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 1day 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 1day 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.
