Time Series Assignment 2

2. Graphs of time series (and transformations)

United States Economy

library(fpp3)
library(latex2exp)

us_economy <- global_economy |> filter(Country == "United States") |>
autoplot(GDP) + labs(title = "United States GDP")
us_economy

This is a plot of United States’ GDP over years 1960 - 2017. It makes sense to apply a transformation to this data, as GDP in a vacuum carries little meaning. Commonly, GDP per-capita is a useful metric to gauge economic change, especially because GDP itself is somewhat dependent on population.

us_economy_adj <- global_economy |> filter(Country == "United States") |>
autoplot(GDP/Population) + labs(title = "United States GDP per-capita")
us_economy_adj

Even still, the plot looks roughly the same!

Victorian Cows

aus_cows <- aus_livestock |> filter(Animal == "Bulls, bullocks and steers") |>
  filter(State == "Victoria") |>
  autoplot(Count) + labs(title = "Victorian Cows Count")
aus_cows

cows_dcmp <- aus_livestock |> filter(Animal == "Bulls, bullocks and steers") |>
  filter(State == "Victoria") |> 
  model(stl = STL(Count))
components(cows_dcmp) |> autoplot()

In the STL decomposition, we can see that the variation in Count from year to year has strong seasonality that seems to be relatively unrelated to the level of Count itself, suggesting no real need for a transformation of the data.

Victorian Electricity

aus_elec <- vic_elec |>
autoplot(Demand) + labs(title = "Victorian Electricity Demand")
aus_elec

elec_dcmp <- vic_elec |>
  model(STL(Demand))
components(elec_dcmp) |> autoplot()

There are several areas of seasonality here: yearly, weekly, daily, and hourly. The first three seem to be dependent on the level of demand, the last dependent on the overall trend. This would suggest that a transformation is in order, we’ll try a Box-Cox transform.

lambda <- vic_elec |>
  features(Demand, features = guerrero) |>
  pull(lambda_guerrero)

aus_elec_t <- vic_elec |>
  autoplot(box_cox(Demand, lambda)) + labs(title = latex2exp::TeX(paste0(
         "Transformed Victorian Electricity Demand with $\\lambda$ = ",
         round(lambda,2))))
aus_elec_t

elec_t_dcmp <- vic_elec |>
  model(STL(box_cox(Demand, lambda)))
components(elec_t_dcmp) |> autoplot()

The transform slightly helped, I guess? The daily seasonality has been almost fixed, same with the monthly. Not the yearly, though. This would tell us that the seasonality is more complex than just multiplicative, and further transforms could probably be used (but I don’t know many useful nonlinear transforms).

Australian Gas

aus_gas <- aus_production[, c(1, 7)] |>
  autoplot(Gas) + labs(title = "Australian Gas Production")
aus_gas

gas_dcmp <- aus_production[, c(1, 7)] |>
  model(STL(Gas))
components(gas_dcmp) |> autoplot()

CLEAR seasonality as the level (and trend) of gas production increase. We’ll try Box-Cox again:

lambda_gas <- aus_production[, c(1, 7)] |>
  features(Gas, features = guerrero) |>
  pull(lambda_guerrero)

aus_gas_t <- aus_production[, c(1, 7)] |>
  autoplot(box_cox(Gas, lambda_gas)) + labs(title = latex2exp::TeX(paste0("Transformed Australian Gas Production with $\\lambda$ = ", round(lambda_gas,2))))
aus_gas_t

gas_t_dcmp <- aus_production[, c(1, 7)] |>
  model(STL(box_cox(Gas, lambda_gas)))
components(gas_t_dcmp) |> autoplot()

The Box-Cox transform successfully fixed the increasing seasonal variance (with year).

7. Australian Gas time series

(a). Plotting the time series

gas <- tail(aus_production, 5*4) |> select(Gas)
autoplot(gas) + labs(title = "Gas Production Over Time")

From an immediate simple plot, we can see a trend of gas production slowly increasing over time, along with a strong seasonality favoring increased production in quarters 2 and 3 relative to quarters 1 and 4.

(b + c). Gas time series decomposition

gas_dcmp <- tail(aus_production, 5*4) |> select(Gas) |>
  model(
    classical_decomposition(Gas, type = "multiplicative")
  ) |> components()
gas_dcmp |> autoplot()

This cleanly matches what was seen in part (a). trend is slowly increasing over time, seasonal is extremely periodic, nearly identical in each 1-year period.

(d). Seasonal adjustment of Gas data

gas_adj = gas
gas_adj[, 1] = gas_adj[, 1]/gas_dcmp$seasonal
autoplot(gas_adj) + labs(title = "Gas Production (Seasonally Adj.)")

(e). Changing the data? Making an outlier?

Lets redo the above, but make a middle point (like 2008 Q1) an outlier. We can do so by artificially adding 300 to it. In theory, this should screw up the moving averages:

gas_weird = gas
gas_weird[11, 1] = gas_weird[11, 1] + 300

gas_weird_dcmp <- gas_weird |>
  model(
    classical_decomposition(Gas, type = "multiplicative")
  ) |> components()
gas_weird_dcmp |> autoplot()

gas_weird_adj = gas_weird
gas_weird_adj[, 1] = gas_weird_adj[, 1]/gas_weird_dcmp$seasonal
autoplot(gas_weird_adj) + labs(title = "Gas Production (With Outlier, Seasonally Adj.)")

Interestingly, adding the outlier makes the seasonal fluctuations want to persist, even after seasonal adjustment. With the massive outlier present, the initial calculation of trend is harmed severely, being based off of moving averages (many of which pick up the outlier). The averages allow the outlier to have more influence than it otherwise would in the trend.

The trend is then separated from the raw data, allowing for seasonal fluctuation calculations. As the outlier overstates its influence in the trend, dividing out the trend from the raw data understates the seasonal component of the data. Thus, when we seasonally adjust the data, the adjustment is not enough, which still leaves seasonal artifacts present (visible in the plot above)

(f). Outlier on the edges

If the outlier is on the edges, fewer moving average moments should contain it, meaning that seasonal adjustment should be better:

gas_weird = gas
gas_weird[1, 1] = gas_weird[1, 1] + 300

gas_weird_dcmp <- gas_weird |>
  model(
    classical_decomposition(Gas, type = "multiplicative")
  ) |> components()
gas_weird_dcmp |> autoplot()

gas_weird_adj = gas_weird
gas_weird_adj[, 1] = gas_weird_adj[, 1]/gas_weird_dcmp$seasonal
autoplot(gas_weird_adj) + labs(title = "Gas Production (With End Outlier, Seasonally Adj.)")

And this is exactly what we see. Less seasonality in the seasonally adjusted data, compared with before, with the strong seasonality.

8. Decompose Chapter 2 Time Series with X-11

set.seed(2468)
myseries <- aus_retail |>
  filter(`Series ID` == sample(aus_retail$`Series ID`,1))

x11_dcmp <- myseries |>
  model(x11 = X_13ARIMA_SEATS(Turnover ~ x11())) |>
  components()
x11_dcmp |> autoplot()

I don’t see any outliers with the aus_retail data selection. In fact, the X-11 decomposition makes the data look incredibly predictable and stable. We see a very clean bounding, and consistent values, in the seasonal component. The trend is stably increasing across the 4 present decades. Additionally, the irregular component is quite minor, indicating few random fluctuations.