11

CHAPTER 4CHAPTER 4

MOVING AVERAGES AND MOVING AVERAGES AND

SMOOTHING METHODSSMOOTHING METHODS

(Page 107)(Page 107)

2

• They are based solely on the most recent information available.

• Is sometimes called the “no change” forecast.• Suitable for very small data sets.• The simplest model is:

tt YY 1ˆ (4.1)

NAÏVE MODELSNAÏVE MODELS

3

Method Pattern of Data

Time Horizon

Type of Model

Minimal Data Requirements

Nonseasonal Seasonal

Naïve Models ST , T , S S TS 1

Simple Averages ST S TS 30

Moving Averages ST S TS 4-20

Double Moving Averages ST S TS 2

Linear (Double) exponential smoothing (Holt’s) T S TS 3

Quadratic exponential smoothing T S TS 4

Seasonal exponential smoothing (Winter’s) S S TS 2 x s

Adaptive filtering S S TS 5 x s

Simple regression T I C 10

Multiple regression C , S I C 10 x V

Classical decomposition S S TS 5 x s

Exponential trend models T I , L TS 10

S-curve fitting T I , L TS 10

Gompertz models T I , L TS 10

Growth curves T I , L TS 10

Census X-12 S S TS 6 x s

ARIMA (Box-Jenkins) ST , T , C , S S TS 24 3 x s

Lading indicators C S C 24

Econometric models C S C 30

Time series multiple regression T , S I , L C   6 x sPattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long termType of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

4

Example 4.1 (Page 108)(Page 108)

• Table 4-1 Sales of Saws for Acme Tool Company, 2000 – 2006.

• Initialization (Fitting) Part: 2000 – 2005.• Test Part: 2006.

5

• The technique can be adjusted to take trend into consideration:

)(ˆ11 tttt YYYY (4.2)

• The rate of change might be more appropriate than the absolute amount of change:

3.4ˆ1

1

t

ttt Y

YYY

6

• An appropriate forecast equation for quarterly data:

31ˆ

tt YY (4.4)

• For monthly data:111

ˆ tt YY

• The analyst can combine seasonal and trend estimates using:

44

....ˆ 43

43131

tt

ttttt

tt

YYY

YYYYYY (4.5)

7

Example 4.1 (cont.)

• Using Equations (4.3, 4.4, and 4.5)

(Pages 110 – 111)(Pages 110 – 111)

88

Forecasting Methods Forecasting Methods Based on AveragingBased on Averaging

(Page 111)(Page 111)

9Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long termType of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

Method Pattern of Data

Time Horizon

Type of Model

Minimal Data Requirements

Nonseasonal Seasonal

Naïve Models ST , T , S S TS 1

Simple Averages ST S TS 30

Moving Averages ST S TS 4-20

Double Moving Averages ST S TS 2

Linear (Double) exponential smoothing (Holt’s) T S TS 3

Quadratic exponential smoothing T S TS 4

Seasonal exponential smoothing (Winter’s) S S TS 2 x s

Adaptive filtering S S TS 5 x s

Simple regression T I C 10

Multiple regression C , S I C 10 x V

Classical decomposition S S TS 5 x s

Exponential trend models T I , L TS 10

S-curve fitting T I , L TS 10

Gompertz models T I , L TS 10

Growth curves T I , L TS 10

Census X-12 S S TS 6 x s

ARIMA (Box-Jenkins) ST , T , C , S S TS 24 3 x s

Lading indicators C S C 24

Econometric models C S C 30

Time series multiple regression T , S I , L C   6 x s

10

Simple AveragesSimple Averages• Uses the mean of all the relevant

historical observations as the forecast of the next period.

• New observation is added:

t

iit Y

tY

11

1(4.6)

111

2

t

YYtY ttt

(4.7)

(Page 111)(Page 111)

11

Is used for stabilized series, and the environment is generally unchanging.

Method Pattern of Data

Time Horizon

Type of Model

Minimal Data Requirements

Nonseasonal Seasonal

Simple averages ST S TS 30

Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long termType of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

12

Example 4.2

Week t Purchases Week t Purchases Week t Purchases

1 275 11 302 21 310

2 291 12 287 22 299

3 307 13 290 23 285

4 281 14 311 24 250

5 295 15 277 25 260

6 268 16 245 26 245

7 252 17 282 27 271

8 279 18 277 28 282

9 264 19 298 29 302

10 288 20 303 30 285

Table 4-2 Gasoline Purchases for the Spokane Transit Authority for Example 4.2

13

Time Series Graph

Week

Purc

hase

s

302520151050

310

300

290

280

270

260

250

240

Time Series Graph

The data seems stationary.

