Data Mining and Knowledge Discovery Handbook, 2 Edition part 59. Knowledge Discovery demonstrates intelligent computing at its best, and is the most desirable and interesting end-product of Information Technology. To be able to discover and to extract knowledge from data is a task that many researchers and practitioners are endeavoring to accomplish. There is a lot of hidden knowledge waiting to be discovered – this is the challenge created by today’s abundance of data. Data Mining and Knowledge Discovery Handbook, 2nd Edition organizes the most current concepts, theories, standards, methodologies, trends, challenges and applications of data mining (DM) and knowledge discovery. | 560 Tao Li Sheng Ma and Mitsunori Ogihara Fig. . 1D TSA Tree Structure X is the input sequence. AXi and DXi are the trend and surprise sequence at level i. Denoising Noise is a random error or variance of a measured variable. Removing noise from data can be considered as a process of identifying outliers or constructing optimal estimates of unknown data from available noisy data. Wavelet techniques provide an effective way to denoise and have been successfully applied in various areas especially in image research. Formally Suppose observation data y yi . yn is a noisy realization of the signal x xi . xn yi xi Ei i 1 . n where e is noise. It is commonly assumed that e are independent from the signal and are independent and identically distributed iid Gaussian random variables. A usual way to denoise is to find x such that it minimizes the mean square error MSE MSE n n i - i xi 2. The main idea of wavelet denoising is to transform the data into a different basis the wavelet basis where the large coefficients are mainly the useful information and the smaller ones represent noise. By suitably modifying the coefficients in the new basis noise can be directly removed from the data. A methodology called WaveShrink for estimating x was developed in Donoho and Johnstone 1998 . WaveShrink includes three steps 1 Transform data y to the wavelet domain 2 Shrink the empirical wavelet coefficients towards zero and 3 Transform the shrunk coefficients back to the data domain. There are three commonly used shrinkage functions the hard the soft and the non-negative garrote shrinkage functions 27 Wavelet Methods in Data Mining 561 0 x A x x A 0 x A 8 x x A x A I A x x A 8h x 0 x - A A x A2 x x A where A e 0 is the threshold. Wavelet denoising is generally different from traditional Altering approaches and it is nonlinear due to a thresholding step. Determining threshold A is the key issue in WaveShrink denoising. Minimax threshold is one of commonly used thresholds. The .
