Business Forecasting

Chapter 6Adaptive Filtering as a Forecasting Technique

Chapter Topics

Introduction How the Model is Used in Forecasting Chapter Summary

Introduction

The adaptive filtering approach also depends heavily on historical observations.

More complicated data patterns such as those with cycles are easily accommodated.

The model makes it possible to learn from past errors and correct for it before making a final forecast.

Introduction

In both the moving average and the exponential smoothing models, the forecaster depended on selecting the right value for the smoothing constant or the weight.

In the adaptive filtering technique, the model seeks to determine the “best” set of weights that needs to be used in making a forecast.

Introduction

The model is capable of generating information about past inaccuracy and corrects itself.

Basic Elements of the Technique

The technique has two distinct phases: The first phase is the adapting or training

of a set of weights with historical data. The second phase uses these weights to

make a forecast. As in the previous methods, we

compute the error for the forecast. The weights are adjusted based on the

errors, and a new forecast is made.

Basic Elements of the Technique

Note how we forecast for period 5, for example, using the weights:

The error will be computed as:

142332415 YwYwYwYwY

555 YYe

Adapting and Training Process Hypothetical Data Used in Adjusting the Weights by Adaptive Filtering Time Observed Data Weights Forecast

1

2

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5

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13

1Y

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

11Y

12Y

1

2

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w

w

w

w

1

2

3

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w

w

w

w

5Y

13Y

Training Phase

In the training phase we use the following equation to adjust or revise the weights:

12' /2 ittii Yyekww

th weightrevised The

:where' iwi

Training Phase

= the old i th weight iw

k = a constant term referred to as the learning constant

2/ yet= the standardized error of forecast in the period

1 itY = the observed value at period t – i +1i = 1, 2, …, p (p = number of weights)

t = p +1, p +2, … , N (N = number of observations)

y = the largest of the most recent N values of Y.

Training Phase

As you can see, the revised set of weights is equal to the old set of weights plus some adjustments made for the error.

The adjustment for each weight is based on:

The observed value The value of the learning constant k The error for that forecast.

Training Phase

The learning constant allows the weights to be changed automatically as the time series changes its patterns.

The steps for this adjustment process involves:

Specifying the number of weights Specifying the learning constant (k).

Training Phase

There are at least two ways of assigning the initial weights:

Forecaster uses his/her judgment to assign the weight.

A statistical approach is used to determine the weight as shown below:

whereNwi /1

N = the number of observations in the data series.

Training Phase

For quarterly data, N would equal 4 and for monthly data, N would equal 12

The minimum number of weights that can be used in adaptive filtering is two.

The learning constant (k) has a value between 0 and 1.

Training Phase

If we choose a higher value for k, then we can expect rapid adjustment to changes in patterns, making it difficult to find the optimal weight.

If we select a small value for k, the number of iterations needed to reach optimal weights may increase significantly.

Training Phase

As a rule, k can be set to equal to 1/p where p is the number of weights.

Alternative values of k are used, and the one that has the smallest standard error is selected.

When data have a great deal of variation, a smaller k is recommended.

ExampleTable 6.3 Quarterly Production Output Period Year Quarter Output 1 2005 I 100 2 II 93 3 III 113 4 IV 125 5 2006 I 110 6 II 95 7 III 124 8 IV 136 9 2007 I 125 10 II 98 11 III 129 12 IV 142

Example

Step 1: Graph the data to see what observations can be made from the scatter plot.

Example

60

80

100

120

140

160

I2005

II III IV I2006

II III IV I2007

II III IV

Time

Pro

duct

ion

Example

Step 2: Select the weight. Given that we have quarterly data, the weight will be:

1/n =0.25 for each quarter.Step 3:

Since there is seasonality in the data, we may want to give more weight to the quarter prior to the one for which the forecast is being made.

