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Exponential smoothing in the telecommunications data

Everette S. Gardner, Jr.

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Exponential smoothing in thetelecommunications 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

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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

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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

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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

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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

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About 1/4 of the series contain an abrupt trend reversal during the fit periods

About 2/3 of the series contain a reasonably consistent trend

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BFirst forecast origin

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Re-examination: Methods tested Holt’s additive trend Damped additive trend Theta method (Assimakopoulos & Nikolopoulos,

IJF, 2000) SES with drift (Hyndman & Billah, IJF, 2003)

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SES with drift

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)(ˆ htX hbt 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.

Fixed drift

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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)

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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

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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.

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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.

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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

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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

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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.

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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

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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

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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 (AS,1978)

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Chatfield’s thoughtful approach toexponential smoothing1. Plot the series and look for trend, seasonality,

outliers, and changes in structure2. Adjust or transform the data if necessary3. Choose an appropriate form of trend and

seasonality4. Fit the method and produce forecasts5. Examine the errors and verify the adequacy of the

method