Process or Product Monitoring and Control_6 ppt

Final table This is the final table:Period Value Centered MA Unfortunately, neither the mean of all data nor the moving average of the most recent M values, when used as forecasts for th

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The question arises: can we use the mean to forecast income if we

suspect a trend? A look at the graph below shows clearly that we should

not do this

6.4.2 What are Moving Average or Smoothing Techniques?

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In summary, we state that

The "simple" average or mean of all past observations is only auseful estimate for forecasting when there are no trends If thereare trends, use different estimates that take the trend into account

1

The average "weighs" all past observations equally For example,the average of the values 3, 4, 5 is 4 We know, of course, that anaverage is computed by adding all the values and dividing thesum by the number of values Another way of computing theaverage is by adding each value divided by the number of values,or

3/3 + 4/3 + 5/3 = 1 + 1.3333 + 1.6667 = 4

The multiplier 1/3 is called the weight In general:

The are the weights and of course they sum to 1

2

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6 Process or Product Monitoring and Control

6.4 Introduction to Time Series Analysis

6.4.2.1 Single Moving Average

Recall the set of numbers 9, 8, 9, 12, 9, 12, 11, 7, 13, 9, 11,

10 which were the dollar amount of 12 suppliers selected at

random Let us set M, the size of the "smaller set" equal to

3 Then the average of the first 3 numbers is: (9 + 8 + 9) /

3 = 8.667

This is called "smoothing" (i.e., some form of averaging) Thissmoothing process is continued by advancing one period and calculatingthe next average of three numbers, dropping the first number

Moving

average

example

The next table summarizes the process, which is referred to as Moving

Averaging The general expression for the moving average is

M t = [ X t + X t-1 + + X t-N+1 ] / N

Results of Moving Average

Supplier $ MA Error Error squared

The MSE = 2.018 as compared to 3 in the previous case

6.4.2.1 Single Moving Average

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6.4.2.1 Single Moving Average

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6.4.2.2 Centered Moving Average

in the middle of the time interval of three periods, that is, next to period

2 This works well with odd time periods, but not so good for even time

periods So where would we place the first moving average when M =

4?

Technically, the Moving Average would fall at t = 2.5, 3.5,

To avoid this problem we smooth the MA's using M = 2 Thus we

smooth the smoothed values!

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Final table This is the final table:

Period Value Centered MA

Unfortunately, neither the mean of all data nor the moving average of

the most recent M values, when used as forecasts for the next period, are

able to cope with a significant trend

There exists a variation on the MA procedure that often does a better job

of handling trend It is called Double Moving Averages for a Linear

Trend Process It calculates a second moving average from the original

moving average, using the same value for M As soon as both single and

double moving averages are available, a computer routine uses theseaverages to compute a slope and intercept, and then forecasts one ormore periods ahead

6.4.2.2 Centered Moving Average

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6.4.3 What is Exponential Smoothing?

weighted equally, Exponential Smoothing assigns exponentially

decreasing weights as the observation get older.

In other words, recent observations are given relatively more weight

in forecasting than the older observations.

In the case of moving averages, the weights assigned to the

observations are the same and are equal to 1/N In exponential smoothing, however, there are one or more smoothing parameters to

be determined (or estimated) and these choices determine the weightsassigned to the observations

Single, double and triple Exponential Smoothing will be described inthis section

6.4.3 What is Exponential Smoothing?

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6.4.3.1 Single Exponential Smoothing

This smoothing scheme begins by setting S2 to y1, where S i stands for

smoothed observation or EWMA, and y stands for the original observation The subscripts refer to the time periods, 1, 2, , n For the third period, S3 = y2 + (1- ) S2; and so on There is no S1; the

smoothed series starts with the smoothed version of the secondobservation

For any time period t, the smoothed value S t is found by computing

This is the basic equation of exponential smoothing and the constant or parameter is called the smoothing constant.

Note: There is an alternative approach to exponential smoothing that

replaces y t-1 in the basic equation with y t, the current observation Thatformulation, due to Roberts (1959), is described in the section on

EWMA control charts The formulation here follows Hunter (1986)

Setting the first EWMA

6.4.3.1 Single Exponential Smoothing

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The first

forecast is

very

important

The initial EWMA plays an important role in computing all the

subsequent EWMA's Setting S2 to y1 is one method of initialization.Another way is to set it to the target of the process

Still another possibility would be to average the first four or fiveobservations

It can also be shown that the smaller the value of , the more important

is the selection of the initial EWMA The user would be wise to try afew methods, (assuming that the software has them available) beforefinalizing the settings

Why is it called "Exponential"?

