the Gas data")
## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

c.

Do the results support the graphical interpretation from part a?

yes the results support the graphical interpretation from part a. The components display an increasing trend with annual seasonality.

d.

Compute and plot the seasonally adjusted data.

gas_decomp |>
  ggplot(aes(x = Quarter)) +
  geom_line(aes(y = Gas, colour = "Data")) +
  geom_line(aes(y = season_adjust,
                colour = "Seasonally Adjusted")) +
  geom_line(aes(y = trend, colour = "Trend")) +
  labs(y = "Gas production",
       title = "Quarterly production of Gas in Australia") +
  scale_colour_manual(
    values = c("gray", "#0072B2", "#D55E00"),
    breaks = c("Data", "Seasonally Adjusted", "Trend")
  )
## Warning: Removed 4 rows containing missing values or values outside the scale range
## (`geom_line()`).

e.

Change one observation to be an outlier (e.g., add 300 to one observation), and recompute the seasonally adjusted data. What is the effect of the outlier?

gas1 = gas
gas1$Gas[10] = gas1$Gas[10] + 300

gas_decomp1 = gas1 |>
  model(classical_decomposition(Gas, type = "multiplicative")) |>
  components()

gas_decomp1 |>
  autoplot() +
  labs(title = "Classical multiplicative decomposition of Gas data with outlier")
## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

gas_decomp1 |>
  ggplot(aes(x = Quarter)) +
  geom_line(aes(y = Gas, colour = "Data")) +
  geom_line(aes(y = season_adjust,
                colour = "Seasonally Adjusted")) +
  geom_line(aes(y = trend, colour = "Trend")) +
  labs(y = "Gas production",
       title = "Quarterly production of Gas in Australia with Outlier") +
  scale_colour_manual(
    values = c("gray", "#0072B2", "#D55E00"),
    breaks = c("Data", "Seasonally Adjusted", "Trend")
  )
## Warning: Removed 4 rows containing missing values or values outside the scale range
## (`geom_line()`).

knit_table(gas_decomp1, 'Seasonally Adjusted Data with Outlier in the Middle')
Seasonally Adjusted Data with Outlier in the Middle
.model Quarter Gas trend seasonal random season_adjust
classical_decomposition(Gas, type = “multiplicative”) 2005 Q3 221 NA 1.0570139 NA 209.0796
classical_decomposition(Gas, type = “multiplicative”) 2005 Q4 180 NA 1.1236007 NA 160.1993
classical_decomposition(Gas, type = “multiplicative”) 2006 Q1 171 200.500 0.8209216 1.0389151 208.3025
classical_decomposition(Gas, type = “multiplicative”) 2006 Q2 224 203.500 0.9984638 1.1024306 224.3446
classical_decomposition(Gas, type = “multiplicative”) 2006 Q3 233 207.000 1.0570139 1.0648904 220.4323
classical_decomposition(Gas, type = “multiplicative”) 2006 Q4 192 210.250 1.1236007 0.8127430 170.8792
classical_decomposition(Gas, type = “multiplicative”) 2007 Q1 187 213.000 0.8209216 1.0694496 227.7928
classical_decomposition(Gas, type = “multiplicative”) 2007 Q2 234 253.625 0.9984638 0.9240415 234.3600
classical_decomposition(Gas, type = “multiplicative”) 2007 Q3 245 293.625 1.0570139 0.7893914 231.7850
classical_decomposition(Gas, type = “multiplicative”) 2007 Q4 505 293.875 1.1236007 1.5293847 449.4479
classical_decomposition(Gas, type = “multiplicative”) 2008 Q1 194 293.750 0.8209216 0.8044928 236.3198
classical_decomposition(Gas, type = “multiplicative”) 2008 Q2 229 256.500 0.9984638 0.8941611 229.3523
classical_decomposition(Gas, type = “multiplicative”) 2008 Q3 249 219.000 1.0570139 1.0756588 235.5693
classical_decomposition(Gas, type = “multiplicative”) 2008 Q4 203 220.375 1.1236007 0.8198260 180.6692
classical_decomposition(Gas, type = “multiplicative”) 2009 Q1 196 221.875 0.8209216 1.0760836 238.7560
classical_decomposition(Gas, type = “multiplicative”) 2009 Q2 238 223.125 0.9984638 1.0683078 238.3662
classical_decomposition(Gas, type = “multiplicative”) 2009 Q3 252 225.125 1.0570139 1.0590004 238.4075
classical_decomposition(Gas, type = “multiplicative”) 2009 Q4 210 226.000 1.1236007 0.8269873 186.8991
classical_decomposition(Gas, type = “multiplicative”) 2010 Q1 205 NA 0.8209216 NA 249.7193
classical_decomposition(Gas, type = “multiplicative”) 2010 Q2 236 NA 0.9984638 NA 236.3631

When placing an outlier in the middle of the gas data, the seasonally adjusted data spikes where the outlier is present. In addition, the seasonally adjusted data shows a lot more variation before and after the outlier. This is due to the increase in variation of the trend and random compoents.

f.

