Introduction Wavelet shrinage methods have been very successful in nonparametric regression. But so far most of the wavelet regression methods have be
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1 Wavelet Estimation For Samples With Random Uniform Design T. Tony Cai Department of Statistics, Purdue University Lawrence D. Brown Department of Statistics, University of Pennsylvania Abstract We show that for nonparametric regression if the samples have random uniform design, the wavelet method with universal thresholding can be applied directly to the samples as if they were equispaced. The resulting estimator achieves within a logarithmic factor from the minimax rate of convergence over a family of Holder classes. Simulation result is also discussed. Keywords: wavelets, nonparametric regression, minimax, adaptivity, Holder class. AMS 99 Subject Classication: Primary 62G07, Secondary 62G20.
2 Introduction Wavelet shrinage methods have been very successful in nonparametric regression. But so far most of the wavelet regression methods have been focused on equispaced samples. There, data are transformed into empirical wavelet coecients and threshold rules are applied to the coecients. The estimators are obtained via the inverse transform of the denoised wavelet coecients. The most widely used wavelet shrinage method for equispaced samples is the Donoho-Johnstone's VisuShrin procedure (Donoho & Johnstone (992), Donoho, Johnstone, Keryacharian & Picard (995)). It has three steps:. Transform the noisy data via the discrete wavelet transform; 2. Denoise the empirical wavelet coecients by \hard" or \soft" thresholding rules with threshold = p 2 log n. 3. Estimate function f at the sample points by inverse discrete wavelet transform of the denoised wavelet coecients. This procedure is adaptive and easy to implement. The computational cost is of O(n). And with high probability, VisuShrin estimator is at least as smooth as the target function. The estimator produced by the procedure achieves minimax convergence rates up to a logarithmic penalty over a wide range of function classes. In many statistical applications, however, the samples are nonequispaced. It is shown that the procedure might produce suboptimal estimator if it is applied directly to nonequispaced samples (Cai, 996). Wavelet methods for samples with nonequispaced designs have been studied by Cai and Brown (998) and Hall and Turlach (997). Cai and Brown (998) introduced a wavelet shrinage method for samples with xed nonequispaced designs based on approximation approach. It is shown that the estimator attains near-minimaxity across a range of piecewise Holder classes. Hall and Turlach (997) proposed interpolation methods for samples with random designs. They used samples with random uniform design as examples for their methods. Despite the asymptotic near-optimality for these nonequispaced methods, the estimators are computationally much harder to implement than VisuShrin for equispaced samples. In the present paper, we consider the special case of samples with random uniform design. We show that in this special case the samples can in fact be treated as if they were equispaced. That is, the VisuShrin procedure of Donoho and Johnstone can be applied directly to the data and the resulting estimator adaptively achieves within a logarithmic factor of the optimal convergence rate across a range of Holder classes. Therefore, we have a fast estimation procedure for samples with random uniform design. Simulation is conducted to evaluate the numerical performance of the method. It is shown that the mean squared error is comparable to that of the samples with truly equispaced designs. In Section 2 we describe the method and state the asymptotic optimality property of the estimator. Section 3 summarizes the simulation results. Some relevant results on wavelet approximation is presented in Section 4. Section 5 contains a concise proof of the main results. 2
