hw 2

Homework 2

Do exercises 3.1, 3.2, 3.3, 3.4, 3.5, 3.7, 3.8 and 3.9 from the online Hyndman book.

library(fpp3)

## Registered S3 method overwritten by 'tsibble':
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##   as_tibble.grouped_df dplyr

## ── Attaching packages ──────────────────────────────────────────── fpp3 1.0.0 ──

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## ✔ lubridate   1.9.3     ✔ fable       0.3.4
## ✔ ggplot2     3.5.1     ✔ fabletools  0.4.2

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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?

• In 2017, the country with the highest GDP per capita was Luxembourg

• The ggplot is misleading and also looks like Luxembourg was the highest through out most of time.

• I also reviewed the top countries in the year 1980, while Luxembourg was eighth highest. In 1908 Monaco was the highest

• I looked at some online resources and finally made the autoplot work; it’s pretty clear that the GDP per cap was a lot lower in the past than in the recent 20 years.

data("global_economy")

GDP_per_cap <- global_economy %>%
  mutate(per_cap = GDP/Population) %>%
  select(Country, Year, per_cap) %>%
  as_tsibble(index = Year, key = Country)

# autoplot(GDP_per_cap, per_cap) not running...facet_wrap(~ Country, scales = "free_y") also not helping...15k+ observations might impact that too....trying to names to country also not working. 
ggplot(GDP_per_cap, aes(x=Year, y=per_cap, colour = Country)) + geom_line(stat = "identity", show.legend = F) 

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

GDP_per_cap %>%
  filter(Year == 1980) %>%
  arrange(desc(per_cap))

GDP_per_cap %>%
  filter(Year == 2017) %>%
  arrange(desc(per_cap))

gdp1 <- global_economy %>%
  mutate(GDP_Per_Capital = GDP/Population) %>%
  filter(GDP_Per_Capital > 45000) %>%# adjusting the legend
  autoplot(GDP_Per_Capital) +   labs(title= "GDP per capital", y = "Currency in US Dollars")
gdp1

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

Transforming is needed ot prepare for decomposition to balance the fluctuations; consider if variation of cycle trend/seasonality is proportional to the level of the time series

United States GDP from global_economy.

• Transforming not needed for this series

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

• The fluctuations could be optimized. I made a graph for the original plot and for box cox transformation; however, other than the scale, there was no effect which might mean it was already normally distributed.

Victorian Electricity Demand from vic_elec.

• Transforming not needed for this series. While I should look more at the errors in my code to change the 30 mins to monthly, consistent seasonality is present in month Nov-Feb (their summers), where electricity use is typically higher.

Gas production from aus_production.

• Transforming not needed for this series; consistent seasonality is present.

global_economy |>
  filter(Country == "United States") |>
     autoplot(GDP/Population)

glimpse(aus_livestock)

## Rows: 29,364
## Columns: 4
## Key: Animal, State [54]
## $ Month  <mth> 1976 Jul, 1976 Aug, 1976 Sep, 1976 Oct, 1976 Nov, 1976 Dec, 197…
## $ Animal <fct> "Bulls, bullocks and steers", "Bulls, bullocks and steers", "Bu…
## $ State  <fct> Australian Capital Territory, Australian Capital Territory, Aus…
## $ Count  <dbl> 2300, 2100, 2100, 1900, 2100, 1800, 1800, 1900, 2700, 2300, 250…

vic_livestock <- aus_livestock %>%
  filter(State == "Victoria", 
         Animal %in% c("Bulls, bullocks and steers"))
vic_livestock 

vic_livestock <- aus_livestock %>%
  filter(State == "Victoria", 
         Animal %in% c("Bulls, bullocks and steers"))
autoplot(vic_livestock, Count)

autoplot(aus_production, Gas)

#hard to see with the default 30 mins; kept getting errors when trying to change it to quarterly/ monthly
autoplot(vic_elec, Demand)

autoplot(aus_production, Gas)

3. Why is a Box-Cox transformation unhelpful for the canadian_gas data? It seems that it is already normally distributed. While transforming the data does change the scale the peaks and troughs look rougly the same, suggesting it is not helpful. The pattern, peaks and troughs are still the same and the seasonality seem to vary in a consistent pattern.

autoplot(canadian_gas, Volume)

lambda <- canadian_gas %>%
  features(Volume, features = guerrero) %>%
  pull(lambda_guerrero)
canadian_gas %>%
  autoplot(box_cox(Volume, lambda))

