Assignment week 38
Exponential smoothing of monthly observations of the General Index of the
Stockholm Stock Exchange.
A. Graphical illustration of data
First, construct a graph of the original series of monthly values.
Stock_Exchange.txt
Time Series Plot of Index
4000
Index
3000
2000
1000
0
1
61
122
183
244
305
Index
366
427
488
549
610
Then construct a graph of the percentage change from month to month.
Time Series Plot of Change
30
20
Change
10
0
-10
-20
1
61
122
183
244
305
Index
366
427
488
549
610
Which smoothing techniques (single, double, Holt-Winters) can be used on the
original series, which can be used on the series of percentage change.
Original series:
Series of percentage change:
Double (Holt’s) (or Holt-Winters’ (Winters’)
method)
Single (or Winters’ without trend)
B. Exponential smoothing with predefined smoothing parameters
Perform single exponential smoothing on the time series of percentage change
(of the General Indices). Set the smoothing parameter, , first to 0.9 and then to
0.1.
Variable Change is not in the list, due to
the initial missing value  Copy the nonmissing values to a new column.
Smoothing Plot for Change_1949_2
Single Exponential Method
30
Variable
Actual
Fits
Change_1949_2
20
Smoothing Constant
Alpha 0.1
Accuracy Measures
MAPE
160.999
MAD
3.514
MSD
23.053
10
0
-10
-20
1
61
122
183
244
305 366
Index
427
488
549
Smoothing Plot for Change_1949_2
Single Exponential Method
30
Variable
Actual
Fits
Change_1949_2
20
Smoothing Constant
Alpha 0.9
Accuracy Measures
MAPE
240.685
MAD
4.382
MSD
35.101
10
0
-10
-20
1
61
122
183
244
305 366
Index
427
488
549
Then study the graphs produced and try to understand how the choice of the
smoothing parameter affects the forecasted values.
 = 0.1 gives very damped predicted values (red curve) wile  = 0.9 gives predicted
values highly responding to the recent changes in original series.
C. Exponential smoothing with automatic parameter setting
Let the program choose an optimal value of the smoothing parameter and calculate
forecasts for a two-year period (24 months) after the last observed time-point.
Construct a graph for the errors in the
one-step-ahead forecasts (residuals) in
the whole time series and try to judge
upon whether the forecasting methods
uses earlier observations in the series in
an efficient way.
Smoothing Plot for Change_1949_2
Single Exponential Method
30
Variable
Actual
Fits
Forecasts
95.0% PI
Change_1949_2
20
Are the earlier observations
used in an efficient way?
Smoothing Constant
Alpha
0.0113270
10
Accuracy Measures
MAPE
141.322
MAD
3.464
MSD
22.392
0
-10
-20
1
63
126
189
252
315 378
Index
441
504
567
630
Versus Order
(response is Change_1949_2)
30
10
Residual
Are the residuals serially
correlated – Make a visual
judgement.
20
0
-10
-20
-30
1
50
100
150
200
250 300 350 400
Observation Order
450
500
550
600
Use also the autocorrelation function on the residual. (MINITAB-Time SeriesAutocorrelation).
Autocorrelation Function for RESI1
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
65
What do you see in the plot you get?
First spike is significantly different from zero, so is also some spikes for larger
lags.
 Residuals seem to be serially correlated.
Exponential smoothing of time series with seasonal variation
A. Forecasting the employment in USA
Perform an exponential smoothing of the time series of monthly employments
figures in USA and calculate forecasts for a two-year period (24 month) after the last
observed time-point.
Labourforce.txt
Time Series Plot of Value
68
66
Value
64
62
60
58
56
1
60
120
180
240
300
Index
360
420
480
540
Time series possesses trend and seasonal variation  Use Winters’ method
Seasonal variation do not seem to change with level  Use additive case
600
Winters' Method Plot for Value
Additive Method
70
Variable
Actual
Fits
Forecasts
95.0% PI
68
Value
66
Smoothing C onstants
A lpha (lev el)
0.2
Gamma (trend)
0.2
Delta (seasonal)
0.2
64
62
Accuracy
MA PE
MA D
MSD
60
Measures
0.370202
0.226105
0.086981
58
56
1
62
124
186
248
310 372
Index
434
496
558
620
Then use a suitable model for time series decomposition to make forecasts for the
same period (additive or multiplicative).
Time Series Decomposition Plot for Value
Additive Model
70.0
Variable
Actual
Fits
Trend
Forecasts
67.5
Value
65.0
A ccuracy
MAPE
MAD
MSD
62.5
60.0
57.5
55.0
1
62
124
186
248
310 372
Index
434
496
558
620
Measures
1.41981
0.86505
1.05524
Print out graphs for observed and forecasted values and compare how the seasonal
effects are described in each method of forecasting. Which method do you prefer in
this case?
