Avoiding boundary uncertainties under prediction

By convolution, the length of the scaled and translated wavelet functions in the DWT is distinct from the signal length, which can lead to additional convolution uncertainties on the boundaries of the signal as stated in [19]. The authors of paper [9], as well as a large number of references which using the orthogonal DWT as auxiliary tool to obtain performance improvement of Time Series Prediction (TSP) models, do not consider that its obtained performance may not hold for the operation of the prediction model. The documented uncertainty may increase during the application of the model to predict unknown future values in real or actual time. The authors applying the TSP models to the vector of the training and the validation set in which the increased boundary uncertainty do only appear on the correction of the last predicted value of these vectors. Some authors applying padding techniques to the orthogonal DWT, which reduce the uncertainties of the reconstructed signal at the signal boundaries. These techniques distort the typical characteristics of the wavelet coefficients and the behavior of the sub-signals at the boundaries to lower the reconstruction uncertainty at the boundary. As the wavelet sub-signals (see section 3.1) are used for the time series prediction models, the local distortion can decrease its performances considerably. Therefore, also the performances of the reconstructed predictions decrease. Furthermore, most of the authors do not consider that if the trained and validated TSP model is applied for the prediction, it is continuously exposed to boundary conditions. At each new prediction, the model input variables or predictors are separated from the estimated output variable, by the right hand side boundary of the sub-signals as also discussed in Renaud [23]. The boundary appears new, because the signal has to be decomposed before each new prediction. By the utilization of wavelet networks as presented in [7] and [8] the boundary uncertainty in time series predictions may not appear due to its adaptive learning feature which can be applied to real time predictions as worked out in [24]. Wavelet networks substitute the sigmoidal activation function by a wavelet function. However, this method presents too many permutations for its optimization due to the selection of the network architecture, the number of neurons in the network layer, the wavelets functions and its scaling and translating counterpart, thus its optimization is a time consuming computational task. The anti-symmetric Haar was used in [23] for time series
predictions. Beside the Haar wavelet the DWT provides also the biorthogonal wavelet functions for the symmetric wavelet transform. The statistical correction in the present article is based on a biorthogonal wavelet function. Due to its symmetry and linear phase characteristic undesirable phase distortions of the sub-signals are avoided [25] and boundary treatment can be simplified [19]. By symmetric padding at the boundaries the bi-orthogonal wavelet transform, is converted in a DWT on a bounded interval as exposed in [26].

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