Assignment2_DATA624

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?

# Load the data
data("global_economy")

# Calculate GDP per capita
global_economy <- global_economy %>%
  mutate(GDP_per_capita = GDP / Population)

# 1. Plot GDP per capita for each country over time
gdp_per_capita_plot <- global_economy %>%
  ggplot(aes(x = Year, y = GDP_per_capita, group = Country)) +
  geom_line(alpha = 0.3) +
  scale_y_log10(labels = scales::comma) +
  labs(title = "GDP per Capita Over Time by Country",
       x = "Year",
       y = "GDP per Capita (log scale)",
       caption = "Source: global_economy dataset") +
  theme_minimal()

print(gdp_per_capita_plot)

## Warning: Removed 3242 rows containing missing values or values outside the scale range
## (`geom_line()`).

# 2. Which country has the highest GDP per capita?
highest_gdp_per_capita <- global_economy %>%
  group_by(Country) %>%
  summarise(max_GDP_per_capita = max(GDP_per_capita, na.rm = TRUE)) %>%
  arrange(desc(max_GDP_per_capita)) %>%
  slice(1:10)  # Top 10 countries

## Warning: There were 3325 warnings in `summarise()`.
## The first warning was:
## ℹ In argument: `max_GDP_per_capita = max(GDP_per_capita, na.rm = TRUE)`.
## ℹ In group 23: `Country = "Afghanistan"` `Year = 1982`.
## Caused by warning in `max()`:
## ! no non-missing arguments to max; returning -Inf
## ℹ Run `dplyr::last_dplyr_warnings()` to see the 3324 remaining warnings.

## Warning: Current temporal ordering may yield unexpected results.
## ℹ Suggest to sort by `Country`, `Year` first.

print("Top 10 countries by maximum GDP per capita:")

## [1] "Top 10 countries by maximum GDP per capita:"

print(highest_gdp_per_capita)

## # A tsibble: 10 x 3 [1Y]
## # Key:       Country [2]
##    Country        Year max_GDP_per_capita
##    <fct>         <dbl>              <dbl>
##  1 Monaco         2014            185153.
##  2 Monaco         2008            180640.
##  3 Liechtenstein  2014            179308.
##  4 Liechtenstein  2013            173528.
##  5 Monaco         2013            172589.
##  6 Monaco         2016            168011.
##  7 Liechtenstein  2015            167591.
##  8 Monaco         2007            167125.
##  9 Liechtenstein  2016            164993.
## 10 Monaco         2015            163369.

# 3. How has the highest GDP per capita changed over time?
# Identify the country with the highest GDP per capita
top_country <- highest_gdp_per_capita$Country[1]

# Plot that specific country
top_country_plot <- global_economy %>%
  filter(Country == top_country) %>%
  ggplot(aes(x = Year, y = GDP_per_capita)) +
  geom_line(color = "red", size = 1) +
  geom_point(color = "darkred", alpha = 0.5) +
  scale_y_continuous(labels = scales::comma) +
  labs(title = paste("GDP per Capita Over Time:", top_country),
       x = "Year",
       y = "GDP per Capita") +
  theme_minimal()

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

print(top_country_plot)

## Warning: Removed 11 rows containing missing values or values outside the scale range
## (`geom_line()`).

## Warning: Removed 11 rows containing missing values or values outside the scale range
## (`geom_point()`).

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. Slaughter of Victorian “Bulls, bullocks and steers” in aus_livestock. Victorian Electricity Demand from vic_elec. Gas production from aus_production.

# 1. United States GDP from global_economy
data("global_economy")

us_gdp <- global_economy %>%
  filter(Country == "United States") %>%
  select(Year, GDP)

# Plot US GDP
p1_raw <- us_gdp %>%
  ggplot(aes(x = Year, y = GDP)) +
  geom_line(color = "steelblue", size = 1) +
  scale_y_continuous(labels = scales::comma) +
  labs(title = "United States GDP (Raw)",
       subtitle = "Clear exponential growth pattern",
       x = "Year",
       y = "GDP (billions USD)") +
  theme_minimal()