3 2 Methodology 2. Wavelets Let and denote the orthogonal father and mother wavelet functions. The functions and are assumed to be compactly supported with associated discrete wavelet transform W. Assume has r vanishing moments and satises R =. Let j (x) = 2 j=2 (2 j x? ); j (x) = 2 j=2 (2 j x? ): And denote the periodized wavelets p j(x) = `2Z j (x? `); j(x) p = `2Z j (x? `) for x 2 [0; ]. For the purposes of this paper, we use the periodized wavelet bases on [0; ]. The collection f p j 0 ; = ; :::; 2 j 0 ; p j; j j 0 ; = ; :::; 2 j g constitutes such an orthonormal basis of L 2 [0; ]. Note that the basis functions are periodized at the boundary. The superscript \p" will be suppressed from the notations for convenience. This basis has an associated exact orthogonal Discrete Wavelet Trasnform (DWT) that transforms data into wavelet coecient domains. For a given square-integrable function f on [0; ], denote j = hf; j i; j = hf; j i: So the function f can be expanded into a wavelet series: f(x) = 2 j 0 = j0 j0 (x) + 2 j j=j 0 = j j (x): () Wavelet transform decomposes a function into dierent resolution components. In (), j0 are the coecients at the coarsest level. They represent the gross structure of the function f. And j are the wavelet coecients. They represent ner and ner structures of the function f as the resolution level j increases. We note that the DWT is an orthogonal transform, so it transforms i.i.d. Gaussian noise to i.i.d. Gaussian noise and it is norm-preserving. This important property of DWT allows us to transform the problem in the function domain into a problem in the sequence domain of the wavelet coecients with isometry of riss. 2.2 The Estimator Consider the nonparametric regression model: y i = f(x i ) + z i (2) 3
4 i = ; 2; :::; n(= 2 J ), x i 's are independently uniformly distributed on [0, ], z i 's are independent N(0; ) variables and independent of x i 's, and the noise level is xed and nown. The function f is an unnown function of interest. We wish to estimate f globally with small integrated mean squared error: R( ^f; f) = E Z 0 ( ^f(x)? f(x)) 2 dx: Let 0 x () < x (2) < ::: < x (n) be the order statistics of the x i 's. Now relabel y i 's and z i 's according the order of the x i 's. For convenience, we use the same label. So, y i = f(x (i) ) + z i : (3) Now we observed (x () ; y ); (x (2) ; y 2 ); ; (x (n) ; y n ) with x i independently uniformly distributed on [0, ]. So x 0 (i) s are not equispaced in general. But we pretend that x (i) is Ex (i) = i=(n + ). That is, we pretend to have an equispaced sample: ( n + ; y 2 ); ( n + ; y n 2); ; ( n + ; y n): We apply Donoho and Johnstone's VisuShrin procedure directly to y = fy ; y 2 ; :::; y n g. Let ~ = W n?=2 y be the discrete wavelet transform of n?=2 y. Write ~ = ( ~ j0 ; ; ~ j0 2 j 0 ; ~ j0 ; ; ~ j0 2 j 0 ; ; ~ ; ; ; ~ ;2 ) T : Here ~ j0 are the empirical coecients of the father wavelets at the lowest resolution level. They represent the gross structure of the function and are usually not thresholded. The coecients ~ j (j = j 0 ; ; J? ; = ; ; 2 j ) are ne structure wavelet terms. The empirical wavelet coecients is denoised via soft thresholding: ^ j = s ( ~ j ) = sgn( ~ j )(j ~ j j? ) + ; where = p 2n? log n. The whole function f is estimated by ^f (x) = 2 j 0 = ~ j0 j0 (x) + 2 j j=j 0 = ^ j j (x): If one is interested in estimating the function at the sample points, apply the inverse discrete wavelet transform to the denoised wavelet coecients. ( d f (x () )) n = = W? n =2 ^: The estimator is adaptive and easy to implement. 4