Figure 4-3 Time Series Plot of Weekly Gasoline Purchases for the Spokane Transit Authority

14

Solution

28

1128 28

1

iiYY

2.281

28

787429 Y

8.202.281302292929 YYe

128

28 128128228

YYY

9.28129

302)2.281(2830

Y

1.39.281285303030 YYe

30

1130 30

1

iiYY

282

30

846131 Y

15

Moving Averages• A moving average of order k is the mean value

of the k most recent observations.

The method does not handle trend or seasonality very well, although it does better than the simple average method.

Method Pattern of Data

Time Horizon

Type of Model

Minimal Data Requirements

Nonseasonal Seasonal

Moving averages ST S TS 4-20

K = number of terms in the moving average.k

YYYYY kttttt

)....( 1211

(4.8)

Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long termType of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

16

Example 4.3 (last data)

Week t Purchases Week t Purchases Week t Purchases

1 275 11 302 21 310

2 291 12 287 22 299

3 307 13 290 23 285

4 281 14 311 24 250

5 295 15 277 25 260

6 268 16 245 26 245

7 252 17 282 27 271

8 279 18 277 28 282

9 264 19 298 29 302

10 288 20 303 30 285

17

CalculationsUsing a five-week moving average (Page 114, 115)

Minitab can be used (See the Minitab Applications section for instructions, Pages: 159-161Pages: 159-161)

18

ResultsWeek Purchases AVER1

RESI11 275 * *2 291 * *3 307 * *4 281 * *5 295 289.8 *6 268 288.4 -21.87 252 280.6 -36.48 279 275.0 -1.69 264 271.6 -11.010 288 270.2 16.411 302 277.0 31.812 287 284.0 10.013 290 286.2 6.014 311 295.6 24.815 277 293.4 -18.616 245 282.0 -48.417 282 281.0 0.018 277 278.4 -4.019 298 275.8 19.620 303 281.0 27.221 310 294.0 29.022 299 297.4 5.023 285 299.0 -12.424 250 289.4 -49.025 260 280.8 -29.426 245 267.8 -35.827 271 262.2 3.228 282 261.6 19.829 302 272.0 40.430 285 277.0 13.0

*****289.8288.4280.6275.0271.6270.2277.0284.0286.2295.6293.4282.0281.0278.4275.8281.0294.0297.4299.0289.4280.8267.8262.2261.6272.0

tY

19

Index

Purc

hase

s

30272421181512963

340

320

300

280

260

240

220

Moving AverageLength 5

Accuracy MeasuresMAPE 7.503MAD 20.584MSD 622.149

Variable

Forecasts95.0% PI

ActualFits

Moving Average Plot for Purchases

Minitab Results

Note: (MSE is called MSD on Minitab output)FIGURE 4 - 4

(Page 115)(Page 115)

Minitab InstructionsMinitab Instructions

Stat > time Series > Moving averagesStat > time Series > Moving averages

20

654321

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

Aut

ocor

rela

tion

LBQTCorrLag

37.46

35.74

21.81

10.13

7.39

7.21

-0.61

-2.06

-2.31

-1.21

0.32

2.53

-0.22

-0.64

-0.60

-0.30

0.08

0.51

6

5

4

3

2

1

Autocorrelation Function for the Residuals

The series is nonrandom

Task: Try a nine-week moving average, it would be better, because the large-order moving average pays very little attention to the large fluctuations in the data series

21

22

23

24

25

26

27

30272421181512963

340

320

300

280

260

240

220

Index

Gallo

ns

Length 5Moving Average

MAPE 7.503MAD 20.584MSD 622.149

Accuracy Measures

ActualFitsForecasts95.0% PI

Variable

Moving Average Plot for Gallons

28

29

* ** ** ** ** *289.8 -21.8288.4 -36.4280.6 -1.6275.0 -11.0271.6 16.4270.2 31.8277.0 10.0284.0 6.0286.2 24.8295.6 -18.6293.4 -48.4282.0 0.0281.0 -4.0278.4 19.6275.8 27.2281.0 29.0294.0 5.0297.4 -12.4299.0 -49.0289.4 -29.4280.8 -35.8267.8 3.2262.2 19.8261.6 40.4272.0 13.0

