A comparison of same slope and exponential smoothing forecasting models.

Pasic, Mugdim ; Bijelonja, Izet ; Bajric, Hadis 等

1. INTRODUCTION

Forecasting is the art and science of predicting future events, and very important factor that influences organizational performances, since forecasting enables efficient use of scarce organizational resources (Nikolopoulos & Assimakopoulos, 2003). Good forecasts are an essential part of efficient service and production and effective planning in both the short and long run depends on a forecast of demand for the company's services or products (Heizer & Render, 2006; Gardner, 2006).

There was a significant development of forecasting models in the last decade. Many different models are developed, and many science works and published papers are dedicated to this area (Sang at al., 2003). Particular development is achieved in the area of time series models (De Gooijer & Hyndman, 2006).

Models of exponential smoothing, which belong to the group of time series models, were introduced in the middle of recent century. From that time until now, many models are developed (Gardner, 2006; Gardner, 1985; Hyndman at al,. 2002).

Research (Hyndman at al., 2002) presents twelve exponential smoothing models with two types of error estimates for each model (additive and multiplicative), which gives total of twenty four different models. Study (Gardner, 2006) suggests fifteen basic exponential smoothing models, from which twelve are already presented (Hyndman at al., 2002). Some models are with additive and some with multiplicative error estimates, and three new models are introduced with dumped multiplicative trend.

Autoregressive Conditional Heteroskedasticity (ARCH) model describes the forecast variance in terms of current observations (Engle, 2004).

Research (Pasic at al., 2007) introduces basics of the same slope forecasting model, and the model is tested on the real time series. In general, results of the research shows better performances of the same slope model than moving average, weighted moving average and simple exponential smoothing models. These models are of the same complexity level as the same slope model.

The aim of this work is to compare performances of the same slope model with fifteen different exponential smoothing models using characteristic time series. Testing results show very good performances of the same slope model in comparison with the above mentioned exponential smoothing models, despite the fact that the same slope model is one-parameter model as compared to multi-parameter exponential smoothing models. Also, further improvement of the same slope model is suggested in this paper.

2. PROBLEM APPROACH AND METHODOLOGY

Mathematical formulation of the same slope forecasting model (Pasic at al., 2007) is expressed by:

[F.sub.t] = [D.sub.t-1] - [beta]{[D.sub.t-2] - [D.sub.t-1]) (1)

where:

[F.sub.t]--Expected value of time series in period t,

[D.sub.t-1]--Value of time series in period t-1,

[D.sub.t-2]--Value of time series in period t-2,

[beta]--Trend coefficient.

Depending on the value of the trend coefficient [beta], the following scenarios are possible:

[beta] < 1--Forecasts smaller change rate in the period t to t-1 than in the prior period.

[beta] = 1--Forecasts the same change rate in the period t to t-1 than in the prior period.

[beta] > 1--Forecasts bigger change rate in the period t to t-1 than in the prior period.

The trend coefficient [beta] can be optimized by minimizing the forecast errors. In this research fifteen exponential smoothing models, (Granger, 2006), are used and listed in Table 1.

Models (Granger, 2006), treat trend component in five different ways and seasonal component in three different ways, which give fifteen different exponential smoothing models.

For an estimation of parameters of the same slope model and exponential smoothing models, nonlinear mathematical programming technique is used. The goal in nonlinear programming is to minimize standard deviation of a measure of an error. Standard deviation of the model is a measure of reliability of a forecasting model and can be calculated using the following equation:

[sigma] = [square root of MSE] = [square root of [[summation].sup.n.sub.i=1] [([D.sub.t] - [F.sub.t]).sup.2]/n] (2)

where MSE denotes the mean square error or the model variance, and n stands for number of periods for which the model error is measured.

Standard deviation of the model will be used for the comparison of model performances.

3. RESULTS

All models are tested on ten time series as follows (source: Statistics Institute FB&H):

TS1--GDP of Federation of Bosnia and Herzegovina (FB&H). Time series from 1996 to 2004.

TS2--Total export of FB&H. Time series from 1996 to 2005.

TS3--Total import of FB&H. Time series from 1996 to 2005.

TS4--Realized investments in FB&H. Time series from 1996 to 2005.

TS5--Number of foreign tourists in FB&H. Time series from January 2002 to December 2004.

TS6--Number of arrivals of foreign tourists in FB&H. Time series from January 2002 to December 2004.

TS7--Live births in FB&H. Time series from January 2001 to December 2003.

TS8--Number of telephone units in FB&H. Time series from January 2002 to November 2005.

TS9--Number of minutes of telephone calls by mobile phones in FB&H. Time series from January 2004 to December 2006.

TS10--Number of calls by public telephones in FB&H. Time series from January 2004 to December 2006.

Time series TS1, TS2, TS3 and TS4 are with very dominant trend component. Time series from TS5 to TS10 are with very dominant seasonality, which means that time series values, in successive periods, constantly change, having values from local minimum to local maximum. The ranks of models are listed in Table 2. In this test, the same slope model is at the ninth position. Because of seasonality and continuous value changes, the same slope model does not fit the best to this kind of time series.

