Module: Computing seasonality¶

The seasonality algorithm’s course¶

To explain the algorithms involved in seasonality, their overview is shown in the following diagram:

Legend to the seasonal indices:

• Sharp indices – unsmoothed seasonal indices without upper and lower limits.
• Strong indices – smoothed seasonal indices without upper and lower limits.
• Weak indices – smoothed seasonal indices with both the lower and upper limits.

Algorithm for a sharp season¶

For this algorithm, the time series must first be adjusted – and it is done in the following steps:

• If the history is longer than 3 years, it gets trimmed to 3 years.
• It is smoothed with trend adjustment, so that weaker years are not penalized.

The time series adjusted in this way is then “overlaid over itself by years” and using an average, seasonal indices are determined. For example, then, the seasonal index for January is calculated as the average of all the Januaries in time series for all years.

Subsequently, a seasonality score is calculated for a time series and seasonal indices; the seasonality score is used in the further calculations. The seasonality score is a metric that indicates the extent to which the seasonal indices match the time series. For the metric, the following statement holds: the higher the metric, the more the seasonal indices correspond to the time series. STOCK normally uses the seasonality score as thresholds; for instance, an earlier seasonal algorithm used the following distribution of intervals to determine seasonality:

• x < -1 – unseasonal
• x >= -1 && x <0.8 – undefined seasonality
• x >= 0.8 – seasonal

Exact procedure to measure the seasonality score

The input for computing the seasonality score are adjusted values of the time series and resulting seasonal indices for each calendar month. To describe the algorithm, we use the following notation:

• G:sub: i – a set of values ​​for a given calendar month; G₂, for instance, is a set of all the values of the time series from February.
• AVG_i – the seasonal index in the i-th calendar month; AVG is used to suggest the computation of seasonal indices as averages of the sets G_i.
• STD_i – a standard deviation of i-th calendar month calculated from the set G_i.

Each seasonal index I, an element of {1..12}, contributes to the seasonality score with its Add_i – that, then, is computed as follows:

If the following condition applies, the contribution is computed as described below; otherwise, the contribution of the seasonal index is Add_i = 0.

• G_i is at least 2
• At least one of the following conditions hold:
  • The number of seasonal indices AVG_i equal to zero is less than or equal to 4
  • AVG_i - STD_i > 0

If the condition is met, the contribution of the seasonal index is then determined as follows:

• If AVG_i> = 1, then the seasonal index contribution is Add_i = AVG_i - STD_i - 1
• If AVG_i < 1, then the seasonal index contribution is Add_i = 1 - (AVG_i + STD_i)

The final seasonality score is equal to the sum of all the contributions of the seasonal indices limited by 5 from above. The seasonality score is thus calculated as:

I is an element of {1..12}

S = sum_i(min(add_i,5))

Using seasonal indices of a sharp season

The seasonal indices computed in the algorithm detecting sharp seasons are used, if one of the following conditions holds:

1. Condition for Easter

• The time series has at least 500 sales.
• The time series must span at least two Easter seasons, i.e., sales are high in the actual Easter month, but in the subsequent month, the sales are weak.

2. Standard condition

In the text for the standard condition, seasonal indices refer to the resulting array of size 12; the i-th index corresponds to the seasonal index of i-th calendar month.

• The seasonality score (described above) is greater than the threshold (now set to 4.0).
• At least one seasonal index is greater than the threshold (now set to 4.0).
• At least 8 seasonal indices are smaller than the threshold (now set to 0.5).
• The seasonal indices contain only “one hill”, i.e., all indices greater than the threshold (now set as 0.05) create a single contiguous block with no gap – or they create two contiguous blocks without gaps and one of them starts in January and the other ends in December. The second condition allows to have a hill between December and January.

The “Classical algorithm”¶

For this algorithm, the time series must first be adjusted – and it is done in the following steps:

• If the history is longer than 3 years, it gets trimmed to 3 years.
• Adjustment to trim very low values.
• Smoothing with weighted moving average (0.25, 0.50, 0.25).
• It is smoothed with trend adjustment, so that weaker years are not penalized.

The time series adjusted in this way is then “overlaid over itself by years” and using an average, seasonal indices are determined. For example, then, the seasonal index for January is calculated as the average of all the Januaries in time series for all years.

The seasonal indices described above are computed for all the time series, but not all time series might have finite seasonal indices. Specifically for nonseasonal series with random high sales, we encounter too high indices, that, in turn, have a bad effect on the operation of certain algorithms in Planning Wizard. Therefore, the time series is subjected to a test to confirm its seasonality. If the test is successful, the result of the whole algorithm are strong indices, otherwise they are weak indices.

• Strong indices – raw indices just as they were computed, without any adjustments.
• Weak indices – indices computed by the algorithms and adjusted so that they acquire only the defined limit values ​​(now set from 0.7 to 1.3). Seasonal effects on these indices is suppressed, but it cannot disappear completely. This, as a matter of fact, is an advantage especially for those time series, that are the limit cases, on the border of seasonality. Limitations of indices, on the other hand, helps with nonseasonal and sporadic time series.

Confirmation of seasonality in the classical algorithm

Again, let us repeat that the result of the seasonality algorithm is either a strong or weak index. Which indices are used is determined by the algorithm that confirms seasonality. This algorithm is standardly governed by the following rules:

1. First, it selects all the peaks of “hills” in the seasonal indices higher than the defined threshold (now set to 2.0).
2. For the seasonality to be confirmed, for each peak (of the hill), there must be at least two years with - seasonal index in the peak month was greater than the defined threshold (now set to 1.5). - sales in the peak month must be greater than one-fifth of the largest sales in a given month in one of the years: this ensures that a random small sale is not accepted as a season.

There is, however, one exception to confirm the seasonality for the months defined in the configuration. For the months set in the configuration, weaker seasonality confirmation rules apply. For example, a single confirmed season is taken as sufficient. This configuration is used to detect seasonality of Christmas more effectively .