30

Double Moving Average

• Equations (4.9) – (4.12) (Pages 116-117)

• Example 4-4

• (Pages 118-119)

3131

Exponential Smoothing Exponential Smoothing

Methods Methods

32

2.1 Simple (Single) Exponential Smoothing

MethodPattern of Data

Time Horizon

Type of Model

Minimal Data Requirements

Nonseasonal Seasonal

Single Exponential smoothing ST S TS 2

Based on averaging (smoothing) past values of a series in a decreasing exponential manner, with more weight being given to the more recent observations.

New forecast = [α x (new observation)] + [(1- α) x (old forecast)]

= smoothing constant (0< <1) ttt YYY)1(1

33

Comparison of Smoothing Constants

ttt YYY)1(1 11 )1( ttt YYY

])1()[1( 111 tttt YYYY

12

1 )1()1( ttt YYY

23

22

1 )1()1()1( tttt YYYY

221 )1( ttt YYY

Period

= 0.1

=0.6Calculations Weight Calculations Weight

t 0.1 0.1 0.6 0.6t-1 0.1 x 0.9 0.09 0.6 x 0.4 0.24t-2 0.1 x 0.9 x 0.9 0.081 0.6 x 0.4 x 0.4 0.096

t-3 0.1 x 0.9 x 0.9 x 0.9 0.073 0.6 x 0.4 x 0.4 0.038

t-4 0.1 x 0.9 x 0.9 x 0.9 x 0.9 0.066 0.6 x 0.4 x 0.4 x 0.4 0.015

All others 0.59 0.011

Totals 1.0 1.0

34

Starting the algorithm

An initial value for the old smoothed series must be set:

To set the first estimate = the first observation.

Another method: To use the average of the first 5 or 6 observations.

35

2000 1 500* 2 350* 3 250* 4 4002001 5 450* 6 350* 7 200* 8 3002002 9 350* 10 200* 11 150* 12 4002003 13 550* 14 350* 15 250* 16 5502004 17 550* 18 400* 19 350* 20 6002005 21 750* 22 500* 23 400* 24 6502006 25 850

Year QuarterstY

• The actual sales for a Company for the years 2000 to 2006 are demonstrated in the Table.

• The data for the first quarter of 2006 will be used as the test part to help determine the best value of α among the two considered.

Example 4.5Example 4.5

36

Results

2000 1 500* 2 350* 3 250* 4 400

Year Quarters tY tY

(α=0.1)

500.000 1) 0.000500.000 -150.000485.000 2) -235.000 3)

461.500 4) -61.500

te

1) Initial value for the smoothed series = first observation = 5001Y

1Y

2)ttt YYY)1(1 2212 )1( YYY

485500)1.01()350(1.03 Y

333 YYe

250-485 =-2353)

4Y

4) =0.1(250)+0.9(485)=461.5

37

Results500.000 1) 0.000500.000 -150.000485.000 2) -235.000 3)

461.500 4) -61.500455.350 -5.350454.815 -104.815444.334 -244.334419.900 -119.900407.910 -57.910402.119 -202.119381.907 -231.907358.716 41.284362.845 187.155381.560 -31.560378.404 -128.404365.564 184.436384.007 165.993400.607 -0.607400.546 -50.546395.491 204.509415.942 334.058449.348 50.652454.413 -54.413448.972 201.028469.075 5)

te500.000 0.000500.000 -150.000410.000 -160.000314.000 86.000365.600 84.400416.240 -66.240376.496 -176.496270.598 29.402288.239 61.761325.296 -125.296250.118 -100.118190.047 209.953316.019 233.981456.408 -106.408392.563 -142.563307.025 242.975452.810 97.190511.124 -111.124444.450 -94.450387.780 212.220515.112 234.888656.045 -156.045562.418 -162.418464.967 185.033575.987 5)

