Time plots and time series patterns

For time series data the most obvious graphical form is a time plot (in which the data are plotted over time). A time plot immediately reveals any trends over time, any regular seasonal behavior, and other systematic features of the data.

Figure 2-1 below shows a time plot of the beer data. This reveals the range of the data and the time at which peaks occur, the relative size of the peaks and the randomness in the time series.

beer %>%
  autoplot() +
  ggtitle("Monthly Australian Beer Production") +
  xlab("Year") +
  ylab("Megalitres") +
  labs(caption = "Figure 2-1: Time plot of monthly Australian beer production (megaliters, Ml) from January 1991–August 1995.")

Seasonal plots

For time series data that have seasonal patterns, it is often useful to also produce a seasonal plot. This consists of the data plotted against the individuals “seasons” in which the data were observed. This is something like a time plot except that the data from each season are overlapped. A seasonal plot enables the underlying seasonal pattern more obvious, as well as more easily identify deviations from this pattern. Figure 2-2 shows a seasonal plot of the beer data.

This can be done using the ggseasonplot() function from the forecast package:

ggseasonplot(beer, col = rainbow(5), year.labels = TRUE) +
  ggtitle('Monthly Australian Beer Production') +
  xlab('Months') + ylab('Megalitres') +
  labs(caption = 'Figure 2-2: A seasonal plot of the Australian beer production data. Note that production peaks in
    November and December in preparation for the southern hemisphere summer and is least in winter.')

Scatterplots

Time plots and seasonal plots are not appropriate for cross-sectional data. Patterns in these data can be readily visualised with scatterplots. Scatterplots help to visualise the relationships between variables of interest in a data set. Figure 2-3 shows a plot of two variables from the auto data set.

auto %>%
  ggplot(aes(x = Mileage, y = Price)) +
  geom_point() +
  xlab("Mileage (mpg)") + ylab("Price ($US)") +
  ggtitle("Price/Mileage Relationship for 45 Automobiles") +
  labs(caption = 'Figure 2-3: A scatterplot of price versus mileage for the automobile data.')

The auto data also has an additional explanatory variable Country. It is a categorical variable so cannot be plotted against Price in the same way as the continuous Mileage variable. However, we can include the Country information in our scatterplot with a shape aesthetic. This is displayed in Figure 2-4 below.

auto %>%
  ggplot(aes(x = Mileage, y = Price, shape=Country)) +
  geom_point(size=2) +
  xlab("Mileage (mpg)") + ylab("Price ($US)") +
  ggtitle("Price/Mileage Relationship for 45 Automobiles") +
  labs(caption = 'Figure 2-4: A scatterplot showing price, mileage, and the country of origin for the automobile data.')

Univariate statistics

Table 2-4 shows the computation of some useful univariate statistics for the Mileage variable in the auto dataset, after filtering the data set to only include cars from Japan. These statistics are defined and explained in the textbook. The following code produces these summary statistics:

auto_japan <- auto %>%
  filter(Country == 'Japan')
auto_japan
#>                            Model Country Mileage Price
#> 6                   Mazda MPV V6   Japan      19 14944
#> 7                   Nissan Van 4   Japan      19 14799
#> 9                Acura Legend V6   Japan      20 24760
#> 11            Mitsubishi Wagon 4   Japan      20 14929
#> 12               Nissan Axxess 4   Japan      20 13949
#> 13           Mitsubishi Sigma V6   Japan      21 17879
#> 14               Nissan Stanza 4   Japan      21 11650
#> 16                  Mazda 929 V6   Japan      21 23300
#> 23              Nissan Maxima V6   Japan      22 17899
#> 27             Toyota Cressida 6   Japan      23 21498
#> 29                Nissan 240SX 4   Japan      24 13249
#> 31               Subaru Loyale 4   Japan      25  9599
#> 33           Mitsubishi Galant 4   Japan      25 10989
#> 37        Honda Prelude Si 4WS 4   Japan      27 13945
#> 38                   Subaru XT 4   Japan      28 13071
#> 40               Mazda Protege 4   Japan      32  6599
#> 43          Honda Civic CRX Si 4   Japan      33  9410
#> 44                Subaru Justy 3   Japan      34  5866
#> 45               Toyota Tercel 4   Japan      35  6488

