| With modern technology massive quantities of data are being collected continuously. The purpose of our research has been to develop a method for data reduction and model selection applicable to large data sets and replicated data. We propose a novel wavelet shrinkage method by introducing a new model selection criterion. The proposed shrinkage rule has at least two advantages over the current shrinkage methods. First, it is adaptive to the smoothness of the signal regardless of whether it has a sparse wavelet representation, since we consider both the deterministic and the stochastic cases. The wavelet decomposition not only catches the signal components for a pure signal, but de-noises and extracts these signal components for a signal contaminated by external influences. Second, the proposed method allows for fine "tuning" based on the particular data at hand. Our simulation study shows that the methods based on the model selection criterion have better mean square error (MSE) over the methods currently known. Two aspects make wavelet analysis the analytical tool of choice. First, the largest in magnitude wavelet coefficients in the discrete wavelet transform (DWT) of the data, extract the relevant information, while discarding the rest eliminates the noise component. Second, the DWT allows for a fast algorithm calculation of computational complexity O(n).;For the deterministic case we derive a bound on the approximation error of the nonlinear wavelet estimate determined by the largest in magnitude discrete wavelet coefficients. Upper bounds for the approximation error and the rate of increase of the number of wavelet coefficients in the model are obtained for the new wavelet shrinkage estimate. When the signal comes from a stochastic process, a bound for the MSE is found, and for the bias of its estimate. A corrected version of the model selection criterion is introduced and some of its properties are studied.;The new wavelet shrinkage is employed in the case of replicated data. An algorithm for model selection is proposed, based on which a manufacturing process can be automatically supervised for quality and efficiency. We apply it to two real life examples. |