Year Quarters

tY tY

(α=0.6)tY

(α=0.1) te

5) The calculation for the first quarter of 2007

2000 1 500* 2 350* 3 250* 4 4002001 5 450* 6 350* 7 200* 8 3002002 9 350* 10 200* 11 150* 12 4002003 13 550* 14 350* 15 250* 16 5502004 17 550* 18 400* 19 350* 20 6002005 21 750* 22 500* 23 400* 24 6502006 25 850

38

Single Exponential Smoothing (α=0.1)

Index

Y

24222018161412108642

800

700

600

500

400

300

200

100

Smoothing ConstantAlpha 0.1

Accuracy MeasuresMAPE 38.9MAD 127.0MSD 24261.7

Variable

Forecasts95.0% PI

ActualFits

Single Exponential Smoothing Plot for Y

39

Single Exponential SmoothingSingle Exponential Smoothing (α=0.6)

Index

Y

24222018161412108642

1000

900

800

700

600

500

400

300

200

100

Smoothing ConstantAlpha 0.6

Accuracy MeasuresMAPE 36.5MAD 134.5MSD 22248.4

Variable

Forecasts95.0% PI

ActualFits

Single Exponential Smoothing Plot for Y

40Index

Y

24222018161412108642

900

800

700

600

500

400

300

200

100

Smoothing ConstantAlpha 0.266357

Accuracy MeasuresMAPE 32.2MAD 117.5MSD 19447.0

Variable

Forecasts95.0% PI

ActualFits

Single Exponential Smoothing Plot for Y

α = 0.266

4141

OptimizationMAPE 32.2MAD 117.5MSD 19447.0

Comparisonαα = 0.6 = 0.6MAPE 36.5MAD 134.5MSD 22248.4

αα = 0.1 = 0.1 MAPE 38.9MAD 127.0MSD 24261.7

Initial smoothed Value = The first observation

Initial Smoothed Value = The Average of the first six Observations

αα = 0.1 = 0.1

MAPE 32.1MAD 115.5MSD 21091.2

αα = 0.6 = 0.6MAPE 36.7MAD 137.1MSD 22152.8

The weight α is selected subjectively or by minimizing an error such as the MSE

4242

654321

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

Aut

ocor

rela

tion

LBQTCorrLag

33.86

23.65

23.65

10.59

10.24

0.40

-1.65

-0.02

2.40

0.41

-2.84

0.59

-0.54

-0.01

0.65

0.11

-0.59

0.12

6

5

4

3

2

1

Autocorrelation Function for Residuals

• Large residual autocorrelations at lags 2 and 4:• Seasonal Variation in the data is not accounted for by simple exponential method.• The large value of LBQ (33.86): series is nonrandom.

43

MethodPattern of Data

Time Horizon

Type of

Model

Minimal Data Requirements

Nonseasonal Seasonal

Linear (Double) exponential smoothing (Holt’s) T S TS 3

Exponential Smoothing Adjusted for Exponential Smoothing Adjusted for

Trend:Trend:

(Holt’s Method)(Holt’s Method) “Holt’s “Holt’s two-parameter method”two-parameter method”

Smoothes the level and slope (trend) using different constants.

Double Exponential Smoothing Double Exponential Smoothing

44

Equations Used:))(1( 11 tttt TLYL

11 )1()( tttt TLLT

ttpt pTLY

1. The current level estimate:

2. The trend estimate:

3. Forecast p periods in the future.

Lt = new smoothed value.

α = smoothing constant for the data.

β = smoothing constant for trend estimate.

Yt = Actual value of series in period t.Tt = trend estimate.

p = periods to be forecast into the future.

= forecast for p periods into the future.

0 ≤ α and β ≤ 1.

ptY

45

Starting the algorithm The weights can be selected as in the single

exponential smoothing method. A grid of values could be developed, then

selecting the ones producing the lowest MSE. To begin the algorithm: One approach is to set

the first estimate equal to the first observation, the trend is then estimated to equal zero. A second approach is to use the average of the first six observations , the trend is the slope of a line fit to these observations.

Minitab develops a regression equation, and uses constants from the equation as initial estimates for the level and the trend.