4. What Box-Cox transformation would you select for your retail data (from Exercise 7 in Section 2.10)? I would think to use features = guerrero since it does it selects the best lamda automatically at 0.2053157

set.seed(123788)
myseries <- aus_retail |>
  filter(`Series ID` == sample(aus_retail$`Series ID`,1))
autoplot(myseries)

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

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

myseries %>%
  autoplot(box_cox(Turnover, lambda))

print(lambda)

## [1] 0.2053157

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.

aus_production %>%
  autoplot(Tobacco)

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

library(latex2exp)
lambda <- aus_production %>%
  features(Tobacco, features = guerrero) %>%
  pull(lambda_guerrero)

aus_production %>%
  autoplot(box_cox(Tobacco, lambda)) +
  labs(y = "", title = TeX(paste0("Transformed Tobacco Production with $\\lambda$ = ",
         round(lambda,2))))

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

#something is wrong here since lambda should be between 0&1
mel_syd <- ansett %>%
  filter(Class == "Economy",
         Airports == "MEL-SYD")

autoplot(mel_syd, Passengers)+
  labs(title = "Economy class Passengers Between Melbourne and Sydney")

lambda <- mel_syd %>%
  features(Passengers, features = guerrero) %>%
  pull(lambda_guerrero)

mel_syd %>%
  autoplot(box_cox(Passengers, lambda)) +
  labs(y = "", title = TeX(paste0("Transformed Number of Passengers with $\\lambda$ = ",
         round(lambda,2))))

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? There is an increasing pattern over the years. Seasonal pattern present lowest at Q1 and highest at Q3.

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

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

as_tsibble(gas) %>% 
     model(classical_decomposition(type ="multiplicative") )%>%
     components() %>% 
     autoplot() 

## Model not specified, defaulting to automatic modelling of the `Gas` variable.
## Override this using the model formula.

## 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, seasonal and trend being accurate to my interpretation. Compute and plot the seasonally adjusted data.

as_tsibble(gas) %>% 
     model(classical_decomposition(type ="multiplicative") )%>%
     components() %>% 
     as_tsibble() %>%
     autoplot() + geom_line(aes(y=season_adjust), colour = "red")

## Model not specified, defaulting to automatic modelling of the `Gas` variable. Override this using the model formula.
## Plot variable not specified, automatically selected `.vars = Gas`

4. 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? 20x2; The outlier, adding 300, natually results in a spike.

gas$Gas[19]<-gas$Gas[19]+300

as_tsibble(gas) %>% 
                model(classical_decomposition(type ="multiplicative") )%>% 
                components() %>%
                as_tsibble() %>%
                autoplot() +geom_line()

## Model not specified, defaulting to automatic modelling of the `Gas` variable. Override this using the model formula.
## Plot variable not specified, automatically selected `.vars = Gas`

gas$Gas[11]<-gas$Gas[11]+300

as_tsibble(gas) %>% 
                model(classical_decomposition(type ="multiplicative") )%>% 
                components() %>%
                as_tsibble() %>%
                autoplot() +geom_line()

## Model not specified, defaulting to automatic modelling of the `Gas` variable. Override this using the model formula.
## Plot variable not specified, automatically selected `.vars = Gas`

5. Does it make any difference if the outlier is near the end rather than in the middle of the time series? Yes where it is located impacts the data and outlier at the end will impact predictions.

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? The peaks in seasonality is decreasing from 2005 and forward, yet the trend is still increasing.

library(seasonal)

## 
## Attaching package: 'seasonal'

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

x11 <- myseries %>%
        model(x11 = X_13ARIMA_SEATS(Turnover ~ x11())) %>%
        components()

autoplot(x11)

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.

Decomposition of the number of persons in the civilian labour force in Australia each month from February 1978 to August 1995.

1. Write about 3–5 sentences describing the results of the decomposition. Pay particular attention to the scales of the graphs in making your interpretation. It shows a upward trend, which is significant when considering the scale of the graphs. The dip during the 1991-1992 recession is evident but relatively minor compared to the overall growth trajectory, indicating that while it impacted the labor force, it did not derail the long-term trend. Seasonality is present, but its scale is much smaller than the trend, suggesting it has less influence on labor force changes.

2. Is the recession of 1991/1992 visible in the estimated components? Even though remainder is noise that is where I see it most; there is also a drip in the first graph ‘value’ during those years. The trend does not decrease but looks like a bit of a plateau during those years.