Observed (and forecasts):
Winters' Method Plot for Value
Time Series Decomposition Plot for Value
Additive Method
Additive Model
70
Variable
Actual
Fits
Forecasts
95.0% PI
66
Smoothing C onstants
A lpha (lev el)
0.2
Gamma (trend)
0.2
Delta (seasonal)
0.2
64
62
Accuracy
MA PE
MA D
MSD
60
Measures
0.370202
0.226105
0.086981
Variable
Actual
Fits
Trend
Forecasts
67.5
65.0
Value
68
Value
70.0
A ccuracy
MAPE
MAD
MSD
62.5
60.0
57.5
58
56
55.0
1
62
124
186
248
310 372
Index
434
496
558
620
1
62
124
186
248
310 372
Index
434
496
558
620
Measures
1.41981
0.86505
1.05524
Forecasts only:
From Winters’ method
From Decomposition
Make a time series plot with both series of forecasts in the same plot
Time Series Plot of FORE1, FORE2
68.5
Variable
FORE1
FORE2
68.0
Data
67.5
67.0
66.5
66.0
2
4
6
8
10
12 14
Index
16
18
20
22
24
B. Forecasting of monthly mean temperature
temperature.txt (title “Stockholm” removed)
Time Series Plot of Temperature
25
20
Temperature
15
10
5
0
-5
-10
1
43
86
129
172
215
Index
258
301
344
387
430
Use exponential smoothing to make forecasts of monthly mean temperatures in
Stockholm. Try single, double (Holt’s method) and Winters’ method.
Study the residuals (the errors in one-step-ahead forecasts) and the forecasts for 24
months after the last observed time-point. Are the one-month-ahead and one-yearahead forecasts realistic?
Single exponential smoothing:
Smoothing Plot for Temperature
Single Exponential Method
Versus Order
10
Smoothing Constant
A lpha
1.39083
0
Accuracy Measures
MAPE
144.034
MAD
3.198
MSD
15.636
10
5
Residual
20
Temperature
(response is Temperature)
Variable
Actual
Fits
Forecasts
95.0% PI
0
-5
-10
1
-10
-20
1
46
92
138
184
230 276
Index
322
368
414
50
100
150
200
250
Observation Order
300
350
400
Double exponential smoothing (Holt’s method):
Smoothing Plot for Temperature
Double Exponential Method
Variable
Actual
Fits
Forecasts
95.0% PI
0
Temperature
-50
Smoothing Constants
A lpha (lev el)
0.76251
Gamma (trend)
1.26707
-100
-150
A ccuracy Measures
MAPE
146.299
MAD
3.211
MSD
16.528
-200
-250
-300
Versus Order
-350
1
46
92
(response is Temperature)
138 184 230 276 322 368 414
Index
20
15
Neither single, nor double exponential
smoothing seems to work.
Residual
10
5
0
-5
Surprising?
-10
1
50
100
150
200
250
Observation Order
300
350
400
Winters’ method:
Note that we do not have any particularly pronounced trend in data and shifts in
level are (if existing) very modest.
Time Series Plot of Temperature
25
20
Temperature
15
10
5
0
-5
-10
1
43
86
129
172
215
Index
258
301
344
387
 Try low values of smoothing
parameters for level and trend
430
Winters' Method Plot for Temperature
Additive Method
20
Variable
Actual
Fits
Forecasts
95.0% PI
10
Smoothing C onstants
Alpha (lev el)
0.05
Gamma (trend)
0.01
Delta (seasonal)
0.20
Forecasts much better here
Accuracy Measures
MA PE
108.557
MA D
2.490
MSD
10.021
0
-10
1
46
92
138
184 230 276 322
Index
368 414
Versus Order
(response is Temperature)
Residuals become positively
correlated
5
Residual
Temperature
30
0
-5
-10
1
50
100
150
200
250
Observation Order
300
350
400
Compare with an analysis with default values on smoothing parameters:
Winters' Method Plot for Temperature
Additive Method
20
Variable
Actual
Fits
Forecasts
95.0% PI
10
Smoothing C onstants
A lpha (lev el)
0.2
Gamma (trend)
0.2
Delta (seasonal)
0.2
Accuracy Measures
MA PE
95.4625
MA D
1.8182
MSD
5.2691
0
-10
1
46
92
138
184
230 276
Index
322
368
Residuals are much better.
414
Forecasts seem to contain an
“artificially” induced trend.
Versus Order
(response is Temperature)
10
We have to keep on trying.
5
Residual
Temperature
30
0
-5
-10
1
50
100
150
200
250
Observation Order
300
350
400
Is there a better way for making
forecasts than applying
exponential smoothing on the
original series?