Wavelets and multiwavelets in data-driven signal denoising

The Efromovich-Pinsker (EP) Block denoising procedures for multiwavelets is the latest in the wavelet denoising procedures to be examined when used with multiwavelets. The procedure can be used without regards to the correlational structure of the observations (or more specifically, the errors). EP block denoising use a hard threshold technique to kill or keep the wavelet coefficients. I will show the asymptotic properties of the EP block method, how it mimics the block oracle, and the rate of convergence on a Besov space. Then, in a search for the optimal block size, I will demonstrate that smaller block sizes are preferable to larger one. In fact the smallest block size allowed (block size = 2) was selected more often than any other in the search for the optimum one. That is, the block size that minimizes the integrated squared error (ISE).; Next, I will use the cristina family of biorthogonal wavelets to demonstrate the ability of the EP estimator to recover the derivative of a noisy signal. The cristina family of wavelets is indexed by the parameter s and s˜, for which I found the optimal value to be −0.2. This is the case where the multiwavelets are orthogonal. The fact that the cristina multiwavelets have a roughened (found through differentiation) and a smoothened (found through integration) form which are the extended members of this family of multiwavelets is the characteristic that allows us to find the derivative of a noisy signal.; And finally, I will compare (with Monte Carlo simulations) different denoising techniques on various intensities of noisy signals to see which has the smallest mean integrated square error. The signal will be contaminated with both normal and Tukey noise, to determine under what conditions will the EP block denoising estimator performs at least as well as the accepted methods. The results indicate the EP estimator performs better than the other methods (with the possible exception of the Sureshrink method in some cases), when reconstructing the blocks, bumps, and doppler signals contaminated with both types of noise. In general, the EP method outperforms the vector thresholding method, the accepted method of denoising using multiwavelets.