A comparison of same slope seasonality and exponential smoothing forecasting models.

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

1. INTRODUCTION

No forecasting method is perfect under all conditions. Also, there is no generally applicable and universally successful forecasting model (Makridakis & Hibon, 2000).

Method efficiency depends on specific situation which is determined by length of time horizon for which forecasting is conducted, number or available data, characteristics of time series itself, required numbers of forecasts and possibility of forecast automation etc. (Makridakis & Hibon, 2000; Sang at al., 2003).

There was a significant development of forecasting models in the last decades. 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.

Models of exponential smoothing, which belong to the group of time series models, were introduced in the middle of last century. From that time until now, many models are developed (Gardner, 2006).

Study (Gardner, 2006) suggests fifteen basic exponential smoothing models. Some models are with additive and some with multiplicative error estimates, while three models are with dumped multiplicative trend. Autoregressive Conditional Heteroscadasticity model describes the forecast variance in terms of current observations (Engle, 2004).

Papers (Pasic at al., 2007, 2008) described settings of the same slope model. In these papers, the same slope model was tested on real time series and compared with fifteen exponential smoothing models. Research results showed that developed model is more efficient with respect to other models of the same complexity. Model developed in these papers is successfully used in forecasting only one step in the future due to the fact that the gradient from the previous season is used to predict the future.

2. SAME SLOPE SEASONALITY MODEL

The same slope seasonality model is based on an idea that time series will have the same gradient for the same seasonal periods in different seasonal cycles (Figure 1). This idea can be mathematically written as:

[[??].sub.t](m) = [X.sub.t] - ([X.sub.t-p] - [X.sub.t-p+m]) (1)

where:

[[??].sub.t](m)--predicted time series value m step(s) forward, forecasted at time t (m [less than or equal to] p),

[X.sub.t]--actual demand of time series in time t,

p--number of periods in a season.

[FIGURE 1 OMITTED]

The main idea of the same slope seasonality model can be extended introducing a gradient correction coefficient [beta]:

[[??].sub.t](1) = [X.sub.t] - [beta]([X.sub.t-p] - [X.sub.t-p+1]) (2)

Equation (2) enables generation of forecasts taking into account increase or decrease of time series variance. With respect to coefficient [beta] following cases can be considered:

[beta] < 1 -- forecasts smaller change rate in the period t to t+m than in the period t-p to t-p+1.

[beta] = 1 -- forecasts the same change rate in the period t to t+m as in the period t-p to t-p+m.

[beta] > 1 -- forecasts bigger change rate in the period t to t+m than in the period t-p to t-p+m.

Coefficient [beta] can be optimized by minimizing errors.

3. RESEARCH METHODOLOGY

Model developed in this paper is tested on ten real time series and its performances are compared to the same slope model and fifteen exponential smoothing models, described in previous research (Granger, 2006). In Table 1. trend component is analysed in five different ways, and seasonal component is modelled in three different ways, which results in fifteen different exponential smoothing models.

Nonlinear mathematical programming technique is used for the estimation of parameters of both same slope models and exponential smoothing models. The goal of nonlinear programming is to minimize standard deviation of a measure of an error. Standard deviation of a forecasting error is adopted as 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.t=1][[[X.sub.t+m] - [[??].sub.t](m)].sup.2]/n] (3)

where MSE denotes mean square error of a model prediction variance and n stands for the number of periods for which the model prediction error is calculated. Standard deviation of the error will be used for the comparison of model performances.

4. RESULTS

All models are tested on ten real time series which are denoted as follows (source: Federal Office of Statistics of the Federation of Bosnia and Herzegovina (FBH)):

TS1 GDP of FBH. Time series from 1996 to 2004.

TS2 Total export of FBH. Time series from 1996 to 2005.

TS3 Total import of FBH. Time series from 1996 to 2005.

TS4 Investments in FBH. Time series from 1996 to 2005.

TS5 Number of foreign tourists in FBH. Time series from January 2002 to December 2004.

TS6 Number of arrivals of foreign tourists in FBH. Time series from January 2002 to December 2004.

