Identifying Patterns in Time Series Data - Analysis of Seasonality
Seasonal dependency (seasonality) is another general component of the time series pattern. The concept was illustrated in the example of the airline passengers data above. It is formally defined as correlational dependency of order k between each i'th element of the series and the (ik)'th element (Kendall, 1976) and measured by autocorrelation (i.e., a correlation between the two terms); k is usually called the lag. If the measurement error is not too large, seasonality can be visually identified in the series as a pattern that repeats every k elements.
| Element Name | Description |
|---|---|
| Autocorrelation correlogram | Seasonal patterns of time series can be examined via correlograms. The correlogram (autocorrelogram) displays graphically and numerically the autocorrelation function (ACF), that is, serial correlation coefficients (and their standard errors) for consecutive lags in a specified range of lags (e.g., 1 through 30). Ranges of two standard errors for each lag are usually marked in correlograms but typically the size of autocorrelation is of more interest than its reliability (see Elementary Concepts) because we are usually interested only in very strong (and thus highly significant) autocorrelations. |
| Examining correlograms | While examining correlograms you should keep in mind that autocorrelations for consecutive lags are formally dependent. Consider the following example. If the first element is closely related to the second, and the second to the third, then the first element must also be somewhat related to the third one, etc. This implies that the pattern of serial dependencies can change considerably after removing the first order autocorrelation (i.e., after differencing the series with a lag of 1). |
| Partial autocorrelations | Another useful method to examine serial dependencies is to examine the partial autocorrelation function (PACF) - an extension of autocorrelation, where the dependence on the intermediate elements (those within the lag) is removed. In other words, the partial autocorrelation is similar to autocorrelation, except that when calculating it, the (auto) correlations with all the elements within the lag are partialled out (Box & Jenkins, 1976; see also McDowall, McCleary, Meidinger, & Hay, 1980). If a lag of 1 is specified (i.e., there are no intermediate elements within the lag), then the partial autocorrelation is equivalent to autocorrelation. In a sense, the partial autocorrelation provides a "cleaner" picture of serial dependencies for individual lags (not confounded by other serial dependencies). |
| Removing serial dependency | Serial dependency for a particular lag of k can be removed by differencing the series, that is converting each i'th element of the series into its difference from the (i-k)''th element. There are two major reasons for such transformations.
First, you can identify the hidden nature of seasonal dependencies in the series. Remember that, as mentioned in the previous paragraph, autocorrelations for consecutive lags are interdependent. Therefore, removing some of the autocorrelations will change other autocorrelations, that is, it may eliminate them or it may make some other seasonalities more apparent. The other reason for removing seasonal dependencies is to make the series stationary, which is necessary for ARIMA and other techniques. |