Example 31.3 Modeling Long Seasonal Patterns

This example illustrates some of the techniques you can use to model long seasonal patterns in a series. If the seasonal pattern is of moderate length and the underlying dynamics are simple, then it is easily modeled by using the basic settings of the SEASON statement and these additional techniques are not needed. However, if the seasonal pattern has a long season length and/or has a complex stochastic dynamics, then the techniques discussed here can be useful. You can obtain parsimonious models for a long seasonal pattern by using an appropriate subset of trigonometric harmonics, or by using a suitable spline function, or by using a block-season pattern in combination with a seasonal component of much smaller length. You can also vary the disturbance variances of the subcomponents that combine to form the seasonal component.

The time series used in this example consists of number of calls received per shift at a call center. Each shift is six hours long, and the first shift of the day begins at midnight, resulting in four shifts per day. The observations are available from December 15, 1999, to April 30, 2000. This series is seasonal with season length 28, which is moderate, and in fact there is no particular need to use pattern approximation techniques in this case. However, it is adequate for demonstration purposes. The plan of this example is as follows. First an initial model with a full seasonal component is created. This model is used as a baseline for comparing alternate models created by the techniques that are being illustrated. In practice any candidate model is first checked for adequacy by using various diagnostic procedures. In this illustration the main focus is on the different ways a long seasonal pattern can be modeled and no model diagnostics are done for the models being entertained. The alternate models are compared by using the sum of absolute prediction errors in the holdout region.

The following DATA step statements create the input data set used in this example.

data callCenter;
   input calls @@;
   label calls= "Number of Calls Received in a 6 Hour Shift";
   start = '15dec99:00:00'dt;
   datetime = INTNX( 'dthour6', start, _n_-1 );
   format datetime datetime10.;
datalines;
  18    122    244    128     19    113    230    119     17    112
 219     93     14     73    139     53     11     32     74     56
  15    137    289    153     20    125    227    106     16    101

   ... more lines ...   

Initial exploration of the series clearly indicates that the series does not show any significant trend, and time of day and day of the week have a significant influence on the number of calls received. These considerations suggest a simple random walk trend model along with a seasonal component of season length 28, the total number of shifts in a week. The following statements specify this model. Note the PRINT=HARMONICS option in the SEASON statement, which produces a table that lists the full set of harmonics contributing to the seasonal along with the significance of their contribution. This table will be useful later in choosing a subset trigonometric model. The BACK=28 and the LEAD=28 specifications in the FORECAST statement create a holdout region of 28 observations. The sum of absolute prediction errors (SAE) in this holdout region are used to compare the different models.

proc ucm data=callCenter;
   id datetime interval=dthour6;
   model calls;
   irregular;
   level;
   season length=28 type=trig
      print=(harmonics);
   estimate back=28;
   forecast back=28 lead=28;
run;

The forecasting performance of this model in the holdout region is shown in Output 31.3.1. The sum of absolute prediction errors SAE = , which appears in the last row of the holdout analysis table.

Now that a baseline model is created, the exploration for alternate models can begin. The review of the harmonic table in Output 31.3.2 shows that all but the last three harmonics are significant, and deleting any of them to form a subset trigonometric seasonal component will lead to a poorer model. The last three harmonics, th, th and th, with periods of 2.333, 2.15 and 2.0, respectively, do appear to be possible choices for deletion. Note that the disturbance variance of the seasonal component is not very insignificant (see Output 31.3.3); therefore the seasonal component is stochastic and the preceding logic, which is based on the final state estimate, provides only a rough guideline.