# The UCM Procedure

### Example 34.3 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
201     92     14     94    187     69     11     59     94     21

... 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 34.3.1. The sum of absolute prediction errors SAE = 516.22, which appears in the last row of the holdout analysis table.

Output 34.3.1: Predictions in the Holdout Region: Baseline Model

Obs datetime Actual Forecast Error SAE
525 24APR00:00 12 -4.004 16.004 16.004
526 24APR00:06 136 110.825 25.175 41.179
527 24APR00:12 295 262.820 32.180 73.360
528 24APR00:18 172 145.127 26.873 100.232
529 25APR00:00 20 2.188 17.812 118.044
530 25APR00:06 127 105.442 21.558 139.602
531 25APR00:12 236 217.043 18.957 158.559
532 25APR00:18 125 114.313 10.687 169.246
533 26APR00:00 16 2.855 13.145 182.391
534 26APR00:06 108 95.202 12.798 195.189
535 26APR00:12 207 194.184 12.816 208.005
536 26APR00:18 112 97.687 14.313 222.317
537 27APR00:00 15 1.270 13.730 236.047
538 27APR00:06 98 85.875 12.125 248.172
539 27APR00:12 200 184.891 15.109 263.281
540 27APR00:18 113 93.113 19.887 283.168
541 28APR00:00 15 -1.120 16.120 299.288
542 28APR00:06 104 84.983 19.017 318.305
543 28APR00:12 205 177.940 27.060 345.365
544 28APR00:18 89 64.292 24.708 370.073
545 29APR00:00 12 -6.020 18.020 388.093
546 29APR00:06 68 46.286 21.714 409.807
547 29APR00:12 116 100.339 15.661 425.468
548 29APR00:18 54 34.700 19.300 444.768
549 30APR00:00 10 -6.209 16.209 460.978
550 30APR00:06 30 12.167 17.833 478.811
551 30APR00:12 66 49.524 16.476 495.287
552 30APR00:18 61 40.071 20.929 516.216

Now that a baseline model is created, the exploration for alternate models can begin. The review of the harmonic table in Output 34.3.2 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, 12th, 13th and 14th, 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 34.3.3); therefore the seasonal component is stochastic and the preceding logic, which is based on the final state estimate, provides only a rough guideline.

Output 34.3.2: Harmonic Analysis of the Season: Initial Model

The UCM Procedure

Harmonic Analysis of Trigonometric Seasons (Based on the Final State)
Name Season Length Harmonic Period Chi-Square DF Pr > ChiSq
Season 28 1 28.00000 234.19 2 <.0001
Season 28 2 14.00000 264.19 2 <.0001
Season 28 3 9.33333 95.65 2 <.0001
Season 28 4 7.00000 105.64 2 <.0001
Season 28 5 5.60000 146.74 2 <.0001
Season 28 6 4.66667 121.93 2 <.0001
Season 28 7 4.00000 4299.12 2 <.0001
Season 28 8 3.50000 150.79 2 <.0001
Season 28 9 3.11111 89.68 2 <.0001
Season 28 10 2.80000 8.95 2 0.0114
Season 28 11 2.54545 6.14 2 0.0464
Season 28 12 2.33333 2.20 2 0.3325
Season 28 13 2.15385 3.40 2 0.1828
Season 28 14 2.00000 2.33 1 0.1272

Output 34.3.3: Parameter Estimates: Initial Model

Final Estimates of the Free Parameters
Component Parameter Estimate Approx
Std Error
t Value Approx
Pr > |t|
Irregular Error Variance 92.14591 13.10986 7.03 <.0001
Level Error Variance 44.83595 10.65465 4.21 <.0001
Season Error Variance 0.01250 0.0065153 1.92 0.0551

The following statements fit a subset trigonometric model formed by dropping the last three harmonics by specifying the DROPH= option in the SEASON statement:

proc ucm data=callCenter;
id datetime interval=dthour6;
model calls;
irregular;
level;
season length=28 type=trig droph=12 13 14;
estimate back=28;
forecast back=28 lead=28;
run;

The last row of the holdout region prediction analysis table for the preceding model is shown in Output 34.3.4. It shows that the subset trigonometric model has better prediction performance in the holdout region than the full trigonometric model, its SAE = 471.53 compared to the SAE = 516.22 for the full model.

Output 34.3.4: SAE for the Subset Trigonometric Model

Obs datetime Actual Forecast Error SAE
552 30APR00:18 61 40.836 20.164 471.534

The following statements illustrate a spline approximation to this seasonal component. In the spline specification the knot placement is quite important, and usually some experimentation is needed. In the following model the knots are placed at the beginning and the middle of each day. Note that the knots at the beginning and end of the season, 1 and 28 in this case, should not be listed in the knot list because knots are always placed there anyway.

proc ucm data=callCenter;
id datetime interval=dthour6;
model calls;
irregular;
level;
splineseason length=28
knots=3 5 7 9 11 13 15 17 19 21 23 25 27
degree=3;
estimate back=28;
forecast back=28 lead=28;
run;

The spline season model takes about half the time to fit that the baseline model takes. The last row of the holdout region prediction analysis table for this model is shown in Output 34.3.5, which shows that the spline season model performs even better than the previous two models in the holdout region, its SAE = 313.79 compared to SAE = 471.53 for the previous model.

Output 34.3.5: SAE for the Spline Season Model

Obs datetime Actual Forecast Error SAE
552 30APR00:18 61 23.350 37.650 313.792

The following statements illustrate yet another way to approximate a long seasonal component. Here a combination of BLOCKSEASON and SEASON statements results in a seasonal component that is a sum of two seasonal patterns: one seasonal pattern is simply a regular season with season length 4 that captures the within-day seasonal pattern, and the other seasonal pattern is a block seasonal pattern that remains constant during the day but varies from day to day within a week. Note the use of NLOPTIONS statement to change the optimization technique during the parameter estimation to DBLDOG, which in this case performs better than the default technique, TRUREG.

proc ucm data=callCenter;
id datetime interval=dthour6;
model calls;
irregular;
level;
season length=4 type=trig;
blockseason nblocks=7 blocksize=4
type=trig;
estimate back=28;
forecast back=28 lead=28;
nloptions tech=dbldog;
run;

This model also takes about half the time to fit that the baseline model takes. The last row of the holdout region prediction analysis table for this model is shown in Output 34.3.6, which shows that the block season model does slightly better than the baseline model but not as good as the other two models, its SAE = 508.52 compared to the SAE = 516.22 of the baseline model.

Output 34.3.6: SAE for the Block Season Model

Obs datetime Actual Forecast Error SAE
552 30APR00:18 61 39.339 21.661 508.522

This example showed a few different ways to model a long seasonal pattern. It showed that parsimonious models for long seasonal patterns can be useful, and in some cases even more effective than the full model. Moreover, for very long seasonal patterns the high memory requirements and long computing times might make full models impractical.