5 Theorem Suppose that the sample (x ; y ); (x 2 ; y 2 ); ; (x n ; y n ) is observed as in (2) and the mother wavelet has r vanishing moments. Then the estimator constructed above achieves within a logarithmic factor of the optimal convergence rate over the a range of Holder classes (M) (dened in Section 4) with =2 r. That is, sup f2 (M ) sup f2 (M ) for all M 2 (0; ) and 2 [=2; r]. E ^f? f 2 2 C ( log n n E d n f (x i )? f(x i ) 2 2 C ( log n n ) 2 +2 (4) ) 2 +2 (5) Remar: The same result holds for hard threshold estimator. The result shows that in the case of random design with uniformly distributed x i 's, we can treat it as if they are xed equispaced design. The constraint =2 is due to the approximation of f(x (i) ) by f(i=(n + )). 3 Simulations A simulation study is conducted to compare the estimator based on random-x samples with the estimator based on truly equispaced samples. The results show that the quality of the estimator based on random-x samples is comparable to the estimator based on equispaced samples. We consider four test functions of Donoho and Johnstone (994) representing dierent level of spatial variability. The test functions are plotted in Figure 2. For each of the four objects under study, we compare the estimators at two noise levels, one with signal-to-noise ratio SNR = 5 and another with SNR = 7. Sample sizes from n = 52 to n = 892 are considered. Table reports the mean squared errors over 200 replications of the four test functions: Doppler, HeaviSine, Bumps and Blocs. The wavelet used is the Symmlet \s8". The conventional t-test is used to test the signicance of the dierences between the MSEs of random design and the equispaced design. Table shows that the MSEs of random-x are worse than those of equispaced-x in 29 out of 40 cases, and are signicantly worse in the case of Doppler function. Random design is better in 6 cases, and the dierences in MSE between the two designs are insignicant in 5 cases at 95% level according to the t-test. The following plots compare the visual quality of the reconstructions. The solid line is the estimator and the dotted line is the true function. The sample size is 024 and SNR = 7. For each function, one is based on a sample with random-x and another is based on a sample with equispaced-x. One can see from the plots, the visual quality of the estimators are comparable, with the random-x reconstruction a little wobblier due to the stochastic nature of the design. For more simulation results, the readers are referred to Brown and Cai (997). 5
6 Random-x Equispaced-x HeaviSine Random-x HeaviSine Equispaced-x Doppler Doppler Figure : Comparisons of the reconstructions 4 Wavelet Approximation Wavelets provide smoothness characterization of function spaces. Many traditional smoothness spaces, for example Holder spaces, Sobolev spaces and Besov spaces, can be completely characterized by wavelet coecients. See Meyer [8]. In the present paper, we consider the estimation problem over a range of Holder classes. Denition We dene the following Holder classes (M): (i): if, (M) = ff : jf(x)? f(y)j M jx? yj g (ii): if >, (M) = ff : jf (bc) (x)? f (bc) (y)j M jx? yj 0 and jf 0 (x)j Mg where bc is the largest integer less than and 0 =? bc. The wavelet coecients of functions in a Holder class (M) decay exponentially as the resolution level j increases (see Daubechies (992)). Lemma Let f 2 (M) and let the wavelet function has r vanishing moments with r. Let j = hf; j i be wavelet coecients of f. Then j j j C 2?j(=2+) (6) where C is a constant depending on M and the wavelet basis only. If one has a sampled function ff(=(n + ))g n = with n = 2 J, one can utilize a wavelet basis to get a good approximation of the entire function f. Denote s() = min(; ). We have the following (also see Daubechies (992)). 6
7 Proposition Suppose that f 2 (M); and let J = hf; J i, then jn?=2 f( n + )? Jj C n?(=2+s()) : (7) According to this result, we may use n?=2 f(=(n + )) as an approximation of J. This means that if an equispaced sampled function is given, we can use a wavelet basis to get an approximation of the entire function f. To be more specic, we can use f n (x) = P n= n?=2 f(=(n + )) J (x) as an approximation of f. Furthermore, the approximation error can be bounded based on the sample size and the smoothness of the function. We quote the following result from Cai (996). Proposition 2 Suppose that f 2 (M). Let f n (x) = P n = n?