Further analysis of the deseasonalised data

The first step in analysing seasonal data is usually to remove the seasonal effect.

Data   =   Seasonal effect   +   Deseasonalised

However we should next examine the deseasonalised data for trend and cyclical features. Using the methods in the first two sections of this chapter, we can obtain trend and cyclical components. What is left after removing these are the residuals.

Deseasonalised  =   Trend   +   Cyclical   +   Residual

Putting these together, we can identify four separate components to the variation in our original time series.

Data  =   Seasonal effect   +   Trend   +   Cyclical   +   Residual

It is important that you understand the distinction between these four components of a seasonal time series:

Seasonal effect
For monthly or quarterly data, this component repeats in exactly the same way each year.
Trend
This component changes smoothly over the whole time period of the data. It is often described by a linear or quadratic model.
Cyclical
This component describes peaks and troughs that last for several data values. The component includes both long-term cycles and short-term autocorrelation of a few adjacent values.
Residual
This component is what is left. It reflects 'unusual' individual data values that cannot be explained by the other effects.

Tourist arrivals in Fiji

You will use the diagram below to help understand the different components of the time series of tourist arrivals in Fiji.

Initially, the blue line in the top half of the diagram connects the raw data points. Click the checkbox Seasonal to delete the seasonal effect from this diagram and display it in the bottom half of the diagram. (The mean has been added to the seasonal effects to allow them to be plotted against the same axis as the data.) The top diagram therefore shows the deseasonalised data and the bottom shows the seasonal effects.

Now also click the checkbox Trend. This removes a quadratic trend from the top and adds it to the bottom.

Finally, click Cyclical to remove the cyclical component from the top diagram and add it to the bottom. What remains in the top is the residuals.

Selecting the components one-at-a-time allows them to be displayed separately in the bottom half of the diagram.