Does it make any difference if the outlier is near the end rather than in the middle of the time series?

gas2 = gas
gas2$Gas[1] = gas2$Gas[1] + 300

gas_decomp2 = gas2 |>
  model(classical_decomposition(Gas, type = "multiplicative")) |>
  components()

gas_decomp2 |>
  autoplot() +
  labs(title = "Classical multiplicative decomposition of Gas data with outlier")
## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

gas_decomp2 |>
  ggplot(aes(x = Quarter)) +
  geom_line(aes(y = Gas, colour = "Data")) +
  geom_line(aes(y = season_adjust,
                colour = "Seasonally Adjusted")) +
  geom_line(aes(y = trend, colour = "Trend")) +
  labs(y = "Gas production",
       title = "Quarterly production of Gas in Australia with Outlier") +
  scale_colour_manual(
    values = c("gray", "#0072B2", "#D55E00"),
    breaks = c("Data", "Seasonally Adjusted", "Trend")
  )
## Warning: Removed 4 rows containing missing values or values outside the scale range
## (`geom_line()`).

knit_table(gas_decomp2, 'Seasonally Adjusted Data with Outlier at the Beginning')
Seasonally Adjusted Data with Outlier at the Beginning
.model Quarter Gas trend seasonal random season_adjust
classical_decomposition(Gas, type = “multiplicative”) 2005 Q3 521 NA 1.1352158 NA 458.9436
classical_decomposition(Gas, type = “multiplicative”) 2005 Q4 180 NA 0.9329010 NA 192.9465
classical_decomposition(Gas, type = “multiplicative”) 2006 Q1 171 238.000 0.8488157 0.8464587 201.4572
classical_decomposition(Gas, type = “multiplicative”) 2006 Q2 224 203.500 1.0830675 1.0163144 206.8200
classical_decomposition(Gas, type = “multiplicative”) 2006 Q3 233 207.000 1.1352158 0.9915329 205.2473
classical_decomposition(Gas, type = “multiplicative”) 2006 Q4 192 210.250 0.9329010 0.9788805 205.8096
classical_decomposition(Gas, type = “multiplicative”) 2007 Q1 187 213.000 0.8488157 1.0343050 220.3070
classical_decomposition(Gas, type = “multiplicative”) 2007 Q2 234 216.125 1.0830675 0.9996669 216.0530
classical_decomposition(Gas, type = “multiplicative”) 2007 Q3 245 218.625 1.1352158 0.9871606 215.8180
classical_decomposition(Gas, type = “multiplicative”) 2007 Q4 205 218.875 0.9329010 1.0039733 219.7446
classical_decomposition(Gas, type = “multiplicative”) 2008 Q1 194 218.750 0.8488157 1.0448171 228.5537
classical_decomposition(Gas, type = “multiplicative”) 2008 Q2 229 219.000 1.0830675 0.9654635 211.4365
classical_decomposition(Gas, type = “multiplicative”) 2008 Q3 249 219.000 1.1352158 1.0015596 219.3415
classical_decomposition(Gas, type = “multiplicative”) 2008 Q4 203 220.375 0.9329010 0.9874115 217.6008
classical_decomposition(Gas, type = “multiplicative”) 2009 Q1 196 221.875 0.8488157 1.0407210 230.9100
classical_decomposition(Gas, type = “multiplicative”) 2009 Q2 238 223.125 1.0830675 0.9848570 219.7462
classical_decomposition(Gas, type = “multiplicative”) 2009 Q3 252 225.125 1.1352158 0.9860487 221.9842
classical_decomposition(Gas, type = “multiplicative”) 2009 Q4 210 226.000 0.9329010 0.9960366 225.1043
classical_decomposition(Gas, type = “multiplicative”) 2010 Q1 205 NA 0.8488157 NA 241.5130
classical_decomposition(Gas, type = “multiplicative”) 2010 Q2 236 NA 1.0830675 NA 217.8996

If the outlier is at the beginning of the series, there is not nearly as much variation within the seasonally adjusted data due to the trend and random components being mostly consistent for majority of the series after the outlier occurs.

3.8

Recall your retail time series data (from Exercise 7 in Section 2.10). Decompose the series using X-11. Does it reveal any outliers, or unusual features that you had not noticed previously?

# x11_dcmp <- myseries |>
#   model(x11 = X_13ARIMA_SEATS(Turnover ~ x11())) |>
#   components()
# 
# autoplot(x11_dcmp) +
#   labs(title =
#     "Decomposition of retail trade turnover using X-11.")

I am getting an error when trying to run this code. I have attempted to troubleshoot and have not been successful in resolving this issue.

3.9

Figures 3.19 and 3.20 show the result of decomposing the number of persons in the civilian labour force in Australia each month from February 1978 to August 1995.

a.

Write about 3–5 sentences describing the results of the decomposition. Pay particular attention to the scales of the graphs in making your interpretation.

The results of the decomposition for the number of persons in the civilian labor force in Australia each month from February 1978 to August 1995 show that the labor force has an increasing trend over time with an annual seasonal pattern. This seasonal pattern peaks about three times a year around March, September, and December. The scale of the seasonal component is the smallest which mean that the variation in this component is smallest compared to the variation in the data. This could be due the increases variance/noise in the remainder component and the sharp decline on the labor force in the early 1990s.

b.

Is the recession of 1991/1992 visible in the estimated components?

Yes, the recession of the 1991/1992 is visible in the remainder component which includes what is left over when the seasonal and trend-cycle components have been subtracted from the data.