Business Forecasting Chapter 5 Forecasting with Smoothing Techniques

Business Forecasting

Chapter 5Forecasting with Smoothing

Techniques

Chapter Topics

Introduction Naïve Model Forecasting with Averaging Models:

Simple Average Model Moving Average Model Double Moving Average Model

Exponential Smoothing: Double Exponential Smoothing

Exponential Smoothing (Continued)

Holt’s Method

Triple Exponential Smoothing

Winters’ Seasonal Exponential Smoothing

Introduction

Elaborate statistical models are not always required to develop accurate forecasts.

The principle of parsimony suggests that the simpler the model the better.

The main advantage of simple models is that they serve as a benchmark with which to gauge applicability, reliability, and necessity of the more sophisticated models.

Introduction

These time series models are good tools in forecasting short-term events.

The cardinal premise underlying all time series models is that the historical pattern of the dependent variable can be used as the basis for developing forecasts.

In these models, historical data for the forecast variable are analyzed in an attempt to discern any underlying pattern(s).

Introduction

Time series or autoregressive forecasting models will be most useful when economic conditions can be expected to remain relatively stable.

Reliance of time series models on analysis and extrapolation of historical patterns carries several important implications with respect to technique selection:

Introduction

1. Time series are best when applied to short-term forecasts.

2. Time series models prove most satisfactory when historical data contain either no systematic data pattern or when the changes are occurring very slowly or consistently.

3. Data requirements and ease of implementation are a function of the specific time series technique selected.

Naïve Model Uses recent past as the best indicator of

the future.

The error associated with this model is computed as:

tt YY 1ˆ

ttt YYe ˆ

Example for the Naïve Model

Week Sales (in $1,000)

Forecast

1 9 -

2 8 9

3 9 8

5 12 9

6 9 12

7 12 9

8 11 12

9 ? 11

Example for the Naïve ModelWeek Sales (in

$1,000)Forecast Error Absolute

ErrorSquared

Error

1 9

2 8 9 −1 1 1

3 9 8 1 1 1

5 12 9 3 3 9

6 9 12 −3 3 9

7 12 9 3 3 9

8 11 12 −1 1 1

Sum 2 12 30

Mean 0.33 2 5

Example for the Naïve Model What you should keep in mind is that,

although MAD is often used as the measurement of error in evaluating a forecast, an alternative criterion is the MSE.

Note that the difference between MAD and MSE is that the latter penalizes a forecast much more for extreme deviations than it does for small ones.

Example for the Naïve Model Whenever a manager evaluates

alternative forecasting techniques in terms of their accuracy, it is necessary to go beyond the computation of error.

Managers are generally concerned with two forms of accuracy: Accuracy of the technique in predicting the

underlying patterns or relationship of past data.

Accuracy of the changes in the pattern. That is, how fast forecasting procedure can respond to that basic change. (We will discuss this in later chapters.)

Averaging Models

The basic premise of these models is that a weighted-average of past observations can be used to smooth the fluctuations in the data in the short term.

Simple Average Model

Similar to the naïve model, this model uses part of the historical data to make a forecast.

n

YY

n

tt

t

1

1ˆ

Moving Average Model

Recent observations play an important role in the forecast.

As new observations become available, a new average is computed.

The choice of using a smaller or larger number of observations has implications for the forecast.

Double Moving Average Model

Used when we have a linear trend in the data.

Two different moving averages are computed in this model.

The idea is to remove the trend.

Double Moving Average Model

Week Sales (in $1,000)

Simple Moving Average

Simple Moving AverageForecast

Double Moving Average

1 9

2 11 10

3 10 10.5 10 10.25

5 14 12 10.5 11.25

6 18 16 12 14

7 22 20 16 18

8 23 22.5 20 21.25

Exponential Smoothing Model

The model relies on the assumption that the data are stationary.

Most recent observations play a more important role than the distant past.

33

22

11 )1()1()1( ttttt YYYYY


The model depends on three pieces of data: Most recent actual Most recent forecast Smoothing constant.

The value of alpha assigned as a smoothing constant is critical to the forecast.

The best alpha should be chosen on the basis of minimal sum of error squared.


Several approaches are followed in selecting the smoothing constant. If a great amount of smoothing is desired, a

small alpha should be chosen. The choice of alpha is affected by the

characteristics of the time series. If sharp ups and downs are noticed in the data, the best smoothing constant is 0.1. That is alpha chosen should equal 0.1.

If the data show that the past is very different from the present, then alpha of 0.9 is appropriate.


Exponential smoothing is used for routine sales forecasting of inventory, production, distribution, and retail planning.

Double Exponential Smoothing Model

Similar to the double moving average model.

Also known as Brown’s double exponential smoothing model.

xbaY ttxt ˆ


The model is represented as:

xbaY ttxt ˆ

xtY ˆ

"A

= forecast value x periods in the future

ta = the difference between the simple

'A

and the double smoothed valuestb = slope in a time series

X = number of periods ahead to be forecasted


To compute the difference between simple and double smoothed values:

'1

' )1( ttt AYA

''1

''' )1( ttt AAA


Once we have computed the simple and double smoothed values, we then compute the intercept and the slope of the forecast line as follows:

The forecast equation is:

'''2 ttt AAa

)(1

'''ttt AAb

xbaY ttxt ˆ

Holt’s Exponential Smoothing Model

To handle linear trend, similar to the Brown’s Method.

The difference is that in this method we smooth the trend and the slope in the time series by using different constants for each.

How do we find the best combination of smoothing constant?

Low values of alpha and beta should be used when there are frequent random fluctuations in the data.

High values of alpha and beta should be used when there is a pattern such as trend in the data.

Holt’s Exponential Smoothing Model

The following equations are used when applying the Holt’s method:

))(1( 11 tttt TAYA

11 )1()( tttt TAAT

ttxt xTAY ˆ

Triple Exponential Smoothing Model

When faced with nonlinear pattern in the data, this model provides a good forecast.

The life cycle model of products, and cost structures are environments where the triple exponential smoothing should be used.

The forecasting equation is:

2)2/1(ˆ xcxbaY tttxt


In this model we have to compute three coefficients:a, b, and c .

Each of the coefficients is computed as follows: 321 33 tttt AAAa

3212

)34()810()56()1(2

tttt AAAb

)2(1

3212

tttt AAAc


You will note that the estimation of the coefficients (a, b, and c) requires us to compute three smoothing values.

11

1 )1( ttt AYA

21

12 )1( ttt AAA

31

23 )1( ttt AAA


Allows for both trend and seasonal patterns to be taken into account.

This is an extension of the Holt’s method of smoothing.

In computing the forecast, we add an equation for seasonality as an index.

The forecast model is:

xLtttxt IxTAY )(ˆ


The Winters’ model has the following components:

))(1( 11

ttLt

tt TA

I

YA

11 )1()( tttt TAAT

Smoothing value

Trend estimate

Ltt

tt I

A

YI )1( Seasonality

estimate

Chapter Summary

Discussed how the naïve model is used in forecasting.

Elaborated on the moving averages model, which included the simple moving average and the double moving average.

Discussed the exponential smoothing models of Brown, Holt, and Winters.

Identified the criteria for using the various models.

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Business Forecasting Chapter 5 Forecasting with Smoothing Techniques