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Wavelets. form of interpolation phenomena of different scales

Wavelets. form of interpolation phenomena of different scales provides both smooth and locally bumpy parts trend. A wavelet model. Y(t) = S(t) +  (t) cp. polynomials, piecewise polynomials, splines, kernels, . mother, , and father, , wavelets

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Wavelets. form of interpolation phenomena of different scales

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  1. Wavelets. form of interpolation phenomena of different scales provides both smooth and locally bumpy parts trend

  2. A wavelet model. Y(t) = S(t) + (t) cp. polynomials, piecewise polynomials, splines, kernels, ...

  3. mother, , and father, , wavelets ((t-b)/a) / a ,  mother e.g. a = 2j , b = k2j jk (t) (t) = (2t) - (2t-1),  father

  4. , 

  5. 

  6. S(t) in L2 wavelet expansion j,kjkjk(t) l0j0(t) + jlkjkjk(t) if orthogonal jk = 0Tjk(t) S(t) dt / 0Tjk (t)2 dt discrete approximation

  7. Estimates. coefficients bjk = 0Tjk(t) Y(t) dt / 0Tjk (t)2 dt shrunken w(bjk/sjk) bjk sjk from higher-order coefficients Sure shrinkage w(b/s) = sign(b)(|b/s| - (2 log T))+

  8. Questions. Which (mother, father) wavelets Which K? Which shrinker? Which software? Approximate distribution? Other cases Irregularly spaced data Spatial Spatial-temporal Long memory

  9. Wavelet software in cran. libraries/packages wavelets, wmtsa, rwt, waveslim, wavethresh

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