1. CHAPTER 4
MOVING AVERAGES AND
SMOOTHING METHODS
(Page 107)

2. NAÏVE MODELS
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:
(4.1)

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 30
Moving Averages
4-20
Double Moving Averages 2
Linear (Double) exponential smoothing (Holt’s) T 3
Quadratic exponential smoothing 4
Seasonal exponential smoothing (Winter’s)
2 x s
Adaptive filtering
5 x s
Simple regression I C 10
Multiple regression
C , S
10 x V
Classical decomposition
Exponential trend models
I , L
S-curve fitting
Gompertz models
Growth curves
Census X-12
6 x s
ARIMA (Box-Jenkins)
ST , T , C , S 24
3 x s
Lading indicators
Econometric models
Time series multiple regression
T , S
Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

4. Example (Page 108)
Table 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:
(4.2)
The rate of change might be more appropriate than the absolute amount of change:

6. An appropriate forecast equation for quarterly data:
(4.4)
The analyst can combine seasonal and trend estimates using:
(4.5)
For monthly data:

7. Example 4.1 (cont.) Using Equations (4.3, 4.4, and 4.5)
(Pages 110 – 111)

8. Forecasting Methods Based on Averaging
(Page 111)

9. Minimal Data Requirements
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 30
Moving Averages
4-20
Double Moving Averages 2
Linear (Double) exponential smoothing (Holt’s) T 3
Quadratic exponential smoothing 4
Seasonal exponential smoothing (Winter’s)
2 x s
Adaptive filtering
5 x s
Simple regression I C 10
Multiple regression
C , S
10 x V
Classical decomposition
Exponential trend models
I , L
S-curve fitting
Gompertz models
Growth curves
Census X-12
6 x s
ARIMA (Box-Jenkins)
ST , T , C , S 24
3 x s
Lading indicators
Econometric models
Time series multiple regression
T , S
Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

10. Simple Averages
Uses the mean of all the relevant historical observations as the forecast of the next period.
(4.6)
New observation is added:
(4.7)
(Page 111)

11. Minimal Data Requirements
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 term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

12. Example 4.2
Table 4-2 Gasoline Purchases for the Spokane Transit Authority for Example 4.2
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 7
252 17
282 27
271 8
279 18 28 9
264 19
298 29 10
288 20
303 30

13. Time Series Graph The data seems stationary.
Figure 4-3 Time Series Plot of Weekly Gasoline Purchases for the Spokane Transit Authority
The data seems stationary.

14. Solution

15. Minimal Data Requirements
Moving Averages
A moving average of order k is the mean value of the k most recent observations.
(4.8)
K = number of terms in the moving average.
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
Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

16. Example (last data)
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 7
252 17
282 27
271 8
279 18 28 9
264 19
298 29 10
288 20
303 30

17. Calculations Using a five-week moving average (Page 114, 115)
Minitab can be used (See the Minitab Applications section for instructions, Pages: )

18. Results Week Purchases AVER1 RESI1 1 275 * * 2 291 * * * 3 307 * *
289.8
288.4
280.6
275.0
271.6
270.2
277.0
284.0
286.2
295.6
293.4
282.0
281.0
278.4
275.8
294.0
297.4
299.0
289.4
280.8
267.8
262.2
261.6
272.0
Week Purchases AVER1 RESI1
* *
* *
* *
* * *

19. Minitab Results FIGURE 4 - 4 (Page 115) Minitab Instructions
Stat > time Series > Moving averages
Note: (MSE is called MSD on Minitab output)
FIGURE 4 - 4
(Page 115)

20. 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

29. * *

30. Double Moving Average Equations (4.9) – (4.12) Example 4-4
(Pages )
Example 4-4
(Pages )

31. Exponential Smoothing Methods

32. 2.1 Simple (Single) Exponential Smoothing
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)
Method
Pattern of Data
Time Horizon
Type of Model
Minimal Data Requirements
Nonseasonal
Seasonal
Single Exponential smoothing ST S TS 2

33. Comparison of Smoothing Constants
Period
= 0.1
=0.6
Calculations
Weight t
0.1
0.6
t-1
0.1 x 0.9
0.09
0.6 x 0.4
0.24
t-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
0.038
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

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. Example 4.5
Year
Quarters * * * * * * * * * * * * * * * * * *
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.

36. Results
Year Quarters
(α=0.1) * * * )
) ) )
1) Initial value for the smoothed series = first observation = 500 2)
=-235 3) 4)
=0.1(250)+0.9(485)=461.5

37. Results 5) The calculation for the first quarter of 2007 Year Quarters
(α=0.1)
(α=0.6) * * * * * * * * * * * * * * * * * * )
) ) ) ) )
5) The calculation for the first quarter of 2007

38. Single Exponential Smoothing (α=0.1)

39. Single Exponential Smoothing (α=0.6)

40. α = 0.266

41. Comparison Optimization α = 0.1 MAPE 38.9 MAD 127.0 MSD 24261.7
Initial smoothed Value = The first observation
α = 0.1
MAPE
MAD
MSD
α = 0.6
MAPE
MAD
MSD
Initial Smoothed Value = The Average of the first six Observations
α = 0.1
MAPE
MAD
MSD
α = 0.6
MAPE
MAD
MSD
Optimization
MAPE
MAD
MSD
The weight α is selected subjectively or by minimizing an error such as the MSE

42. 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. Minimal Data Requirements
Exponential Smoothing Adjusted for Trend: (Holt’s Method) “Holt’s two-parameter method”
Double Exponential Smoothing
Smoothes the level and slope (trend) using different constants.
Method
Pattern of Data
Time Horizon
Type of Model
Minimal Data Requirements
Nonseasonal
Seasonal
Linear (Double) exponential smoothing (Holt’s) T S TS 3

44. Equations Used: 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.

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.

46. Example 4.9 (Last data) Year Quarters 2000 1 500 * 2 350 * 3 250 * * * * * * * * * * * * * * * * * *
Year
Quarters
Example 4.9 (Last data)

47. Comparison: Single: α = 0.266 Holts: α = 0.3, β = 0.1
Gamma = β
Comparison: Single: α = Holts: α = 0.3, β = 0.1
MSE
MAPE

48. LBQ (36.33) shows that the series is nonrandom

49. Minimal Data Requirements
Exponential Smoothing Adjusted for Trend and Seasonal Variations: Winters’ Method
Method
Pattern of Data
Time Horizon
Type of Model
Minimal Data Requirements
Nonseasonal
Seasonal
Seasonal exponential smoothing
(Winter’s) S TS
2 x s

50. The equations used : 1. The exponential smoothed series:
2. The trend estimate:
3. The seasonality estimate:
4. Forecast p periods into the future:
Lt = new smoothed value.
α = smoothed constant for the level.
Yt = actual observation in period t
β = 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

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:

53. Minitab
develops 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) Year Quarters 2000 1 500 * 2 350 * 3 250 * * * * * * * * * * * * * * * * * *
Year
Quarters
Example 4.10 (Last data)

55. Minitab Instructions: STAT > TIME SERIES > WINTERS’ METHOD.
Better than the other 2 models in terms of minimizing MSE.

56. Autocorrelation Functions for the Residuals
None of the coefficients appear to be significantly larger than zero, and
the small value of LBQ (5.01) shows that the series is random.

57. THE END