Once you’ve created the time-series object, you can use functions like start(), end(), and frequency() to return its properties c. You can also use the window() function to create a new time series that’s a subset of the original d.

15.2 Smoothing and seasonal decomposition

Just as analysts explore a dataset with descriptive statistics and graphs before attempting to model the data, describing a time series numerically and visually should be the first step before attempting to build complex models. In this section, we’ll look at smoothing a time series to clarify its general trend, and decomposing a time series in order to observe any seasonal effects.

15.2.1Smoothing with simple moving averages

The first step when investigating a time series is to plot it, as in listing 15.1. Consider the Nile time series. It records the annual flow of the river Nile at Ashwan from 1871– 1970. A plot of the series can be seen in the upper-left panel of figure 15.3. The time series appears to be decreasing, but there is a great deal of variation from year to year.

Time series typically have a significant irregular or error component. In order to discern any patterns in the data, you’ll frequently want to plot a smoothed curve that damps down these fluctuations. One of the simplest methods of smoothing a time series is to use simple moving averages. For example, each data point can be replaced with the mean of that observation and one observation before and after it. This is called a centered moving average. A centered moving average is defined as

St = (Yt-q + … + Yt + … + Yt+q) / (2q + 1)

where St is the smoothed value at time t and k = 2q + 1 is the number of observations that are averaged. The k value is usually chosen to be an odd number (3 in this example). By necessity, when using a centered moving average, you lose the (k – 1) / 2

observations at each end of the series.

Several functions in R can provide a simple moving average, including SMA() in the TTR package, rollmean() in the zoo package, and ma() in the forecast package. Here, you’ll use the ma() function to smooth the Nile time series that comes with the base R installation.

The code in the next listing plots the raw time series and smoothed versions using k equal to 3, 7, and 15. The plots are given in figure 15.3.

Listing 15.2 Simple moving averages

library(forecast)

opar <- par(no.readonly=TRUE) par(mfrow=c(2,2))

ylim <- c(min(Nile), max(Nile)) plot(Nile, main="Raw time series")

plot(ma(Nile, 3), main="Simple Moving Averages (k=3)", ylim=ylim) plot(ma(Nile, 7), main="Simple Moving Averages (k=7)", ylim=ylim) plot(ma(Nile, 15), main="Simple Moving Averages (k=15)", ylim=ylim) par(opar)

As k increases, the plot becomes increasingly smoothed. The challenge is to find the value of k that highlights the major patterns in the data, without underor oversmoothing. This is more art than science, and you’ll probably want to try several values of k before settling on one. From the plots in figure 15.3, there certainly appears to have been a drop in river flow between 1892 and 1900. Other changes are open to interpretation. For example, there may have been a small increasing trend between 1941 and 1961, but this could also have been a random variation.

For time-series data with a periodicity greater than one (that is, with a seasonal component), you’ll want to go beyond a description of the overall trend. Seasonal decomposition can be used to examine both seasonal and general trends.

15.2.2Seasonal decomposition

Time-series data that have a seasonal aspect (such as monthly or quarterly data) can be decomposed into a trend component, a seasonal component, and an irregular component. The trend component captures changes in level over time. The seasonal component captures cyclical effects due to the time of year. The irregular (or error) component captures those influences not described by the trend and seasonal effects.

The decomposition can be additive or multiplicative. In an additive model, the components sum to give the values of the time series. Specifically,

Yt = Trendt + Seasonalt + Irregulart

where the observation at time t is the sum of the contributions of the trend at time t, the seasonal effect at time t, and an irregular effect at time t.

In a multiplicative model, given by the equation Yt = Trendt * Seasonalt * Irregulart

the trend, seasonal, and irregular influences are multiplied. Examples are given in figure 15.4.

The plot in the lower portion of figure 15.5 displays the time series created by taking the log of each observation. The variance has stabilized, and the logged series looks like an appropriate candidate for an additive decomposition. This is carried out using the stl() function in the following listing.

Listing 15.3 Seasonal decomposition using stl()

remain the same across each year (using the s.window="period" option). The trend is monotonically increasing, and the seasonal effect suggests more passengers in the summer (perhaps during vacations). The grey bars on the right are magnitude guides—each bar represents the same magnitude. This is useful because the y-axes are different for each graph.

The object returned by the stl() function contains a component called time.series that contains the trend, season, and irregular portion of each observation d. In this case, fit$time.series is based on the logged time series. exp(fit$time.series) converts the decomposition back to the original metric. Examining the seasonal effects suggests that the number of passengers increased by 24% in July (a multiplier of 1.24) and decreased by 20% in November (with a multiplier of .80).

Two additional graphs can help to visualize a seasonal decomposition. They’re created by the monthplot() function that comes with base R and the seasonplot() function provided in the forecast package. The code

par(mfrow=c(2,1))

library(forecast)

monthplot(AirPassengers, xlab="", ylab="") seasonplot(AirPassengers, year.labels="TRUE", main="")

produces the graphs in figure 15.7.