Wavelet transforms are linear in nature. Then how good they are for time series predictions. Can they analyze nonlinear relations in data well??
Wavelets can be useful feature detectors, and one of their standard advantage is to remove some trend or background effects, due to the vanishing moment property.
So first, this can act, sometimes, as a very low level non-linear effect remover.
Second, one of the successes of wavelets is the nonlinear approximation, where coefficients are selected upon their (highest) amplitude. The shrinkage/thresholding approach is nonlinear, as the shrinkage of a sum of two signals is not the same as the sum of the shrinkage of two signals.
Third, more generally, wavelet coefficients are often used as the input of nonlinear predictors, for instance neural networks.