HW2 Data 624 Predictive Analytics 2024 Spring Term
Melissa Bowman
2024-02-06
Do exercises 3.1, 3.2, 3.3, 3.4, 3.5, 3.7, 3.8 and 3.9 from the online Hyndman book. (https://oteinsom/fpp3/)
library(fpp3)
library(latex2exp)
library(seasonal)Question 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?
Solve for GDP per capita
global_economy_add_GDP_per_capita <- global_economy %>%
mutate(GDP_per_capita = GDP/Population)Plot the GDP per capita for each country over time.
autoplot(global_economy_add_GDP_per_capita, GDP_per_capita) +
theme(legend.position = "none")
Country has the highest GDP per capita
head(global_economy_add_GDP_per_capita %>%
select(Country, Year, GDP_per_capita)%>%
arrange(desc(GDP_per_capita)),10)## # A tsibble: 10 x 3 [1Y]
## # Key: Country [2]
## Country Year 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.In 2014, Monaco boasted the highest GDP per capita. However, over time, this figure declined, and by the following year, Liechtenstein surpassed Monaco to claim the highest GDP per capita.
Question 3.2
For each of the following series, make a graph of the data. If transforming seems appropriate, do so and describe the effect.
a.United States GDP from global_economy.
global_economy %>%
filter(Country == "United States") %>%
autoplot(GDP) +
labs(title= "GDP")
Data is influenced by changes in population, and therefore, it
should be adjusted on a per capita basis.
global_economy %>%
filter(Country == "United States") %>%
autoplot(GDP/Population) +
labs(title= "GDP per capita")
b.Slaughter of Victorian “Bulls, bullocks and steers” in aus_livestock.
aus_livestock %>%
filter(State == "Victoria" &
Animal == "Bulls, bullocks and steers") %>%
autoplot(Count) 
Did not detect any need for data transformation.
c.Victorian Electricity Demand from vic_elec.
autoplot(vic_elec, Demand)
Calendar adjustment is required for computing the average daily demand for total electricity in each month.
average_demand_per_month <- vic_elec %>%
group_by(Date)%>%
mutate(Daily_demand = sum(Demand))%>%
distinct(Date, Daily_demand)%>%
mutate(Month = yearmonth(Date))%>%
group_by(Month)%>%
mutate(Average_Monthly_Demand = mean(Daily_demand, na.rm = TRUE))%>%
distinct(Month, Average_Monthly_Demand)%>%
as_tsibble(index = Month) autoplot(average_demand_per_month, Average_Monthly_Demand)
d.Gas production from aus_production.
autoplot(aus_production, Gas)
There is too much seasonal variation. Will use Box-Cox
transformation to standardize the seasonal patterns.
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))))
### Question 3.3 Why is a Box-Cox transformation unhelpful for
the canadian_gas data?
autoplot(canadian_gas, Volume)
lambda <- canadian_gas |>
features(Volume, features = guerrero) |>
pull(lambda_guerrero)
canadian_gas |>
autoplot(box_cox(Volume, lambda)) 
The Box-Cox transformation is not suitable for the Canadian_gas data because a good value of lambda is one that equalizes the size of the seasonal variation across the entire series. Neither this lambda nor any other lambda will achieve that.
Question 3.4
What Box-Cox transformation would you select for your retail data (from Exercise 7 in Section 2.10)?
