Seasonality

Here we consider a model where the time series being modeled can be expressed as an index that depends on the period multiplied by either a constant time series or a time series with a linear trend. Consider a constant model. The index might be a series that describes seasonal data but in more general use might be any multiplier that depends on time. We illustrate the analysis of a seasonal model. A seasonal model describes a time series that changes in regular way with a given cycle. When the cycle is a year, the variation might describe the annual seasons, fall, winter, spring and summer. The cycle can be any time interval, such as the seven days of the week or the four weeks of a month. The cycle includes a fixed number of observations. Say we are observing the daily hits on a web site page. A history of usage indicates that hits occur with different frequencies on the days of the week. Based on historical data we determine an index that describes the relative frequency of visits on each day, as shown in the table below. The numbers represent the proportion of the average daily use that occur on each of the days. The most active day is Thursday that shows 130% of the average and the quietest day is Saturday with only 64%, Day Index Friday 0.99 Saturday 0.64 Sunday 0.75 Monday 1.10 Tuesday 1.05 Wednesday 1.17 Thursday 1.30 We could try to fit the daily hit data with a time series, but the data clearly does not represent either a constant or linear trend model. Alternatively, we correct the data using the indices above to remove as much as possible of the daily variation due to the cyclic effect. The correction is performed by dividing the data by the index. The prime on the notation below indicates adjusted data. Assume that the adjusted data is adequately described by a constant model. We can use the moving average or exponential smoothing method to forecast the single parameter of the model for the adjusted data. Here we use a moving average. The forecast of the cyclic time series is obtained by multiplying the estimated model by the index for the forecasted period. The worksheet below shows a moving average forecast for a season of seven days using the indices above. The data in column B represents actual hits on the page. The indices are placed in the Factor column (C) and are repeated every seven days. The adjusted data in column D is obtained by dividing the data by the index. The moving averages of 14 days of adjusted data are computed in column E. The 1-day forecasts are in column F. The last observed day is day 13 with 63 hits. The moving average of the adjusted data for that day is in cell E35 and has the value 42.9774. The 1-day forecast is in cell F36. Since we are assuming a constant model, this forecast is also 42.9774. To obtain the forecast in terms of the original measure (hits) we transform this result by multiplying by the index for day 14, 1.30. Then our forecast in cell H36 is 55.74 or 56 when rounded.
	Any of the forecasting methods could have been applied to the adjusted data. The same data is modeled with a double exponential smoothing model on the page discussing the Forecasting add-in. The method assumes that the indices have fixed values. Clearly this is not true as the seasonal indices may be changing over time. A reasonable approach is to separately forecast the seasonal indices. The actual observed indices will be used to adjust historical observations and forecasted indices used to adjust future values. Indices may be useful in other contexts. Say we want to construct a model of the cost of some commodity in currency values experiencing inflation. In this case a measure of relative prices such as the Consumer's Price Index (CPI) could be used to adjust observed costs before applying a forecasting method. Observed values of the CPI would adjust historical data and forecasted CPI would adjust forecasted values. The method used on this page is a simple example of decomposing the model into independent factors. The seasonal effect is modeled using a multiplicative factor. Other forecasting methods treat seasonality using an additive term.