# Log transformation for US GDP
p1_log <- us_gdp %>%
  ggplot(aes(x = Year, y = GDP)) +
  geom_line(color = "darkgreen", size = 1) +
  scale_y_log10(labels = scales::comma) +
  labs(title = "United States GDP (Log Scale)",
       subtitle = "Linear trend indicates constant growth rate",
       x = "Year",
       y = "GDP (billions USD, log scale)") +
  theme_minimal()

print(p1_raw)

print(p1_log)

# 2. Slaughter of Victorian "Bulls, bullocks and steers" from aus_livestock
data("aus_livestock")

victoria_beef <- aus_livestock %>%
  filter(State == "Victoria", 
         Animal == "Bulls, bullocks and steers") %>%
  select(Month, Count)

# Plot raw data
p2_raw <- victoria_beef %>%
  ggplot(aes(x = Month, y = Count)) +
  geom_line(color = "brown", size = 0.7) +
  labs(title = "Victorian Beef Slaughter (Raw)",
         subtitle = "Strong seasonality with potential variance changes",
         x = "Year",
         y = "Number of animals") +
  theme_minimal()

# Log transformation to stabilize variance
p2_log <- victoria_beef %>%
  ggplot(aes(x = Month, y = Count)) +
  geom_line(color = "darkorange", size = 0.7) +
  scale_y_log10() +
  labs(title = "Victorian Beef Slaughter (Log Scale)",
         subtitle = "More stable variance, seasonality still present",
         x = "Year",
         y = "Number of animals (log scale)") +
  theme_minimal()

# Seasonal plot to better visualize pattern
p2_seasonal <- victoria_beef %>%
  gg_season(Count, period = "year") +
  labs(title = "Victorian Beef Slaughter - Seasonal Plot",
       y = "Number of animals") +
  theme_minimal()

## Warning: `gg_season()` was deprecated in feasts 0.4.2.
## ℹ Please use `ggtime::gg_season()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

print(p2_raw)

print(p2_log)

print(p2_seasonal)

# 3. Victorian Electricity Demand from vic_elec
data("vic_elec")

# Take a subset - first 7 days (336 half-hour periods)
elec_subset <- vic_elec %>%
  filter(Time <= min(Time) + days(7))

# Plot raw data
p3_raw <- elec_subset %>%
  ggplot(aes(x = Time, y = Demand)) +
  geom_line(color = "darkblue", size = 0.3) +
  labs(title = "Victorian Electricity Demand (First Week)",
       subtitle = "Daily seasonality visible",
       x = "Time", 
       y = "Demand (MW)") +
  theme_minimal()

# Log transformation
p3_log <- elec_subset %>%
  ggplot(aes(x = Time, y = Demand)) +
  geom_line(color = "purple", size = 0.3) +
  scale_y_log10() +
  labs(title = "Victorian Electricity Demand (Log Scale, First Week)",
       x = "Time", 
       y = "Demand (MW, log scale)") +
  theme_minimal()


print(p3_raw)

print(p3_log)

# 4. Gas production from aus_production
data("aus_production")

gas <- aus_production %>%
  select(Quarter, Gas)

# Plot raw data
p4_raw <- gas %>%
  ggplot(aes(x = Quarter, y = Gas)) +
  geom_line(color = "darkred", size = 0.8) +
  labs(title = "Australian Gas Production (Raw)",
       subtitle = "Exponential growth with seasonality, increasing variance",
       x = "Year",
       y = "Gas production (petajoules)") +
  theme_minimal()

# Log transformation
p4_log <- gas %>%
  ggplot(aes(x = Quarter, y = Gas)) +
  geom_line(color = "forestgreen", size = 0.8) +
  scale_y_log10() +
  labs(title = "Australian Gas Production (Log Scale)",
       subtitle = "Linear trend, stabilized variance, seasonal pattern clearer",
       x = "Year",
       y = "Gas production (petajoules, log scale)") +
  theme_minimal()

# Seasonal decomposition to examine components
gas_tsibble <- gas %>%
  mutate(Gas_log = log(Gas)) %>%
  as_tsibble(index = Quarter)

# Try to fit a seasonal decomposition model
tryCatch({
  gas_decomp <- gas_tsibble %>%
    model(classical_decomposition(Gas_log, type = "additive")) %>%
    components()
  
  p4_decomp <- autoplot(gas_decomp) +
    labs(title = "Gas Production - Seasonal Decomposition (Log Scale)")
  print(p4_decomp)
}, error = function(e) {
  print("Could not create decomposition - try STL instead:")
  gas_stl <- gas_tsibble %>%
    model(STL(Gas_log ~ season(window = "periodic"))) %>%
    components()
  p4_stl <- autoplot(gas_stl) +
    labs(title = "Gas Production - STL Decomposition (Log Scale)")
  print(p4_stl)
})

## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

print(p4_raw)

print(p4_log)

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

canadian_gas %>%
  autoplot(Volume) +
  labs(title = "Canadian Gas Production",
       subtitle = "Constant variance, stable seasonal amplitude")

# Find optimal lambda for Canadian gas
lambda_canadian <- canadian_gas %>%
  features(Volume, features = guerrero) %>%
  pull(lambda_guerrero)

# Apply Box-Cox transformation
canadian_gas %>%
  autoplot(box_cox(Volume, lambda_canadian)) +
  labs(title = paste("Canadian Gas - Box-Cox Transformation (λ =", 
                     round(lambda_canadian, 3), ")"),
       y = "Box-Cox transformed Volume")

# Compare with original
library(patchwork)
p1 <- canadian_gas %>% autoplot(Volume) + labs(title = "Original")
p2 <- canadian_gas %>% autoplot(box_cox(Volume, lambda_canadian)) + 
      labs(title = paste("Box-Cox (λ =", round(lambda_canadian, 3), ")"))
p1 + p2

Looking at canadian_gas:

• The series shows strong, stable seasonality with consistent amplitude throughout
• There’s no increasing or decreasing variance as the level changes
• The seasonal pattern is additive, not multiplicative

Box-Cox transformations are designed to stabilize variance when it changes with the level of the series (e.g., exponential growth, multiplicative seasonality). Since canadian_gas already has relatively constant variance, applying Box-Cox would:

• Provide no benefit - variance is already stable
• Complicate interpretation - changes the scale unnecessarily Potentially distort the clean additive pattern

3.4 What Box-Cox transformation would you select for your retail data (from Exercise 7 in Section 2.10)? 7. Monthly Australian retail data is provided in aus_retail. Select one of the time series as follows (but choose your own seed value):

set.seed(12345678) myseries <- aus_retail |> filter(Series ID == sample(aus_retail$Series ID,1)) Explore your chosen retail time series using the following functions:

autoplot(), gg_season(), gg_subseries(), gg_lag(),

ACF() |> autoplot()

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

# Set seed and select random retail series
set.seed(12345678)
myseries <- aus_retail %>%
  filter(`Series ID` == sample(aus_retail$`Series ID`, 1))

# View the series
myseries %>%
  autoplot(Turnover) +
  labs(title = "Selected Retail Series: Original")

# Find optimal lambda using Guerrero's method
lambda <- myseries %>%
  features(Turnover, features = guerrero) %>%
  pull(lambda_guerrero)

# Apply Box-Cox transformation
myseries %>%
  autoplot(box_cox(Turnover, lambda)) +
  labs(title = paste("Box-Cox Transformed (λ =", round(lambda, 3), ")"),
       y = "Box-Cox transformed Turnover")

# Compare original vs transformed
library(patchwork)
p1 <- myseries %>% autoplot(Turnover) + labs(title = "Original")
p2 <- myseries %>% autoplot(box_cox(Turnover, lambda)) + 
      labs(title = paste("Box-Cox (λ =", round(lambda, 3), ")"))
p1 + p2

# View lambda value
print(paste("Optimal lambda:", round(lambda, 3)))

## [1] "Optimal lambda: 0.083"

lambda = 0.083 is close to 0 so log transformation is a good option

# Use log transformation (simpler and interpretable)
myseries %>%
  autoplot(log(Turnover)) +
  labs(title = "Australian Retail - Log Transformation",
       y = "Log Turnover")

# Or lets use the exact Box-Cox value
myseries %>%
  autoplot(box_cox(Turnover, 0.083)) +
  labs(title = "Australian Retail - Box-Cox (λ = 0.083)",
       y = "Box-Cox transformed Turnover")

#### 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.

# 1. Tobacco from aus_production
data("aus_production")

tobacco <- aus_production %>%
  select(Quarter, Tobacco)