set.seed(100)
myseries <- aus_retail |>
filter(`Series ID` == sample(aus_retail$`Series ID`,1))autoplot(myseries,Turnover)
lambda <- myseries |>
features(Turnover, features = guerrero) |>
pull(lambda_guerrero)
myseries |>
autoplot(box_cox(Turnover, lambda)) +
labs(y = "",
title = latex2exp::TeX(paste0(
"Transformed gas production with $\\lambda$ = ",
round(lambda,2))))
Question 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
autoplot(aus_production, Tobacco)
lambda <- aus_production |>
features(Tobacco, features = guerrero) |>
pull(lambda_guerrero)
aus_production |>
autoplot(box_cox(Tobacco, lambda)) +
labs(y = "",
title = latex2exp::TeX(paste0(
"Transformed gas production with $\\lambda$ = ",
round(lambda,2))))
2. Economy class passengers between Melbourne and Sydney from
ansett
ansett %>%
filter(Airports == "MEL-SYD",
Class == "Economy") %>%
autoplot(Passengers) 
lambda <- ansett %>%
filter(Airports == "MEL-SYD",
Class == "Economy")%>%
features(Passengers, features = guerrero) %>%
pull(lambda_guerrero)
ansett %>%
filter(Airports == "MEL-SYD",
Class == "Economy")%>%
autoplot(box_cox(Passengers, lambda)) +
labs(y = "",
title = latex2exp::TeX(paste0(
"Transformed gas production with $\\lambda$ = ",
round(lambda,2))))
3. Pedestrian counts at Southern Cross Station from
pedestrian
pedestrian %>%
filter(Sensor == "Southern Cross Station") %>%
autoplot(Count) 
lambda <- pedestrian %>%
filter(Sensor == "Southern Cross Station")%>%
features(Count, features = guerrero) %>%
pull(lambda_guerrero)
pedestrian %>%
filter(Sensor == "Southern Cross Station")%>%
autoplot(box_cox(Count, lambda)) +
labs(y = "",
title = latex2exp::TeX(paste0(
"Transformed gas production with $\\lambda$ = ",
round(lambda,2))))
Question 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?
autoplot(gas, Gas)
The trend is increasing over the years, and the cycle occurs
annually.
b.Use classical_decomposition with type=multiplicative to calculate the trend-cycle and seasonal indices.
gas %>%
model(
classical_decomposition(Gas, type = "multiplicative")
) %>%
components() %>%
autoplot() +
labs(title = "Classical multiplicative decomposition of Gas")
c.Do the results support the graphical interpretation from part a? The results do support the graphical interpretation. The trend increases over time, and there is an annual cycle occurrence.
d.Compute and plot the seasonally adjusted data.
dcmp <- gas %>%
model(classical_decomposition(Gas, type = "multiplicative"))
components(dcmp) %>%
as_tsibble() %>%
autoplot(Gas, colour = "gray") +
geom_line(aes(y=season_adjust), colour = "#0072B2") +
labs(y = "Persons (thousands)",
title = "Total employment in US retail")
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?
gas_random <- tail(aus_production, 5*4) %>%
select(Gas)random_row_index <- sample(nrow(gas_random), 1) # Randomly select one row index
gas_random[random_row_index, "Gas"]<-
gas_random[random_row_index, "Gas"] + 300set.seed(200)
dcmp <- gas_random %>%
model(classical_decomposition(Gas, type = "multiplicative"))
components(dcmp) %>%
as_tsibble() %>%
autoplot(Gas, colour = "gray") +
geom_line(aes(y=season_adjust), colour = "#0072B2") +
labs(y = "Persons (thousands)",
title = "Total employment in US retail")
The outlier creates a spike in the plot, making it difficult
to determine seasonality and trend-cycles.
f.Does it make any difference if the outlier is near the end rather than in the middle of the time series?
gas_last <- tail(aus_production, 5*4) %>%
select(Gas)last_index <- nrow(gas_last ) # Get the index of the last row
gas_last [last_index, "Gas"] <- gas_last [last_index, "Gas"] + 300dcmp <- gas_last %>%
model(classical_decomposition(Gas, type = "multiplicative"))
components(dcmp) %>%
as_tsibble() %>%
autoplot(Gas, colour = "gray") +
geom_line(aes(y=season_adjust), colour = "#0072B2") +
labs(y = "Persons (thousands)",
title = "Total employment in US retail")
It does not make any difference whether the outlier in the
data is in the middle or at the end. It still becomes hard to
interpret.
Question 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?
set.seed(300)
x11_dcmp <- myseries |>
model(x11 = X_13ARIMA_SEATS(Turnover ~ x11())) |>
components()
autoplot(x11_dcmp) +
labs(title =
"Decomposition using X-11.")
There seem to be significant spikes in the irregular plot, indicating outliers in the data.
Question 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 of the data, representing the civilian labor force in Australia each month from February 1978 to August 1995, shows a consistent trend increase over the years. Seasonally, the labor force experiences significant declines at the beginning of the year and fluctuates throughout the year. Additionally, there are noticeable spikes in the remainder plot, suggesting the presence of outliers in the data. These outliers may potentially impact the accuracy of any analysis.
b.Is the recession of 1991/1992 visible in the estimated components?
Yes, that is the noticeable downward spike in the remainder plot.