The final rank of the one-parameter same slope model, in comparison to other multi-parameter exponential smoothing models shows very good performances. These parameters present variables in a nonlinear programming model. The goal of this nonlinear programming model is to minimize forecast error of the model. Complexity of exponential smoothing models is not remarkable just because of number of parameters, but because of the fact that for every component of time series there is a different particular equation. So, all of the exponential smoothing models have two or three equations, and N-N model only has just one, as the same slope model has. Also, the same slope model uses a few historical records, precisely only two, while some of exponential smoothing models require more records in order to make good forecast. In some cases more than twenty historical records have to be memorized in order to make a forecast.

4. CONCLUSION

The same slope forecasting model is a very simple and easy to use model. It is very sensitive on trend changes and uses only two historical records no matter the type of time series.

Research results show that the same slope model is very suitable when dealing with time series where trend component dominates. However, the same slope model does not perform well when treating seasonal time series in a sense it uses only two last historical values of a time series. In general, this model is more suitable than simple exponential smoothing model, the only model of the same complexity used in this research.

Future research should be focused on improving the same slope model when dealing with seasonal time series. The model should be improved in a sense to recognize and eliminate nonstandard observations in time series which could significantly influence forecasting results. The model should also have capabilities to take into account non-equidistant time series data.

In the case that n-step forecast is needed the same slope model shows significant disadvantage due to the fact it forecasts time series values using the gradient of its previous time step.

Model developed in this paper could be successfully used in forecasting only one step in the future. Also this shortcoming will be overcome by taking into account seasonal time series effect in the model. This means that the gradients from the past seasons will be used to predict the future.

5. REFERENCES

De Gooijer, J.G. & Hyndman, R.J. (2006). 25 years of time series forecasting, International Journal of Forecasting, Vol. 22, No. 3, (July-Sept. 2006) pp. 443-473, ISSN 0169-2070

Engle, R. (2004). Risk and Volatility: Econometric Models and Financial Practice, The American Economic Review, Vol. 94, No. 3, (June, 2004) pp. 405-420, ISSN 0002-8282

Gardner, E. (2006). Exponential smoothing: The state of the art--Part II, International Journal of Forecasting, Vol. 22, No. 4, (Oct.-Dec., 2006) pp. 637-666, ISSN 0169-2070

Gardner, E. (1985). Exponential smoothing: The state of the art--Part I, International Journal of Forecasting, Vol. 4, No. 1, (January-March, 1985) pp. 1-28, ISSN 0277-6693

Hyndman, R. J.; Koehler, A. B.; Snyder, R. D. & Grose, S., (2002). A state space framework for automatic forecasting using exponential smoothing methods, International Journal Forecasting, Vol. 18, No. 3, (July-September 2002) pp. 439-454, ISSN 0169-2070

Heizer, J. & Render, B. (2006). Operations Management, Pearson Education, Inc. 0-13-185755-X, New Jersey

Nikolopoulos, K. & Assimakopoulos, V. (2003), Theta intelligent forecasting information system, Industrial Management & Data Systems, Vol. 103, No. 9, pp. 711-726, ISSN 0263-5577

Pasic, M., Bijelonja, I., Sunje, A. & Bajric, H. (2007). Same Slope Forecasting Method, Proceedings of the 18th International DAAAM Symposium, Katalinic, B. (Ed.), pp. 547-548, ISBN 3-901509-58-5, Vienna, Austria, October 2007, DAAAM International, Vienna

Sang, C. S.; Sam, I. S., Dan, L. & Jingmiao, G. (2004). A new Insight Into Prediction Modeling System, Journal of Integrated Design and Process Science, Vol. 8, No. 2, (April 2004) pp. 85-104, ISSN:1092-0617

Tab 1. Exponential smoothing models.

 Seasonality

 Trend None Additive Multiplicative

None N-N N-A N-M
Additive A-N A-A A-M
Damped additive DA-N DA-A DA-M
Multiplicative M-N M-A M-M
Damped multiplicative DM-N DM-A DM-M

Tab 2. Models ranks.

Model TS1 TS2 TS3 TS4 TS5 TS6

 N-N 9 10 9 9 11 13
 N-A 5 9 10 7 9 7
 N-M 8 11 11 8 3 2
 A-N 7 1 3 5 10 11
 A-A 3 3 5 2 7 6
 A-M 2 4 6 3 1 1
DA-N 6 2 2 4 5 10
DA-A 1 5 1 1 6 8
DA-M 13 6 12 13 2 4
 M-N 14 12 13 16 16 16
 M-A 16 15 15 14 15 15
 M-M 15 13 14 15 14 9
DM-N 12 16 16 12 12 12
DM-A 11 8 8 11 8 5
DM-M 10 14 7 10 4 3
 SS 4 7 4 6 13 14

Model TS7 TS8 TS9 TS10 Rank

 N-N 9 5 5 7 10
 N-A 5 4 11 3 7
 N-M 1 11 3 1 5
 A-N 8 8 4 8 6
 A-A 6 3 8 5 3
 A-M 2 12 2 4 1
DA-N 7 6 6 6 4
DA-A 4 2 10 2 2
DA-M 3 13 13 10 11
 M-N 15 15 16 11 15
 M-A 13 14 14 13 16
 M-M 11 16 15 16 14
DM-N 16 7 7 12 13
DM-A 14 1 12 14 12
DM-M 12 9 1 15 8
 SS 10 10 9 9 9