# Find optimal lambda
lambda_tobacco <- tobacco %>%
  features(Tobacco, features = guerrero) %>%
  pull(lambda_guerrero)

# Visualize
p1 <- tobacco %>%
  autoplot(Tobacco) +
  labs(title = "Tobacco Production: Original")

p2 <- tobacco %>%
  autoplot(box_cox(Tobacco, lambda_tobacco)) +
  labs(title = paste("Tobacco: Box-Cox (λ =", round(lambda_tobacco, 3), ")"))

# 2. Economy class passengers between Melbourne and Sydney from ansett
data("ansett")

melsydecon <- ansett %>%
  filter(Airports == "MEL-SYD", Class == "Economy") %>%
  select(Week, Passengers)

# Find optimal lambda
lambda_ansett <- melsydecon %>%
  features(Passengers, features = guerrero) %>%
  pull(lambda_guerrero)

# Visualize
p3 <- melsydecon %>%
  autoplot(Passengers) +
  labs(title = "MEL-SYD Economy Passengers: Original")

p4 <- melsydecon %>%
  autoplot(box_cox(Passengers, lambda_ansett)) +
  labs(title = paste("MEL-SYD: Box-Cox (λ =", round(lambda_ansett, 3), ")"))

# 3. Pedestrian counts at Southern Cross Station from pedestrian
data("pedestrian")

southercross <- pedestrian %>%
  filter(Sensor == "Southern Cross Station") %>%
  select(Date_Time, Count)

# Find optimal lambda
lambda_ped <- southercross %>%
  features(Count, features = guerrero) %>%
  pull(lambda_guerrero)

# Visualize
p5 <- southercross %>%
  autoplot(Count) +
  labs(title = "Southern Cross Pedestrians: Original")

p6 <- southercross %>%
  autoplot(box_cox(Count, lambda_ped)) +
  labs(title = paste("Pedestrians: Box-Cox (λ =", round(lambda_ped, 3), ")"))

# Display results
print(paste("Tobacco lambda:", round(lambda_tobacco, 3)))

## [1] "Tobacco lambda: 0.926"

print(paste("MEL-SYD Economy lambda:", round(lambda_ansett, 3)))

## [1] "MEL-SYD Economy lambda: 2"

print(paste("Southern Cross Pedestrian lambda:", round(lambda_ped, 3)))

## [1] "Southern Cross Pedestrian lambda: -0.25"

# Arrange plots
library(patchwork)
(p1 | p2) / (p3 | p4) / (p5 | p6)

## Warning: Removed 24 rows containing missing values or values outside the scale range
## (`geom_line()`).
## Removed 24 rows containing missing values or values outside the scale range
## (`geom_line()`).

Tobacco (λ = 0.926)

Nearly 1.0 → Almost no transformation needed Variance is already relatively stable

MEL-SYD Economy Passengers (λ = 2)

λ = 2 → Square transformation (y²) graph shows zeros in the data, as the Guerrero method requires strictly positive values A practical approach would be adding a small constant before applying a log transformation (λ = 0)

Southern Cross Pedestrians (λ = -0.25) Negative lambda → Reciprocal transformation Variance decreases as pedestrian counts increase for interpretability, a log transformation (λ = 0) would be sufficient and more practical

# Ansett data - MEL-SYD Economy class
ansett %>%
  filter(Airports == "MEL-SYD", Class == "Economy") %>%
  select(Week, Passengers) %>%
  autoplot(Passengers) +
  labs(title = "MEL-SYD Economy Passengers: Original",
       subtitle = "Contains zero/negative values causing Guerrero failure")

# Add constant and transform
melsydecon <- ansett %>%
  filter(Airports == "MEL-SYD", Class == "Economy") %>%
  mutate(Passengers_adj = Passengers + 1)  # Add constant to handle zeros/negatives

# Find optimal lambda on adjusted data
lambda_adj <- melsydecon %>%
  features(Passengers_adj, features = guerrero) %>%
  pull(lambda_guerrero)

# Before and after transformation
p_before <- melsydecon %>%
  autoplot(Passengers) +
  labs(title = "MEL-SYD Economy Passengers: Original",
       subtitle = "Raw data with zeros/negatives")

p_after <- melsydecon %>%
  autoplot(box_cox(Passengers_adj, lambda_adj)) +
  labs(title = "MEL-SYD Economy Passengers: Transformed",
       subtitle = paste("Box-Cox on (Passengers+1), λ =", round(lambda_adj, 3)))

p_before | p_after

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? Use classical_decomposition with type=multiplicative to calculate the trend-cycle and seasonal indices. Do the results support the graphical interpretation from part a? Compute and plot the seasonally adjusted data. 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? Does it make any difference if the outlier is near the end rather than in the middle of the time series?

