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Unformatted text preview: Week 11 Lecture 20 An Introduction to Wavelet regression De&nition: Wavelet is a function such that f 2 j= 2 & 2 j & k ;j;k 2 Z g is an orthonormal basis for L 2 ( R ) . This function is called &mother wavelet&, which can be often constructed from father wavelet& . The father wavelet & is not a wavelet, but we can construct wavelets from it, so it is equally important as mother wavelet. 1. Example : Haar wavelet (A. Haar, Math. Ann. (1910)) ( x ) = 1 x 2 [0 ; 1 = 2) 1 x 2 [1 = 2 ; 1) and 2 j= 2 & 2 j & k = 2 j= 2 x 2 2 & j k; 2 & j k + 2 & ( j +1) 2 j= 2 x 2 2 & j k + 2 & ( j +1) ; 2 & j k + 2 & j . Let ( t ) is the indicator function over the interval [0 ; 1) . Lets dene V j = span n jk = 2 j= 2 & 2 j & k , k 2 Z o Since ( x ) = (2 x 1) + (2 x ) , the following four properties are satised (i) ... V & 2 V & 1 V V 1 V 2 :::; (ii) f ( x ) 2 V j () f (2 x ) 2 V j +1 ; (iii) [ j 2 Z V j = L 2 ( R ) , (iv) there is a function such that f ( & k ) ;k 2 Z g is an orthonormal basis for V . An equivalent form of property (iii) : Recall that b f ( & ) = Z 1 &1 f ( x ) exp ( i&x ) dx It can be shown that, for any the property (iii) is equivalent to b (0) 6 = 0 and j b j is continuous at . A sketch of the proof is as follows: First, [ j 2 Z V j is invariance under all translations. Second, if g is orthogonal to [ j 2 Z V j , which implies g ( x ) is orthogonal to j ( x + t ) for all t , then Plancherel formula implies b g ( & ) b & 2 & j & = 0 a.s.. We know b 6 = 0 around . Let j ! 1 , we then conclude b g ( & ) = 0 a.s.. 1 From to : Observe that ( x ) = (2 x ) & (2 x & 1) and h ; i = 0 . If we de&ne W j = span n 2 j= 2 & 2 j & k , k 2 Z o then V L W = V 1 and more generally V j L W j = V j +1 We also see that J L j = &1 W j = V J +1 ! L 2 ( R ) , J ! 1 , in other words, f jk = 2 j= 2 & 2 j & k , j;k 2 Z g is an orthonormal basis for L 2 ( R ) : 2. Multiresolution analysis (MRA) &a general framework to con struct wavelet functions More generally, if there is such that f ( & k ) ;k 2 Z g is an orthonormal system, and 1 p 2 x 2 = X h k ( x & k ) and j b j is continuous at with b (0) 6 = 0 . Lets de&ne V j = span n 2 j= 2 & 2 j & k , k 2 Z o Then the following four properties are satis&ed i) ... V & 2 V & 1 V V 1 V 2 :::; ii) f ( x ) 2 V j () f (2 x ) 2 V j +1 ; iii) [ j 2 Z V j = L 2 ( R ) iv) there is a function such that f ( & k ) ;k 2 Z g is an orthonormal basis for V ....
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 '09
 HarrisonH.Zhou

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