1. Comparison of Wavelet Families with Application to WiMAX Traffic Forecasting Cristina Stolojescu, Ion Railean, Sorin Moga and Alexandru Isar Faculty of Electronics and Telecommunications, Timisoara, Romania TELECOM Bretagne, Brest, France 1OPTIM 2010

2. Contents 2OPTIM 2010 ETC Timisoara – TELECOM Bretagne Wavelets. The Stationary Wavelet Transform Conclusions Time-frequency localization Results Subject.Objectives

3. Subject 3OPTIM 2010 ETC Timisoara – TELECOM Bretagne ARIMA based forecasting ANN based forecasting

4. Objectives to propose a strategy for the selection of the mother wavelet used in one of the steps of our algorithm. to evaluate the WiMAX traffic prediction accuracy by using different types of mother wavelets. 4OPTIM 2010 ETC Timisoara – TELECOM Bretagne

5. Wavelets Provide a useful decomposition of the time series, in terms of both time and frequency. One of the main properties of wavelets is that they are localized in time (or space) which makes them suitable for analyzing non-stationary signals. 5OPTIM 2010 ETC Timisoara – TELECOM Bretagne

6. The Stationary Wavelet Transform (SWT) two parameters: –the mother wavelet which generates the decomposition. –the number of decomposition levels. 6OPTIM 2010 ETC Timisoara – TELECOM Bretagne The SWT decomposition tree The Multi-resolution Analysis (MRA) the à trous algorithm, which corresponds to the computation of the SWT.

7. Time-frequency localization Two measures are introduced: A measure of the time-frequency localization of a given signal can be obtained by the product : 7OPTIM 2010 ETC Timisoara – TELECOM Bretagne

8. Time-frequency localization In the case of the SWT, both time and frequency localizations depend on the scale factor. It is noticed that the time localization is getting worse with the increasing of the scale factor, while frequency localization improves with the increasing of the scale factor. 8OPTIM 2010 ETC Timisoara – TELECOM Bretagne

9. Results wavelet families: Daubechies, Coiflet, Symmlet, Biorthogonal and Reverse Biorthogonal. 9OPTIM 2010 ETC Timisoara – TELECOM Bretagne

10. Conclusions Mother wavelets selection must be realized searching the best time localization (e.g. Haar). Mother wavelets with good time-frequency localization meaning reduced number of vanishing moments, (e.g rbio1.1 or db3) are also a good choice. 10OPTIM 2010 ETC Timisoara – TELECOM Bretagne

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12. Quality Evaluation Mean absolute error (MAE): the analysis of variance (ANOVA), Symmetric Mean Absolute Percent Error (SMAPE): Root Mean Square Error (RMSE). Mean Square Error (MSE). OPTIM 201012