library(patchwork)

# a) Last five years of Gas data - proper tsibble
gas <- aus_production %>%
  tail(5*4) %>%
  select(Quarter, Gas) %>%
  as_tsibble(index = Quarter)

# Plot time series
p1 <- gas %>%
  autoplot(Gas) +
  labs(title = "Australian Gas Production - Last 5 Years",
       subtitle = "Clear seasonal pattern with upward trend",
       y = "Gas production (petajoules)")
print(p1)

# b) Classical multiplicative decomposition  
gas_decomp <- gas %>%
  model(
    classical_decomposition(Gas, type = "multiplicative")
  ) %>%
  components()

# Plot decomposition components
p2 <- autoplot(gas_decomp) +
  labs(title = "Multiplicative Decomposition of Gas Production")
print(p2)

## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

# c) Seasonally adjusted data
gas_seas_adj <- gas_decomp %>%
  mutate(season_adjusted = Gas / seasonal) %>%
  as_tsibble(index = Quarter)  # Re-establish as tsibble

p3 <- gas_seas_adj %>%
  autoplot(season_adjusted, color = "steelblue", size = 0.8) +
  geom_line(data = gas, aes(x = Quarter, y = Gas), 
            color = "gray50", alpha = 0.5, inherit.aes = FALSE) +
  labs(title = "Seasonally Adjusted Gas Production",
       subtitle = "Gray line = original data, Blue line = seasonally adjusted",
       y = "Gas production (petajoules)")
print(p3)

# d) Add outlier in middle
gas_outlier_mid <- gas
gas_outlier_mid$Gas[10] <- gas_outlier_mid$Gas[10] + 300

gas_decomp_mid <- gas_outlier_mid %>%
  model(classical_decomposition(Gas, type = "multiplicative")) %>%
  components()

gas_seas_adj_mid <- gas_decomp_mid %>%
  mutate(season_adjusted = Gas / seasonal) %>%
  as_tsibble(index = Quarter)  # Add this line

p4 <- gas_seas_adj_mid %>%
  autoplot(season_adjusted, color = "steelblue", size = 0.8) +
  geom_line(data = gas_outlier_mid, aes(x = Quarter, y = Gas), 
            color = "gray50", alpha = 0.5, inherit.aes = FALSE) +
  labs(title = "Seasonally Adjusted with Outlier (Middle)",
       subtitle = "Outlier at quarter 10",
       y = "Gas production (petajoules)")
print(p4)

# e) Add outlier near end
gas_outlier_end <- gas
gas_outlier_end$Gas[19] <- gas_outlier_end$Gas[19] + 300

gas_decomp_end <- gas_outlier_end %>%
  model(classical_decomposition(Gas, type = "multiplicative")) %>%
  components()

gas_seas_adj_end <- gas_decomp_end %>%
  mutate(season_adjusted = Gas / seasonal) %>%
  as_tsibble(index = Quarter)  # Add this line

p5 <- gas_seas_adj_end %>%
  autoplot(season_adjusted, color = "steelblue", size = 0.9) +
  geom_line(data = gas_outlier_end, aes(x = Quarter, y = Gas), 
            color = "gray50", alpha = 0.6, inherit.aes = FALSE) +
  labs(title = "Seasonally Adjusted with Outlier (Near End)",
       subtitle = "Outlier at quarter 18",
       y = "Gas production (petajoules)")
print(p5)

# Display all plots
p1 / p2 / p3 / p4 / p5

## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

Key Findings: a. Clear seasonal pattern with upward trend c. Multiplicative classical decomposition also shows upward trend and seasonality. In trend data is missing at the ends because of the moving averages. d. Outlier clearly reflects in seasonally adjusted data e. Even when the outlier is near the end it reflects in seasonally adjusted graph.

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?

