The Smoothing Process
Each month all three smoothing statistics are updated by the most recent month's sales.
The process of updating is called smoothing because a fixed percent, alpha (), of the most recent
sales is added to (1-) times the old single smoothing statistic. For example, if were .3 and sales
were X, a new single smoothing statistic St (X) would be calculated by taking .3X + (1 - .3) times St-
  
1 (X) = St (X). Note that St-1 (X) refers to last month's single smoothing statistic. If t-2 were used in
the statistic, it would refer to the statistic two months ago. Next month's statistic would be t + 1. The
t in St X is the current statistic. This is a convenient notational technique to designate what statistic
is being referred to. The  in St does not refer to an exponential power, but rather identifies that
this is the single smooth statistic.
St (X) refers to the double smoothed statistic. The (X) in both statistics signifies that the
statistic is calculated from the time series being examined (sales).
It was explained in Chapter 2 that to signify our forecast, we useX t+1. The ^ is referred to as
hat. This signifies that the value under the hat is an estimate rather than an actual value.X t+1 is an
estimate of X. The t+1 signifies that the equation is the estimate for t, the current period plus 1,
which is next month. The error of estimation is calculated for next month by (Xt - Xt). In one month,
Xt+1 will have become Xt.
The double smooth statistic is calculated like the single smooth statistic with one exception:
actual sales are replaced by the new single smoothing statistic. Last month's double smoothing
statistic St-1 (X) is multiplied by (1-) and added to St-1 (X), giving St (X). The box 2, , indicates
  
this is the double smoothing statistic.
The triple smoothing statistic continues the process. The new triple smoothed statistic is
calculated by adding St (X) to (1-) St-1 (X). The smoothing process takes into account trend and
cycles but ignores seasonality. Therefore, a seasonality variable is included in the forecasting
equation. To keep from confounding or combining seasonal effects and trends or cycles, we divide
the actual sales data (Xt) by the seasonal factor for month t before it is used in the smoothing
process. If the seasonal factor is signified by t, the single smoothed statistic for seasonal data is
S t (X) = d + (1 - ) S t -1 (X) 1
The procedure for calculating seasonal factors is given in an earlier section of this chapter.
The Forecasting Equations
The three forecasting equations are designed to generate forecasts that follow three
different patterns. The constant equation models a flat, constant pattern. This forecasting equation
= t+1(St (X)). Figure 3-15 shows the patterns of time series forecasted best by the constant
Figure 3-15 Here
= t+1[d1St (X)-d2St ]. The patterns that this equation models
The linear equation isX t+1
best are those showing a linear trend. In the equation, d1 and d2 are lag factors. The smoothing
(1 - ) (1 - )
statistics lag the actual values. The value of d1 is 2+ 2. The lag for d2 is 1+ 3.
Figure 2-15 also illustrates the patterns of time series forecasted best by the linear equation. The
number of future periods for which the forecast is being made is indicated by . If we are
forecastingX t+1, equals one; if we are forecasting Xt+2, equals 2; etc.
The quadratic forecasting equation is designed to forecast time series with nonlinear trend.
The equation is as follows:
X t+1 = t+1 T 1 S t (X) - T 2 S t (X) + T 3 S t (X) . 4
ˆ   
T1, T2, and T3 are lag correction variables. The calculation of lag coefficients is somewhat
mathematically complicated and is as follows:
6(1 - )2 + (6 - 5 ) + 2 2
T1= 5 (4)
2(1 - )2
6(1 - )2 + 2(5 - 4 ) + 2 2
T2= 6 (5)
2(1 - )2
2(1 - )2 + (4 - 3 ) + 2 2
T3= 7 (6)
2(1 - )2
An example of exponential smoothing forecasting can be made using the seasonal factors in
figure 2-13. If the initial smoothing statistics, after smoothing in the December 1971, were
  
S t (X)= 184,000 , S 1 (X)= 185,000 , and S t (X)= 190,000 8
and the smoothing constant =.1 and January seasonal factor equals .80, the January 1972
forecasts would be:
X t+1 = t+1 ( S t (x))= .80(190,000) = 152,000 9
X t+1 = t+1 ( d 1 S t (X) - d 2 ( S t (X))
ˆ  
= .80(2.1111 (190,000)- 1.1111(85,000))10 (8)
= .80(401,10 - 205,553)= 156,444
X t+1 = t+1 ( T 1 S X - T 2 S X + T 3 S X)
ˆ   
t t t
= .80(3.3456 x 190,000 - 3,58025 x 185,000 + 1.23457 x 184,000)11
= .80(633,56 - 662,344+ 227,160)= .80(198,39 = 158,705
If actual sales for January 1972 were 159,097, to make a forecast for February 1972, the
forecasting program would smooth in the January sales:
S t = + (1 - )( S t (X))
) + .9(190,000 12 (10)
= 19,887 + 171,000= 190,887
S t (X)= S t (X)+ (1 - ) S t -1 (X)
  