Expanded

equation for

S 5

For example, the expanded equation for the smoothed value S5 is:

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From the last formula we can see that the summation term shows that

the contribution to the smoothed value S t becomes less at eachconsecutive time period

-> towards past observations

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Example Let us illustrate this principle with an example Consider the following

data set consisting of 12 observations taken over time:

Error squared

The MSE was again calculated for = 5 and turned out to be 16.29, so

in this case we would prefer an of 5 Can we do better? We couldapply the proven trial-and-error method This is an iterative procedurebeginning with a range of between 1 and 9 We determine the bestinitial choice for and then search between - and + Wecould repeat this perhaps one more time to find the best to 3 decimalplaces

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6.4.3.2 Forecasting with Single Exponential

Smoothing

Forecasting Formula

Forecasting

the next point

The forecasting formula is the basic equation

This can be written as:

where t is the forecast error (actual - forecast) for period t.

In other words, the new forecast is the old one plus an adjustment forthe error that occurred in the last forecast

where y origin remains constant This technique is known as

bootstrapping.

6.4.3.2 Forecasting with Single Exponential Smoothing

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Example of Bootstrapping

Example The last data point in the previous example was 70 and its forecast

(smoothed value S) was 71.7 Since we do have the data point and the

forecast available, we can calculate the next forecast using the regularformula

= 1(70) + 9(71.7) = 71.5 ( = 1)But for the next forecast we have no data point (observation) So now

Single Exponential Smoothing with Trend

Single Smoothing (short for single exponential smoothing) is not verygood when there is a trend The single coefficient is not enough

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Sample data

set with trend

Let us demonstrate this with the following data set smoothed with an

of 0.3:

Data Fit

6.45.6 6.47.8 6.28.8 6.711.0 7.311.6 8.416.7 9.415.3 11.621.6 12.722.4 15.4

The resulting graph looks like:

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6.4.3.3 Double Exponential Smoothing

As in the case for single smoothing, there are a variety of schemes to set

initial values for S t and b t in double smoothing

S1 is in general set to y1 Here are three suggestions for b1:

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Meaning of

the

smoothing

equations

The first smoothing equation adjusts S t directly for the trend of the

previous period, b t-1 , by adding it to the last smoothed value, S t-1 This

helps to eliminate the lag and brings S t to the appropriate base of thecurrent value

The second smoothing equation then updates the trend, which isexpressed as the difference between the last two values The equation issimilar to the basic form of single smoothing, but here applied to theupdating of the trend

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6.4.3.4 Forecasting with Double

to determine an with minimum MSE) The chosen starting values are

S1 = y1 = 6.4 and b1 = ((y2 - y1) + (y3 - y2) + (y4 - y3))/3 = 0.8

For comparison's sake we also fit a single smoothing model with =0.977 (this results in the lowest MSE for single exponential smoothing).The MSE for double smoothing is 3.7024

The MSE for single smoothing is 8.8867

6.4.3.4 Forecasting with Double Exponential Smoothing(LASP)

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results for

the example

The smoothed results for the example are:

5.6 6.6 (Forecast = 7.2) 6.47.8 7.2 (Forecast = 6.8) 5.68.8 8.1 (Forecast = 7.8) 7.811.0 9.8 (Forecast = 9.1) 8.811.6 11.5 (Forecast = 11.4) 10.916.7 14.5 (Forecast = 13.2) 11.615.3 16.7 (Forecast = 17.4) 16.621.6 19.9 (Forecast = 18.9) 15.322.4 22.8 (Forecast = 23.1) 21.5

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This graph indicates that double smoothing follows the data much closerthan single smoothing Furthermore, for forecasting single smoothingcannot do better than projecting a straight horizontal line, which is notvery likely to occur in reality So in this case double smoothing ispreferred.

Finally, let us compare double smoothing with linear regression:

This is an interesting picture Both techniques follow the data in similarfashion, but the regression line is more conservative That is, there is aslower increase with the regression line than with double smoothing

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The selection of the technique depends on the forecaster If it is desired

to portray the growth process in a more aggressive manner, then oneselects double smoothing Otherwise, regression may be preferable Itshould be noted that in linear regression "time" functions as the

independent variable Chapter 4 discusses the basics of linear regression,and the details of regression estimation

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6.4.3.5 Triple Exponential Smoothing

What happens if the data show trend and seasonality?

The basic equations for their method are given by:

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