DATA624 HW2

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?

• 2017: U.S. had the highest GDP per capita.
• 2000s-Present: Singapore saw rapid growth, overtaking many developed nations, including the U.S.
• 2008s-Present: UK shown down trends.
• 1990-Present: China and India had very low GDP per capita but have shown strong upward trends, particularly since the 1990s.
• 1980-1995: Japan experienced rapid growth during this period.

global_economy <- global_economy
global_economy %>%
  filter(Country %in% c("United States", "China", "India", "Germany", "Japan", "United Kingdom","Singapore","Malaysia","Hong Kong SAR, China","France","Italia","Brazil")) %>%
  ggplot(aes(x = Year, y = GDP / Population, color = Country)) +
  geom_line(size = 1) +
  labs(title = "GDP per Capita Over Time",
       y = "GDP per Capita (USD)",
       x = "Year") +
  theme_minimal()

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.

• US GDP gshows rows exponentially in time series plot.
• After apply a log transformation, a straight-line trend in a log plot suggests constant percentage growth over time rather than absolute increases.

global_economy %>%
  filter(Country == "United States") %>%
  autoplot(GDP) +
  labs(title = "United States GDP Over Time",
       y = "GDP (USD)",
       x = "Year")

global_economy %>%
  filter(Country == "United States") %>%
  mutate(log_GDP = log(GDP)) %>%
  autoplot(log_GDP) +
  labs(title = "Log-Transformed United States GDP",
       y = "Log(GDP)",
       x = "Year")

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

• I cannot identify a specific season or period of the year where slaughter is consistently higher, but it did show a Long-term Decline.
• It is possible to connect changes in slaughter to external factors like economic policies, weather patterns, or disease outbreaks when slaughter is high during a certain time period.

aus_livestock %>%
  filter(State == "Victoria", Animal == "Bulls, bullocks and steers") %>%
  autoplot(Count) +
  labs(title = "Slaughter of Bulls, Bullocks, and Steers in Victoria",
       y = "Number Slaughtered",
       x = "Year")

aus_livestock %>%
  filter(State == "Victoria", Animal == "Bulls, bullocks and steers") %>%
  gg_season(Count) +
  labs(title = "Seasonal Plot: Victorian Slaughter of Bulls, Bullocks, and Steers")

aus_livestock %>%
  filter(State == "Victoria", Animal == "Bulls, bullocks and steers") %>%
  mutate(log_Count = log(Count)) %>%
  autoplot(log_Count) +
  labs(title = "Log-Transformed Slaughter Data")

Victorian Electricity Demand from vic_elec.

• November to April shows higher electricity demand in the seasonal plot, it suggests a seasonal peak.

vic_elec %>%
  mutate(Month = yearmonth(as_date(Time))) %>%
  index_by(Month) %>%  
  summarise(Demand = mean(Demand)) %>%
  autoplot(Demand) +
  labs(title = "Monthly Average Electricity Demand in Victoria",
       y = "Electricity Demand (MW)",
       x = "Month")

vic_elec %>%
  mutate(Month = month(Time, label = TRUE)) %>%
  gg_season(Demand) +
  labs(title = "Seasonal Plot: Victorian Electricity Demand by Month",
       y = "Electricity Demand (MW)")

vic_elec %>%
  mutate(Month = yearmonth(as_date(Time)),
         log_Demand = log(Demand)) %>%
  index_by(Month) %>%
  summarise(log_Demand = mean(log_Demand)) %>%
  autoplot(log_Demand) +
  labs(title = "Log-Transformed Victorian Electricity Demand (Monthly Avg)",
       y = "Log(Demand)",
       x = "Month")

Gas production from aus_production.

• The plot shows that Australian Gas production has exponentially increased over the year.
• The seasonal plot shows that Q3 has the highest production of Gas.
• The log transformation stabilizes the variance. In 1970, there was a huge increase in production. The box-cox transformation adjusts the variance optimally, as the textbook explains.

aus_production %>%
  autoplot(Gas) +
  labs(title = "Australian Gas Production Over Time",
       y = "Gas Production",
       x = "Time")

aus_production %>%
  gg_season(Gas) +
  labs(title = "Seasonal Plot: Australian Gas Production")

aus_production %>%
  mutate(log_Gas = log(Gas)) %>%
  autoplot(log_Gas) +
  labs(title = "Log-Transformed Australian Gas 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))))

3.3

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

• I think The Box-Cox transformation is unhelpful for the canadian_gas data because the primary issue with the series is trend and seasonality, rather than non-constant variance.