TS7 Live births in FBH. Time series from January 2001 to December 2003.

TS8 Number of telephone time units used in FBH. Time series from January 2002 to November 2005.

TS9 Number of minutes of telephone calls by mobile phones in FBH. Time series from January 2004 to December 2006.

TS10 Number of calls by public telephones in FBH. Time series from January 2004 to December 2006.

Table 2. shows model performance ranks. Time series TS1, TS5, TS6 and TS7 are with very dominant trend and those are time series where the same slope (SS) and the same slope seasonality (SSS) model predictions are very appropriate to use. The other time series are dominated by excessive seasonal oscillations and frequent irregularity where the same slope model shows weak performances, but the same slope seasonality model has respectable performances.

In this test the same slope seasonality model is ranked as the third as shown in Table 2., which is very good result having in mind simplicity of the model. The same slope and the same slope seasonality models are one-parameter models only in comparison to other exponential smoothing models which are two or more parameters models. Complexity of exponential smoothing models in comparison with same slope 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. All of the exponential smoothing models have two or three equations, except N-N model which has only one parameter.

5. CONCLUSION

The same slope seasonality forecasting model is shown as very reliable forecasting model. It has same simplicity as the same slope model. But the same slope seasonality forecasting model overcomes disadvantages of the same slope forecasting model.

The same slope seasonality forecasting model, unlike the same slope model, shows its reliability in the case of time series with dominated seasonal component, as well as with time series with excessive trend component.

The same slope seasonality forecasting model is one- parameter model with just one equation, while all other exponential smoothing models with trend and seasonal components have three parameters and three equations. This property gives big comparative advantage to the same slope seasonality forecasting model with respect to a parameter optimization.

Disadvantage of model presented in this paper is a lack of smoothing component in a case of presence of outliers. Future research will be focused on eliminating this disadvantage.

6. REFERENCES

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, (October-December 2006) pp. 637-666, ISSN 0169-2070

Makridakis, S. & Hibon, M. (2000). The M3-Competition: results, conclusions and implications, International Journal of Forecasting, Vol. 16, No. 4, (October 2000) pp. 451-476, ISSN 0169-2070

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, Zadar, Croatia, October 2007, DAAAM International, Vienna

Pasic, M.; Bijelonja, I. & Bajric, H. (2008). A Comparison of Same Slope and Exponential Smoothing Forecasting Models, Proceedings of the 19th International DAAAM Symposium, Katalinic, B. (Ed.), pp. 1031-1032, ISBN 978-3-901509-68-1, Trnava, Slovakia, October 2008, 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

Trend Seasonality

 None Additive Multiplicative
 (N) (A) (M)

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

Tab. 2. Model performance rank

Model TS1 TS2 TS3 TS4 TS5 TS6

N-N 10 12 6 10 11 10
N-A 6 10 4 5 10 11
N-M 9 3 12 1 12 12
A-N 8 11 9 9 1 3
A-A 3 8 3 6 3 6
A-M 2 1 13 2 4 7
DA-N 7 6 7 7 2 2
DA-A 1 7 2 4 5 1
DA-M 14 2 14 3 6 13
M-N 15 17 16 16 13 14
M-A 16 16 15 14 16 16
M-M 17 15 17 12 14 15
DM-N 13 13 8 17 17 17
DM-A 12 9 1 15 9 9
DM-M 11 4 10 13 15 8
SS 4 14 11 11 7 4
SSS 4 5 5 8 7 4

 Average
Model TS7 TS8 TS9 TS10 Rank

N-N 10 14 6 8 12
N-A 8 8 12 3 8
N-M 9 3 3 1 6
A-N 5 12 5 9 7
A-A 2 7 9 6 4
A-M 3 2 2 5 1
DA-N 4 11 7 7 5
DA-A 1 9 11 2 2
DA-M 14 5 14 11 11
M-N 17 17 17 12 17
M-A 15 16 15 14 16
M-M 16 10 16 17 15
DM-N 13 13 8 13 14
DM-A 12 6 13 15 13
DM-M 11 4 1 16 10
SS 6 15 10 10 9
SSS 6 1 4 4 3