7. Monthly Australian retail data is provided in aus_retail. Select one of the time series as follows (but choose your own seed value):

set.seed(12345678) myseries <- aus_retail |> filter(Series ID == sample(aus_retail$Series ID,1)) Explore your chosen retail time series using the following functions:

autoplot(), gg_season(), gg_subseries(), gg_lag(),

ACF() |> autoplot()

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

#install.packages("seasonal")

library(seasonal)

## 
## Attaching package: 'seasonal'

## The following object is masked from 'package:tibble':
## 
##     view

# Set seed and select random retail series
set.seed(12345)
myseries <- aus_retail %>%
  filter(`Series ID` == sample(aus_retail$`Series ID`, 1))

# View the series
myseries %>%
  autoplot(Turnover) +
  labs(title = "Selected Retail Series: Original",
       subtitle = paste("Series ID:", unique(myseries$`Series ID`),
                        "-", unique(myseries$State),
                        "-", unique(myseries$Industry)))

# X-11 Decomposition
myseries_x11 <- myseries %>%
  model(X_13ARIMA_SEATS(Turnover ~ x11())) %>%
  components()

# Plot X-11 components
autoplot(myseries_x11) +
  labs(title = "X-11 Decomposition of Retail Turnover")

# Focus on seasonal and irregular components to spot outliers/features
myseries_x11 %>%
  select(Month, seasonal, irregular) %>%
  pivot_longer(c(seasonal, irregular), 
               names_to = "component", 
               values_to = "value") %>%
  ggplot(aes(x = Month, y = value)) +
  geom_line() +
  facet_grid(component ~ ., scales = "free_y") +
  labs(title = "X-11: Seasonal and Irregular Components",
       subtitle = "Check for unusual patterns or outliers")

# Highlight potential outliers in irregular component
myseries_x11 %>%
  autoplot(irregular) +
  geom_hline(yintercept = c(-2, 2), color = "red", linetype = "dashed") +
  labs(title = "X-11 Irregular Component",
       subtitle = "Red lines at ±2 indicate potential outliers")

# Identify specific outliers
outliers <- myseries_x11 %>%
  as_tibble() %>%
  select(Month, irregular) %>%
  filter(abs(irregular) > 2)  # Common threshold

print("Potential outliers detected by X-11:")

## [1] "Potential outliers detected by X-11:"

print(outliers)

## # A tibble: 0 × 2
## # ℹ 2 variables: Month <mth>, irregular <dbl>

# Compare with original series
myseries %>%
  autoplot(Turnover) +
  geom_vline(data = outliers, aes(xintercept = Month), 
             color = "red", alpha = 0.5) +
  labs(title = "Retail Series with Potential Outliers Highlighted",
       subtitle = "Red lines indicate months with |irregular| > 2")

X-11 Decomposition of Retail Time Series:

The X-11 decomposition did not reveal any statistically significant outliers (|irregular| > 2) in the selected retail series. This indicates that:

No unusual one-off events - There were no unexpected spikes or drops beyond normal seasonal and trend variation Stable seasonal pattern - The seasonal component is consistent and well-captured by the decomposition Smooth irregular component - Random fluctuations are within expected ranges

Gradual seasonal shift - The seasonal pattern shows slight evolution over time especially after 2007-2008, which is not obvious in the original plot Trend-cycle smoothness - The decomposition shows the underlying trend more clearly than the raw data, revealing steady growth with no abrupt structural breaks

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 trend (scale ~4000-9000) shows a strong upward trajectory over the entire period, with a slight dip around 1991 that is barely visible at this scale but confirmed by increased variation in the original and remainder component
• The seasonal component (scale ±100) exhibits clear and stable monthly patterns, though there is a noticeable change in the seasonal amplitude and pattern after 1990—particularly visible in the later years of the seasonal sub-series plot.
• The remainder (scale ±100) appears largely random with no obvious pattern, though slightly increased volatility around 1991 corresponds with the trend dip.
• This is additive decomposition

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

Yes, the 1991/1992 recession is clearly visible in the components. * The trend component shows a slight decline around 1991, interrupting the otherwise consistent upward growth. * More noticeably, the remainder component exhibits increased volatility and a cluster of negative values during this period. * The seasonal sub-series plot reveals that certain months—particularly March, April, and August show noticeably lower values after 1990 compared to their historical seasonal patterns, indicating the recession’s disproportionate impact on specific times of the year.

This combination of trend stagnation, negative remainder spikes, and altered seasonal amplitudes makes the recession effects unmistakable in the decomposed components even though they were partially obscured in the original data.