= .1(190,887 + (185,000) 13 (11)
= 19,089+ 166,500= 185,589
S t (X)= S t (X)+ (1 - ) S t -1 (X)
  
= .1(185,589 + .9(184,000 ) 14 (12)
= 18,559+ 165,000= 184,159
February forecasts would be,
.82(190,88 = 156,527 15
.82[(2.11111)(190,887) - (1.11111)(185,589)] = .82(1996,774)
68)(190,88 - 3.580251(1
.82[(3.345 7) 85,589) + 1.23457(184,159)] = 165,27117
Actual sales in February 1972 were 191,424.
To forecast March 1972 sales, the program would smooth in the February sales and then
make a new forecast. For April, (Xt+2), the lag coefficients d1, d2, T1, T2, and T3 would be calculated
Gardner (1985) discusses various methods used to estimate initial smoothing statistic
values. Makridakis and Hibon (1991) have looked at the effect that initial smoothing statistic values
have on forecasting accuracy.
Alpha is selected by a simulation method. Various alphas between .01 and .99 are used to
forecast an initial period. Usually, the same years used to calculate seasonal factors are also used
to determine which alpha predicts best. The criterion for determining the alpha is accuracy. Usually
the constant linear and quadratic equations are tested with various alphas. The forecast with the
lowest error determines which alpha and model to use for future forecasts. The larger the alpha, the
greater the weight of recent data and the less the weight of earlier data. The formula for determining
the weight of data (past months) is (1-) , where K is the data's age. The weight of this month is
equal to in the smoothing statistic since k=1; the weight of this month in the smoothing statistics is
(1-). Theoretically, this way of calculating smoothing statistics is appropriate because the most
recent data have the greatest impact on the forecast.
Figure 3-16 Here
Figure 3-16 shows the weights for various values of alpha. Note how quickly the effects of
past data diminish as alpha increases. When alpha equals .1, the data affect the smoothing statistic
for eighteen months. This is in contrast to an alpha of .3, which only uses nine months of data to
calculate the statistic.
The initial values of the smoothing statistic also are often determined with the initial data
from which seasonal factors were calculated. The procedure is to start each of the three smoothing
statistics at the earliest month's sales value and then to smooth in actual sales up to the period for
which forecasts will be made. The resulting smoothing statistics are used to start forecasting.
Another way of establishing smoothing statistics is to use the average monthly sales for all three
smoothing statistics as the initial smoothing statistic values. A third procedure is to forecast
backwards the past values from a point in time--this is called backcasting. The firm then forecasts
from the last backcasted value to the present value. The forecasting system updates smoothing
values ('s) each time new forecasts are made.
Broze and Me'lard (1990) have suggested a maximum likelihood approach to estimating
alpha. The alpha used is an important part of building an accurate forecasting model. Research has
been done to determine the best alpha level (for example, see Newbold and Bos  and Gardner
If a forecast does not have an average error value of zero over time, the forecast is biased.
The forecast is consistently either an over-forecast or an under-forecast. Adding or subtracting a
constant increases forecasting accuracy. Exponential smoothing models often include the addition
of an exponentially smoothed error term to the forecast to correct bias. The process smooths the
errors, just as the actual data is smoothed. An error smoothing statistic is calculated. Each time a
forecast is made, an error is calculated by subtracting the actual forecast, and the error is smoothed
into the old error smoothing statistic. This error smoothing statistic is then added to the next forecast.
If the model underforecasts, the errors are positive and the error smoothing statistic is also positive;
so the next forecast will be increased. If the forecasts are consistently higher than the actual, the
errors are negative, and the error smoothing statistic is negative. Thus, adding the error smooth
statistic will reduce the next forecast. Usually only the constant (single smoothed) error smoothing
statistic is calculated, and usually a small alpha, such as .1, is used. When a forecast is biased over
time, it is most often or usually because the