3.5

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

• For aus_production (Tobacco) and ansett (Economy Class Passengers), the Box-Cox transformation is not very impactful since variance can naturally stabilize over time.
• For pedestrian counts, which have high-frequency data with spikes, aggregating into weekly totals + Box-Cox transformation helps improve trend detection and variance stabilization.

aus_production %>%
  autoplot(Tobacco) +
  labs(title = "Australian Tobacco Production Over Time",
       y = "Tobacco Production",
       x = "Time")

lambda_tobacco <- aus_production %>%
  features(Tobacco, features = guerrero) %>%
  pull(lambda_guerrero)

aus_production %>%
  mutate(Tobacco_transformed = box_cox(Tobacco, lambda_tobacco)) %>%
  autoplot(Tobacco_transformed) +
  labs(title = "Box-Cox Transformed Tobacco Production",
       y = "Transformed Production Volume")

ansett %>%
  filter(Class == "Economy", Airports == "MEL-SYD") %>%
  autoplot(Passengers) +
  labs(title = "Economy Class Passengers (MEL-SYD)",
       y = "Passengers")

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

ansett %>%
  filter(Class == "Economy", Airports == "MEL-SYD") %>%
  mutate(Passengers_transformed = box_cox(Passengers, lambda_passengers)) %>%
  autoplot(Passengers_transformed) +
  labs(title = "Box-Cox Transformed Economy Class Passengers (MEL-SYD)",
       y = "Passengers")

pedestrian %>%
  filter(Sensor == "Southern Cross Station") %>%
  autoplot(Count) +
  labs(title = "Southern Cross Station's Pedestrian Counts",
       y = "Pedestrian")

pedestrian_weekly <- pedestrian %>%
  filter(Sensor == "Southern Cross Station") %>%
  index_by(week = yearweek(Date_Time)) %>%
  summarise(Weekly_Count = sum(Count, na.rm = TRUE))

lambda_pedestrian <- pedestrian_weekly %>%
  features(Weekly_Count, features = guerrero) %>%
  pull(lambda_guerrero) %>%
  first()  

pedestrian_weekly %>%
  mutate(Transformed_Count = box_cox(Weekly_Count, lambda_pedestrian)) %>%
  autoplot(Transformed_Count) +
  labs(title = "Box-Cox Transformed Weekly Southern Cross Station's Pedestrian Counts",
       y = "Transformed Weekly Count")

3.7

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

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

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

• The plot shows seasonal fluctuations and a trend.

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

autoplot(gas, Gas) +
  labs(title = "Gas Production Over the Last 5 Years",
       y = "Gas Production",
       x = "Time")

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()

Do the results support the graphical interpretation from part a?

• Yes, The classical_decomposition with type=multiplicative prefectly show the trend line and the seasonal patterns.

Compute and plot the seasonally adjusted data.

gas_sa <- gas %>%
  mutate(Adjusted = Gas / components(gas_decomp)$seasonal)

autoplot(gas_sa, Adjusted)

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?

• The Outlier distorts the trend component, making it unreliable.

gas_outlier <- gas
gas_outlier$Gas[10] <- gas_outlier$Gas[10] + 300

gas_outlier_decomp <- gas_outlier %>%
  model(classical_decomposition(Gas, type = "multiplicative"))

components(gas_outlier_decomp) %>%
  autoplot()

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

• Those outlier always affect the trend, especially those near the end of the time series, can heavily influence trend estimation and future predictions.

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?

• No significant outliers were detected in the irregular component.

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

autoplot(myseries)

## Plot variable not specified, automatically selected `.vars = Turnover`

gg_season(myseries)

## Plot variable not specified, automatically selected `y = Turnover`

gg_subseries(myseries)

## Plot variable not specified, automatically selected `y = Turnover`

gg_lag(myseries, geom = "point")

## Plot variable not specified, automatically selected `y = Turnover`

ACF(myseries, Turnover) |> autoplot()

myseries_x11 <- myseries |>
  model(X_13ARIMA_SEATS(Turnover ~ x11())) |> 
  components()
autoplot(myseries_x11)

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.

STL decomposition value = trend + season_year + remainder

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 Value and trend plots are shows a clear upward trend. The seasonal plot suggesting consistent seasonal patterns in employment. The remainder plot showing some noticeable dips, like after 1991 Jan.

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

• No, There isn’t a strong indication of the 1991–1992 recession in the decomposition plots. The trend component continues to show stable growth around this period, without a clear decline or stagnation that would strongly signal a recession. The seasonal component remains consistent, suggesting that seasonal employment patterns were unaffected. While the irregular component does show some signs of weakness in employment after 1990, the fluctuations are not strong enough to definitively indicate the recession. Overall, the decomposition does not provide strong evidence of the 1991–1992 recession for me.