Table 2-4

auto_japan %>%
  summarise(mean = mean(Mileage),
            median= median(Mileage),
            MAD = sum(abs(Mileage - mean(Mileage)))/n(),
            MSD = sum((Mileage - mean(Mileage))^2)/n(),
            Variance = var(Mileage),
            Std_Dev = sd(Mileage))
#>       mean median      MAD      MSD Variance  Std_Dev
#> 1 24.68421     23 4.371191 27.05817  28.5614 5.344287

Bivariate statistics

This section introduces the concepts of covariance, correlation and auto-correlation. Table 2-6 below shows the calculation of these bivariate statistics for the auto data with Country == "Japan" (the same data used in Table 2-5). The variables being considered are Price and Mileage. For this section the units of Price will be converted to thousands of dollars. Calculation of these statistics is given below:

Table 2-6

auto_japan %>%
  mutate(Price = Price/1000) %>%
  summarise(mean_milage = mean(Mileage),
            mean_price = mean(Price),
            sd_mileage = sd(Mileage),
            sd_price = sd(Price),
            covariance = cov(Price, Mileage),
            correlation = cor(Price, Mileage))
#>   mean_milage mean_price sd_mileage sd_price covariance correlation
#> 1    24.68421   13.93805   5.344287 5.400006  -21.00154  -0.7277246

Autocorrelation

The covariance and correlation coefficient are statistics that measure the extent of the linear relationship between two variables and can be used to identify explanatory relationships. Autocovariance and autocorrelation are comparable measures that serve the same purpose for a single time series.

For example, if we compare \(Y_t\) (the observation at time \(t\)) with \(Y_{t−1}\) (the observation at time \(t−1\)), then we see how consecutive observations are related. The observation \(Y_{t−1}\) is described as lagged by one period.

Together the autocorrelations at lags 1, 2, …, make up the autocorrelation function or ACF. It is much easier to understand the autocorrelations by plotting them against the lag. This plot is known as a correlogram. Figure 2-6 shows the ACF for the beer data:

ggAcf(beer) +
  ggtitle('ACF of Beer Production') +
  labs(caption = 'Figure 2-6: The correlogram (or ACF plot) for the beer production data.')

Mathematical transformations

Figure 2-10 below shows a plot of the elec data set.

elec %>%
  autoplot() +
  ggtitle("Australian Monthly Electricity Production") +
  xlab("Year") + ylab("million kWh") +
  labs(caption = "Figure 2-10: Monthly Australian electricity production from January 1956 to August 1995.
       Note the increasing variation as the level of the series increases.")

This time plot shows that the size of the annual seasonal variation increases as the level of the series increases. The variation at the start (towards the left) is about 300 million kWh, while in more recent years (to the right) the variation exceeds 2500 kWh. A mathematical transformation is a convenient method for the necessary task of including this increasing variation into the forecasts for this data.

One possible transformation is the square root function. This transform is plotted below:

elec %>%
  sqrt() %>%
  autoplot() +
  ggtitle("Square Root of Electricity Production") +
  xlab("Year") + ylab("sqrt(million kWh)")

So we can see that this transformation has helped in reducing the variation in the seasonal cycles. This will make the forecasting task easier than the un-transformed data shown in Figure 2-10.

There are a number of other useful transformations. One of the most common is the logarithm, as it is relatively easy to interpret. Section 2/7 of the textbook gives the formulation for the set of power transformations. The plots below show a selection of these transformations.

cbind(
    `Square root` = sqrt(elec),
    `Cube root` = elec^(1/3),
    `Log` = log(elec),
    `Inverse` = -1/elec) %>%
  autoplot(facet=TRUE) +
    xlab("Year") +
    ggtitle("Transformations of the electricity production data") +
    labs(caption="Figure 2-11: Transformations of the electricity production data.")

Calendar adjustments

Some of the variation in a time series may be due to the variation in the number of days (or trading days) each month. It is a good idea to adjust for this known source of variation to allow study of other interesting features.

Month length can make quite an impact since number of days in a month can differ by (31-28)/30 = 10%. If this is not removed, seasonal patterns become hard to interpret. Section 2/7/2 in the textbook shows how to make this adjustment.

Figure 2-12 shows the milk data set with and without being adjusted for month length. The simpler pattern will lead to better forecasts and easier identification of unusual observations.

cbind(
    Milk = milk,
    `Milk per day` = milk/monthdays(milk)
  ) %>%
  autoplot(facet=TRUE) +
  ggtitle("Monthly Milk Production per Cow") +
  xlab("Months") + ylab("Pounds") +
  labs(caption="Figure 2-12: Monthly milk production per cow over 14 years.
    The second graph shows the data adjusted for the length of the month.")