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A Brief History of Prediction Intervals for the Exponential Smoothing Methods

• M. Sc. Sergio David Madrigal Espinoza

Abstract

For the schedules of many organizations, prediction intervals (PIs) are as desired as the forecast point. For example in inventory control they help to design policies for customers' service. In this review we will check the historical evolution of the PI for the exponential smoothing methods and its current state

• f the art.

Exponential Smoothing Methods

These forecasting methods give a greater exponential weights to the most recent

• bservations. Since there creation by

Brown (1956), such methods have been widely used for forecasters. Nevertheless, since then, the lack of a statistical base has been notorious. Nowadays, that SB is almost complete.

Statistical Base

Likelihood Function Selection Criteria with Parameter Penalization Stability Conditions Prediction Intervals

Damped Trend Additive Trend No Trend Multiplicative Seasonality Additive Seasonality No Seasonality Multiplicativ e Trend

The State of the Art

Pegels (1969) and Hyndmand (2002) Taxonomy

Prediction Intervals

PIs are measures of the uncertainty of forecasting methods. For practitioners, they are the most important part of the statistical base. Forecast can not be perfect and PIs emphasize this.

Real Data

Fitted Forecasting Model Forecast Prediction Intervals

Historical Evolution of PI

A classical approach to PIs is to obtain the variance of the forecasting errors σ2 and to assume that a good measure of the prediction interval is y(h)=µ(h)±k σ. The problem is that the PIs does not increase its size over time.

Historical Evolution of PI

Another way to obtain PIs is the form y(h)=µ(h)±k root(h)σ. This approach is more sophisticated than the first one and is a function of the forecast lead time (h). Nevertheless, this variance was “inspired” by one obtained analytically for the random walk model and can be seriously misleading.

Historical Evolution of PI

At the 80’s researchers found out that some ESM have an equivalent ARIMA process and thereby, all the PI of those, hold for the ESM too.

Damped Trend Additive Trend No Trend Multiplicative Seasonality Additive Seasonality No Seasonality Multiplicative Trend

Historical Evolution of PI

Nevertheless, in those very years, Abraham and Ledolter (1986) demonstrated that the multiplicative Holt- Winters method had no equivalent ARIMA process.

Damped Trend Additive Trend No Trend Multiplicative Seasonality Additive Seasonality No Seasonality Multiplicative Trend

PI for the MHWM

Archivald (1990) proposed a variation on the MHWM Chatfield and Yar (1991) showed that the variance of the MHWM should depend on the season and the level

Innovation State Space Models

• In 1997, Ord et al. developed the

Innovation State Space Models (ISSM)

• They also provide two classes of

forecasting models: the heterosedastic and the homosedastic error models

• The ISSM underlies the MHWM

yt=h xt−1,αk  xt−1,αεt , xt= f  xt−1,αg xt−1,αεt .

State Space Models Assumptions

• ARIMA and ISSM assume NID
• They assume perfect information too

Innovation State Space Models

• The ISSM will be useful for obtaining PI not only

for the MHWM but also for all the ESM. But before that can happen…

The Quantile Regression Approach

In 1999, Taylor and Bunn used a quantile regression on fitted errors to generate forecast that are functions of the lead

• time. These methods avoid the

assumption of NID and perfect information and have had excellent results in simulated and real data.

Damped Trend Additive Trend No Trend Multiplicative Seasonality Additive Seasonality No Seasonality Multiplicative Trend

PI for the MHWM

In 2001, Koehler et al. developed analytical PIs for the MHWM PIs are available for the heterosedastic and the homosedastic cases They used the ISSM and thereby, assumed NID and perfect information

PI for all the ESM

In 2002, Hyndmand et al. showed that the ISSM underlies all the known ESM Based on this, they obtained simulated PIs for all the ESM Again, they assumed NID and perfect information In real data, they obtain 83% of coverage using PIs with a confidence level of 95%

PI for all the ESM

In 2005, Hyndmand et al. developed analytical approaches of PIs for most of the ESM known These approaches are equivalent to simulations (and their assumptions) and

• nly simplifies the calculation of PIs

Damped Trend Additive Trend No Trend Multiplicative Seasonality Additive Seasonality No Seasonality Multiplicative Trend

Taylor's Work

In 2003, Taylor proposed a new class of ESM called “Double Seasonal Exponential Smoothing Methods” Nevertheless, Taylor does not provide PIs for his method.

Taken from Taylor J.W. (2003) Short-term electricity demand forecasting using double seasonal exponential smoothing. Journal of the Operation Research Society, 54, 799-805.

National Car Sales

Following the approach of Ord, Koehler, Hyndman and Snyder, Madrigal (2006) presented not only analytical PIs for the Taylor’s method but also a complete statistical base for it.

National Car Sales

National Car Sales

However, in the trial period, our PIs turn

• ut to be too narrow. This happens

because the assumptions of NID and perfect information seems to be too restrictive.

National Car Sales

A Taxonomy for PIs

Linear regression y(h)=µ(h)±k σ µ(h)±k root(h)σ NID assumption Quantile regression approach Nether parameters nor assumptions on distributions All the ESM and ARIMA processes NID and Perf. Inf. assumptions Based on underlying statistical models Parameter-free Intuitive Based

References

• Abraham, Ledolter, J. (1986) Forecast functions implied by

autoregressive integrated moving average and other related forecast procedures. International Statistical Review 54, 51-66.

• Brown, R.G. (1956) Exponential smoothing for predicting
• demand. Tenth national meeting of the Operation Research

Society of America, San Francisco, 16 November 1956.

• Chatfield, C. and Yar, M. (1991) Prediction intervals for

multiplicative Holt-Winters. International Journal of Forecasting, 7, 31--37.

• Hyndman, R.J., A.B. Koehler, J.K. Ord and R.D. Snyder (2005)

Prediction intervals for exponential smoothing state space

• models. Journal of Forecasting, 24, 17--37.

References

• Hyndman, R.J., A.B. Koehler, R.D. Snyder and S. Grose (2002)

A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting, 18, 439--454.

• Koehler, A.B., R.D. Snyder and J.K. Ord (2001) Forecasting

models and prediction intervals for the multiplicative Holt-Winters

• method. International Journal of Forecasting, 17, 269--286.
• Madrigal E. S. D. (2006) Modelos de espacio de estados subyacente al

método multiplicativo de Holt-Winters con múltiple estacionalidad. Tesis de maestría, PISIS, UANL.

References

• Ord, J.K., A.B. Koehler and R.D. Snyder (1997) Estimation and

prediction for a class of dynamic nonlinear statistical models. Journal of American Statistical Association, 92, 1621--1629.

• Pegels, C.C. (1969) Exponential smoothing: some new
• variations. Management Science, 12, 311--315.
• Taylor J.W. (2003) Short-term electricity demand forecasting

using double seasonal exponential smoothing. Journal of the Operation Research Society, 54, 799-805.

• Winters, P.R. (1960) Forecasting sales by exponentially

weighted moving averages. Management Science, 6, 324--342.