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Exploring Wavelets: Interpolation and Modeling at Varying Scales

Dive into wavelets – a versatile form of interpolation capturing smooth and bumpy trends. Learn about models, polynomials, splines, wavelet families, expansion, and coefficients estimation techniques. Discover wavelet software, distribution approximations, and applications on irregularly spaced, spatial, temporal, and long-memory data. Explore wavelet packages in CRAN libraries.

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Exploring Wavelets: Interpolation and Modeling at Varying 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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