How wavelet analysis works as a dimension reduction techniques? The approximations coefficients at higher level of decomposition are the fewer retained coefficients from the originial ?Is it correct?
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Well, not really. The wavelet transform (the discrete to be precise) ist just a base transformation of a signal. However, it can be used to compress signals (dimension reduction in some sense), due to the fact that the energy of most natural signals is compactified under the wavelet transform. The continuous wavelet transform does even the opposite. It blows up your signal and introduces a lot of redundancy.
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$\begingroup$ I did not get your answer very well, but, for example, to decompose the original signal(0) into level higher >0, is translated by getting few coefficients = (aj= a0/j*2), so it is similar to dimension reduction ? $\endgroup$ Commented Jul 15, 2012 at 9:23
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$\begingroup$ If you have a special signal together with a special wavelet, it in fact does a dimension reduction, but, as I said, not in general. For instance, a pure sinusoid reduced with a shannon wavelet would be such a special case. In that case a simple fourier transform does the same, btw. But most signals are folded in a nonlinear way and can not be reduced by linear transform. $\endgroup$ Commented Jul 16, 2012 at 7:25