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 = 2 j, b = k2 j  jk (t)  (t) =  (2t) -  (2t-1),  father

4. , 

5. 

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

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

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

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