The Exponential Smoothing Model 3/3/03

This model has received widespread acceptance among American business

firms that employ sales forecasts for managerial planning and control. Most

computer software packages for inventory control use exponential smoothing for

forecasting. Exponential smoothing is generally the most accurate of the time

series forecasting models.

Exponential smoothing models use special weighted moving averages and a

seasonal factor that is multiplied by the weighted moving average to calculate the

forecast. These weighted moving averages are referred to as smoothing statistics.

The exponential smoothing models are an extension of the running average model.

Generally, exponential smoothing uses three smoothed statistics that are

weighted, so that the more recent the data, the more weight given the data in

producing a forecast. These three averages are referred to as single, double, and

triple smoothing statistics and are running averages that are weighted in an

exponential declining method.

Most forecasting systems use three separate forecasting equations: one

model called a constant model, a second called a linear model, and a third called a

quadratic model. The constant forecast uses only the single smoothed statistic and

is best when the time series has little trend. The linear forecast model uses the

single and double smoothed statistics and is best when there is a linear trend in the

time series. The quadratic forecast model uses all three statistics--single, double,

and triple smoothed.

A forecasting system using the exponential smoothing model continually

evaluates forecasting accuracy and selects the forecasting equation with the most

accurate forecasting history. If the equation no longer forecasts as accurately as

one of the other equations because the environment has changed, the system

automatically switches to the equation that will forecast best. This automatic

switching of equations is called adaptive forecasting. Other adapting techniques

sometimes built into the model are adapting the weighing scheme (changing the

smoothing constant used) and smoothing past errors into future forecasts to reduce

bias. If a forecast is consistently underforecasted, adding the average past error

into the forecast, will eliminate the bias. An example of exponential smoothing is

given in both table 3-6 and Table 3-7.

Table 3-6 is the retail department store sales for Phoenix from January 1974

to December 1980. These values are used to establish an exponential smoothing

model to forecast future retail department store sales. Table 3-7 shows the

forecast's results. When the actual sales for 1981 are compared with the

forecasted sales, it is clear that the model can accurately forecast future

department store sales.

Table 3-6

DATA USED TO BUILD THE MODELS

(Phoenix Department Store Sales)

Date Actual Sales Date Actual Sales

JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctober NovemberDecember

JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember

JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember

273562616232315343353626732430305223392129202343803913262252

273562667833820330953761633733314023389530903372994341274401

31884312823824239452398833820437335375633527440087

JulyAugustSeptemberOctoberNovemberDecember

1978JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember

1979JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember

1980January

40222 40880 39368 44684 54551 94653

40454 38627 49760 47368 49245 49616 44119 46646 45027 49026 63309106068

47214 45922 56728 58598 56725 55832 52031 57071 54709 58701 74400119907

52168

JanuaryFebruaryMarchAprilMayJune

4823984307

347383630044711439174276540288

FebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember

53264 59909 60377 62341 56022 53273 58744 55393 62947 74187124151

Table 3-7

Department Store Forecast and Actual Sales

Actual Forecasted Error%Error sales sales Xt+1

Jan 54169.0 54309.4 140.4 0.26Feb 54739.0 52566.0 -2173.0 -3.97Mar 69447.0 65526.8 -3920.2 -5.64Apr 69628.0 67644.1 -1983.9 -2.85May 68359.0 68500.5 141.5 0.21Jun 64448.0 63625.9 -822.1 -1.28Jul 59743.0 59907.5 164.5 0.28Aug 63727.0 63641.4 -85.6 -0.13Sep 58453.0 59163.4 710.4 1.22Oct 67251.0 65978.7 -1272.3 -1.89Nov 78180.0 78506.6 326.6 0.42

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[1](X) would be calculated by taking .3X + (1 - .3) times St-1

[1](X)

= St[1](X). Note that St-1

[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 S t[1]X is the current statistic. This is a

convenient notational technique to designate what statistic is being referred to. The

[1] in St[1] does not refer to an exponential power, but rather identifies that this is the

single smooth statistic.