Example

Suppose we arbitrarily give the following weights for each quarter:Q1 = 0.2Q2 = 0.2Q3 = 0.2Q4 = 0.4

Example

Step 4: The forecast for the first quarter of 2006 will be:

107)102)(4.0()93)(2.0()113)(2.0()125)(2.0(5 Y

Step 5: The error associated with this forecast will be:

Error = Observed value in period 5 – Forecast in period 5

= 110−107= 3.0

Example

Step 6: Compute a new adjusted (or optimized) set of weights.To do this, we have to know the learning constant (k).We can set k to equal 1/p, which in this case would be 0.25, since we have used four weights.

Example

Step 7: Compute a new adjusted set of weights:

2120.0125 125/107110)25.0(22.0 2'1 w

2108.0113 125/107110)25.0(22.0 2'2 w

1999.093 125/107110)25.0(22.0 2'3 w

4098.0102 125/107110)25.0(24.0 2'4 w

Example

Step 8: The forecast for period 6, the second quarter of 2006, is now based on the new weights that we have just computed.

110)93)(4098.0()113)(1999.0()125)(2108.0()110)(2120.0(6ˆ Y

Example

Step 9: This process of using the newly adjusted weights is used to compute the forecasts for the subsequent quarters (Table 6.4).

ExampleTable 6.4 Forecast of Quarterly Production and Adjusted Weights for the 12 Periods Period Year/Quarter Output Forecast

t tY tF 1W 2W 3W 4W 1 2005: I 102 2 II 93 3 III 113 4 IV 125 0.2 0.2 0.2 0.4 5 2006: I 110 107 0.2120 0.2108 0.1999 0.4098 6 II 95 110 0.1579 0.1493 0.1443 0.3641 7 III 124 91 0.2594 0.2669 0.2779 0.4849 8 IV 136 149 0.2090 0.2283 0.2332 0.4341 9 2007 I 125 127 0.2030 0.2228 0.2327 0.4336 10 II 98 126 0.1093 0.1209 0.1398 0.3624 11 III 129 90 0.2132 0.2534 0.2840 0.4939 12 IV 142 155 0.1678 0.2189 0.2400 0.4461

Example

Step 10:Refine the weights by substituting those weights that are generated at the end of the training period:

1678.0'1 w

2189.0'2 w

2400.0'3 w

4461.0'4 w

Example

Step 11:Using the computer program provided with this book, this process is continued until we have minimized the standard error and no more reduction is noted with repeated iterations.

ExampleAdjusted Weights and the Standard Error after 200 Iterations Iteration Standard Number Error 1W 2W 3W 4W

1 22.4013 0.1682 0.2188 0.2454 0.4419

2 20.8818 0.1213 0.2179 0.2771 0.4712

3 20.2727 0.0766 0.2121 0.3063 0.5023

4 19.7485 0.0344 0.2021 0.3331 0.5353

5 19.2626 -0.0051 0.1886 0.3572 0.5698

…… ……. …… …. …… …..

120 4.6367 −0.0990 0.1435 −0.3307 1.3484

130 4.4662 −0.1082 0.1694 −0.3328 1.3451

140 4.4153 −0.1227 0.1804 −0.3274 1.3507

150 4.4179 −0.1321 0.1785 −0.3250 1.3613

Example

Step 11:The forecast for period 13 will be:

60.142)125)(3507.1()98)(3274.0()129)(1804.0()142)(1227.0(13 Y

Chapter Summary

The adaptive filtering approach depends heavily on historical observations.

But, in this technique more complicated data patterns such as those with cycles are easily accommodated.

Additionally, adaptive filtering makes it possible to learn from past errors and correct for it before making a final forecast.

Chapter Summary

In comparison with the moving average and the exponential smoothing models, the adaptive filtering model seeks to determine the “best” set of weights that needs to be used in making a forecast.

The model is capable of generating information about past inaccuracy and corrects itself.

Chapter Summary

The technique has two distinct phases. The first phase is the adapting or training of a set of weights with historical data, and the second is to use these weights to make a forecast.

We discussed how the weights are selected at the beginning and how finally the revised weights are used in making a forecast.

Chapter Summary

Two important elements of the technique are the selection of the weights and the learning constant (k).

Use of the computer program specially designed for this chapter makes the process of forecasting very easy.