4646

Example 4.9Example 4.9 (Last data)

2000 1 500* 2 350* 3 250* 4 4002001 5 450* 6 350* 7 200* 8 3002002 9 350* 10 200* 11 150* 12 4002003 13 550* 14 350* 15 250* 16 5502004 17 550* 18 400* 19 350* 20 6002005 21 750* 22 500* 23 400* 24 6502006 25 850

Year QuarterstY

tY

47

Index

Y

24222018161412108642

900

800

700

600

500

400

300

200

100

Smoothing ConstantsAlpha (level) 0.3Gamma (trend) 0.1

Accuracy MeasuresMAPE 35.4MAD 125.3MSD 20515.5

Variable

Forecasts95.0% PI

ActualFits

Double Exponential Smoothing Plot for Y

Gamma = β

Comparison: Single: α = 0.266 Holts: α = 0.3, β = 0.1

MSE 19447 20515.5

MAPE 32.2 35.4

48

654321

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

Aut

ocor

rela

tion

LBQTCorrLag

36.33

23.08

22.92

11.63

11.47

0.23

-1.90

-0.22

2.18

0.27

-3.06

0.45

-0.62

-0.07

0.60

0.07

-0.63

0.09

6

5

4

3

2

1

Autocorrelation Function for Residuals

LBQ (36.33) shows that the series is nonrandom

49

Method Pattern of Data

Time Horizon

Type of

Model

Minimal Data Requirements

Nonseasonal Seasonal

Seasonal exponential smoothing(Winter’s) S S TS 2 x s

Exponential Smoothing Exponential Smoothing Adjusted for Trend and Adjusted for Trend and Seasonal Variations: Winters’ Seasonal Variations: Winters’ MethodMethod

50

11 )1()( tttt TLLT 2. The trend estimate:

stt

tt S

L

YS )1( 3. The seasonality estimate:

1. The exponential smoothed series: ))(1( 11

ttst

tt TL

S

YL

pstttpt SpTLY )(

4. Forecast p periods into the future:

The equations used :

Lt = new smoothed value.

α = smoothed constant for the level.Yt = actual observation in period tβ = smoothed constant for trend.Tt = trend estimate.

= smoothing constant for seasonality. St = seasonal estimate.

p = periods to be forecast in the future.s = length of seasonality. = forecast for p periods in the future

ptY

51

Choosing the Weights

• , and can be selected subjectively or by minimizing an error such as MSE.• A common approach: a nonlinear optimization algorithm to find optimal constants.

52

Starting the Procedure

One approach is to set the first estimate equal to the first observation, the trend is then estimated to equal zero, and the seasonal indices are set to 1. A second approach is to use the average of the first season or s observations , the trend is the slope of a line fit to these observations, and the seasonal indices are:

ttt LYS /

53

Minitabdevelops a regression equation, and

uses constants from the equation as initial estimates for the level and the trend. The seasonal components are obtained from a dummy variable regression using detrended data.

54

Example 4.10 (Last data)(Last data)

2000 1 500* 2 350* 3 250* 4 4002001 5 450* 6 350* 7 200* 8 3002002 9 350* 10 200* 11 150* 12 4002003 13 550* 14 350* 15 250* 16 5502004 17 550* 18 400* 19 350* 20 6002005 21 750* 22 500* 23 400* 24 6502006 25 850 26 600 27 450 28 700

Year QuarterstY

55

Actual

Predicted

Forecast

Actual

Predicted

Forecast

20100

900

800

700

600

500

400

300

200

100

Y

Time

MSD:MAD:MAPE:

Delta (season):

Gamma (trend):Alpha (level):

Smoothing Constants

7636.86 63.55 15.21

0.300

0.1000.400

Winters' Multiplicative Model for Y

Minitab Instructions: STAT > TIME SERIES > WINTERS’ METHOD.

Better than the other 2 models in terms of minimizing MSE.

56

654321

1.00.80.60.40.20.0

-0.2-0.4-0.6-0.8-1.0

Au

toco

rre

latio

n

LBQTCorrLag

5.01

4.97

2.80

2.44

2.43

2.41

0.14

-1.15

-0.48

0.08

-0.14

1.46

0.03

-0.26

-0.11

0.02

-0.03

0.30

6

5

4

3

2

1

Autocorrelation function for Example 4.6 Residuals

Autocorrelation Functions for the Residuals

None of the coefficients appear to be significantly larger than zero, andthe small value of LBQ (5.01) shows that the series is random.

57

THE END