St[2](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[2](X) is multiplied by (1-) and added to

St-1[1](X), giving St

[2](X). The box 2, [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[2](X) to (1-) St-1

[3](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 calculated

1 (2)

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 isX t+1 = t+1(St[1](X)). Figure 3-15 shows the

patterns of time series forecasted best by the constant equation.

The linear equation isX t+1 = t+1[d1St[1](X)-d2St

[2]]. The patterns that this

equation models best are those showing a linear trend. In the equation, d1 and d2

are lag factors. The smoothing statistics lag the actual values. The value of d1 is

2. The lag for d2 is 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:

4 (3)

T1, T2, and T3 are lag correction variables. The calculation of lag coefficients

is somewhat mathematically complicated and is as follows:

5 (4)

6 (5)

7 (6)

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

8

and the smoothing constant =.1 and January seasonal factor equals .80, the

January 1972 forecasts would be:

Constant:

9 (7)

Linear:

10 (8)

Quadratic:

(9)

11

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:

12 (10)

13 (11)

14 (12)

February forecasts would be,

Constant:

15

Linear:

16

Quadratic:

17

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 with =2.

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.

Selecting Alpha

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-)k, 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

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 [1989] and Gardner [1985]).

CORRECTING BIAS

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 wrong model or wrong parameter

values are used. Thus the forecaster should use a different model or different

parameter values when a bias exists. However, when the bias is small, the

forecaster may have the best model and best parameter values. The error

correction smoothing statistic makes minor adjustments that modestly improve

forecasts. Cipra (1992) has suggested a method of making exponential smoothing

robust to outliers which reduces some bias errors.

Advantages of Exponential Smoothing Forecasting

The advantages of the exponential model are many. Although the process

is somewhat tedious when done by hand, the process is straightforward and easy

when programmed for a computer. Many computer programs have been written

that use exponential smoothing, and they are readily available. Chapter 11

discusses many computer programs that use exponential smoothing forecasting

models. Exponential smoothing models adapt to environmental changes and are

self-correcting. The firm with many products needs only one program with three

equations to forecast all its products' sales. The procedure has proved effective for

a wide spectrum of product forecasts. Only past data are used; thus the problem

and expense of collecting external data are eliminated.

SUMMARY

Time series data contain lots of information about the sales processes of the

company. Some factors hidden in the data are seasonality, trend, and cyclicality of

the sales. Seasonality is dependant on the characteristics of the product and the

people buying it. Managers attempt to decrease the seasonality of sales. A time

series forecast can mathematically explain products' seasonality and show how the

seasonality changes over time. Trends and cycles also occur with sales. Trends

can be modeled as linear, segmented to linear, or nonlinear. (Cyclical sales can be

modeled by an equation using sine or cosine functions).

These sales patterns make up the underlying process of sales for the

company. Each component of the sales pattern can be detected with the time

series data detection which more fully describes the underlying process of sales.

Three approaches to breaking the time series data into components are (1) to plot

the data and estimate the components, (2) to do a spectral analysis on the data,

and (3) to use ARIMA forecasting models (see Chapter Four).

Two models explained in the chapter are the running average forecasting

technique and the exponential smoothing model. The running average forecast

calculates seasonal factors, which are multiplied by the average sales of the

previous three months to determine the monthly sales forecast. A trend factor is

used to increase the accuracy of the running average method.

The exponential smoothing technique is a self-adjusting weighted running

averages model that uses a special weighted moving average factor and the

seasonal factor to calculate the forecast. The three types of exponential smoothing

forecasts are the constant forecast, the linear forecast, and the quadratic forecast.

Each uses a different equation with a different weighted moving average

(smoothing statistic). Alpha values and bias correction factors adjust exponential

smoothing forecasts.