Exponential smoothing in the
telecommunications data
Everette S. Gardner, Jr.
1
Exponential smoothing in the
telecommunications data
 Empirical research in exponential smoothing
 Summary of Fildes et al. (IJF, 1998)
 Data analysis
 Re-examination of the smoothing methods
 Conclusions
 Chatfield’s thoughtful approach to exponential
smoothing
Empirical research in exponential smoothing:
1985-2005
 Total of 65 coherent empirical studies

Excluding M-competitions and related papers

Excluding studies based on simulated data
 Some form of exponential smoothing
performed well in all but 7 studies

All of these exceptions should be re-examined
 The most surprising exception is Fildes et al.’s
(IJF, 1998) study of telecommunications data
Fildes et al. (IJF, 1998)
 Study of 261 monthly, nonseasonal series




71 observations each
Forecasts through 18 steps ahead from origins
23, 31, 38, 45, and 53
Steady trends with negative slopes
Numerous outliers
 Methods tested


Robust trend
Exponential smoothing


Holt’s additive trend
Damped additive trend
The robust trend method
 Underlying model
 ARIMA (0, 1, 0) with drift
 Drift term
 Estimated by median of the differenced data
 Subject to complex adjustments
Fildes et al. (IJF, 1998) continued
 Robust trend was the most accurate
method
 Holt’s additive trend was more accurate
than the damped trend


Contrary to theory and all other empirical studies
in the literature
Armstrong (IJF, 2006) recommends replication
1700
1600
B
1500
About 1/4 of the series contain
an abrupt trend reversal during
the fit periods
Number of circuits
First forecast origin
1400
1300
A
1200
1100
About 2/3 of the series contain a
reasonably consistent trend
1000
900
800
700
1
13
25
37
Observation number
49
61
Re-examination: Methods tested
 Holt’s additive trend
 Damped additive trend
 Theta method (Assimakopoulos & Nikolopoulos,
IJF, 2000)
 SES with drift (Hyndman & Billah, IJF, 2003)
SES with drift
 t   t 1  b   t
Xˆ t (h)   t  hb
Fixed drift
SES with drift is equivalent to the Theta method when
drift equals ½ the slope of a classical linear trend.
However, Hyndman and Billah recommend
optimization of the drift term.
Re-examination: Fitting the methods
 Two sets of data were fitted through each
forecast origin


Original data
Trimmed data – observations prior to an early trend
reversal were discarded
 Initial values for smoothing methods
 Intercept and slope of a classical linear trend
 Fit criteria
 MSE
 MAD (to cope with outliers)
Fitting continued
 Parameter choice
 Usual [0,1] interval
 Full range of invertibility
 SES with drift
 Initial level and drift were optimized
simultaneously with smoothing parameter
 Theta method
 Initial level only was optimized simultaneously
with smoothing parameter
 Drift fixed at half the slope of the fit data
Effects of model-fitting on the damped trend
Fit
Parameters
Data
Criterion
MAPE
1
Approximate
Original
MSE
9.7
2
Optimal
Original
MSE
7.8
3
Optimal
Trimmed
MSE
7.2
4
Optimal
Trimmed
MAD
6.8
Note: MAPE is the average of all forecast origins and
horizons.
Effects of model-fitting on the Holt method
Fit
Parameters
Data
Criterion MAPE
1
Approximate
Original
MSE
8.1*
2
Optimal
Original
MSE
8.1
3
Optimal
Trimmed
MSE
7.9
4
Optimal
Trimmed
MAD
7.4
* We were unable to replicate Fildes et al.’s Holt results.
Effects of model-fitting on SES with drift
Fit
Parameters
Data
Criterion MAPE
1
Optimal
Original
MSE
7.4
2
Optimal
Trimmed
MSE
7.1
3
Optimal
Trimmed
MAD
6.2
Why did SES with drift perform so well?

Trends in most series are so consistent that
there is no need to change initial estimates
obtained by least-squares regression
 Smoothing parameter was fitted at 1.0 almost
half the time

This produces an ARIMA (0,1,0) with drift, the
underlying model for the robust trend
Revised empirical comparisons
Method
MAPE
Robust trend
6.2
SES with drift
6.2
Damped additive trend
6.8
Holt’s additive trend
7.4
Theta method
7.6
All methods except robust trend fitted to
trimmed data to minimize the MAD.
Conclusions
 Contrary to Fildes et al., the damped trend is
in fact more accurate than the Holt method.
 SES with drift:

Simplest method tested

Drift term should be optimized


More accurate than the Theta method
About the same accuracy as the robust trend
Recommendations
 Trim irrelevant early data
 Use a MAD fit to cope with outliers
 Optimize smoothing parameters
 Follow Chatfield’s (AS, 1978) “thoughtful”
approach to exponential smoothing
Chatfield (AS,1978)


Re-examination of Newbold and Granger (JRSS,
1974), who found the Box-Jenkins procedure was
far more accurate than exponential smoothing
Findings
Newbold and Granger’s empirical comparisons were
biased
 It was easy to improve the performance of
exponential smoothing

Chatfield’s thoughtful approach to
exponential smoothing
1. Plot the series and look for trend, seasonality,
outliers, and changes in structure
2. Adjust or transform the data if necessary
3. Choose an appropriate form of trend and
seasonality
4. Fit the method and produce forecasts
5. Examine the errors and verify the adequacy
of the method