=2 f(=(n+)) J (x) Then the approximation error satises 5 Proof f n? f 2 2 Cn?2s() : (8) We need some preparations before we prove the theorem. First some well nown results on the order statistics of uniform variables. Lemma 2 Let x i be iid uniform random variables on [0, ]. And let 0 x () < x (2) < ::: < x (n) be the order statistics. Then x () is distributed as Beta(, n - + ). In particular, Ex () = n + ; Ex2 () = + 2 (n + )(n + 2) ; V arfx ()g = (n + )? 2 (n + ) 2 (n + 2) : Now let us consider the noiseless case. We want to simply use f(x (i) ) as an approximation of f(i=(n + )) and wish to now the approximation error. Denote E the conditional expectation given x ; x 2 ; ; x n and denote E x the expectation with respect to x ; x 2 ; ; x n. Lemma 3 The upper bound of the approximation error is sup f2 (M ) n Ex (f(x () )? f( n + ))2 Cn?s() : (9) Proof: For a xed f 2 (M), we have jf(x)? f(y)j C jx? yj s(). Some algebra shows that n E x (f(x () )? f( n + ))2 C n C(n + ) +s() n(n + ) s() (n + 2) s() Cn?s() : 7 [E x (x ()? n + )2 ] s()
8 To prove the main result, we also need the following upper bound of the ris of threshold estimator of a univariate normal mean. Similar bound holds for hard threshold. The proof can be found in Cai (996). Lemma 4 Suppose that y N(; n? 2 ). Then ^ = s (y) with = p 2n? log n satises E(^? ) 2 (2 2 + n?2 2 ) ^ (2 log n + )n? 2 : (0) Proof of Theorem : We give the proof of (4) only. The proof of (5) is similar. First, some notations. We use j as coecients of j (the \father wavelets"), and use j as coecients of j (the \mother wavelets"). The ~ j0 are the empirical coecients at the coarsest level. They represent the gross structure of the function and they are usually not thresholded. The discrete wavelet transform W is an orthogonal transform, so it is norm-preserving. Let f(x) ~ = P i n?=2 y i J (x). Then f(x) ~ can be written as ~f(x) = i = [ Ji + A z } { (n? i 2 f( B z } { R z } { n + )? Ji) + (n? 2 f(x(i) )? n? i 2 f( n + )) + n? 2 zi ] Ji (x) [ j0 + ~a j0 + ~ b j0 + ~r j0 ] j0 (x) + j [ j + a j + b j + r j ] j (x): Here the j0 and j are the discrete wavelet transform of Ji, and liewise ~a j0 and a j the transform of the term A, ~ b j0 and b j the transform of B and ~r j0 and r j the transform of R. Let ~ j0 = j0 + ~a j0 + ~ b j0 + ~r j0 be the coecients of gross structure terms and set ^ j0 = ~ j0. Let 0 j = j + a j + b j and let ~ j = 0 j + r j be the noisy empirical wavelet coecients. Then ~ j N( 0 j; n? 2 ). Now denote = p 2n? log n and let ^ j = sgn( ~ j )(j ~ j j? ) +. Now the estimator of the regression function f is given by ^f (x) = and the ris function can be written as E ^f? f 2 2 = Lemma yields that Also we have ^ j0 j0 (x) + E x (E (^ j0? j0 ) 2 ) + j=j E (^ j0? j0 ) 2 = 2 j 0 n? 2 + j=j 0 j=j 0 ^ j j (x) E x (E (^ j? j ) 2 ) + j=j 2 j: 2 j = O(n?2 ): () (~a j0 + ~ b j0 ) 2 2 j 0 n? ~a 2 j ~ b 2 j0 : (2)
9 Applying Lemma 4 to the term E (^ j? j ) 2, we have j; E (^ j? j ) 2 2E (^ j? j) a 2 j + 2b 2 j 8j 2 ^ 3n? 2 log n + 0a 2 j + 0b 2 j + n?2 2 : Let J be an integer satisfying 2 J (n= log n)=(+2). Then, E (^ j? j ) 2 J? j=j 0 3n? 2 log n + C(n? log n) j=j j=j 0 It follows from Proposition and Lemma 3 that E x ( ~a 2 j 0 + ~ b 2 j0 + For 2, s() Acnowledgment j=j 0 j=j 0 a 2 j = i b 2 j) = n 8 2 j + 0 a 2 j + 0 j=j 0 j=j 0 (a 2 j + b 2 j) + n? 2 b 2 j: (3) (n? 2 f( i n + )? Ji) 2 C n?2s() (4) n i= 2. Now it follows from () { (5) that +2 E ^f? f 2 2 C ( log n n ) 2 +2 : i E x (f(x (i) )? f( n + ))2 Cn?s() : (5) The authors than the referee for the helpful comments which improve the presentation of this wor. References [] Brown, L.D. & Cai, T. (997). Wavelet Regression For Random Uniform Design, Technical Report #97-5, Department of Statistics, Purdue University. [2] Cai, T. & Brown, L.D. (998). Wavelet Shrinage For Nonequispaced Samples. Ann. Statist., to appear. [3] Cai, T. (996). Nonparametric Function Estimation via Wavelets. Ph.D Thesis, Cornell University. [4] Daubechies, I. (992). Ten Lectures on Wavelets SIAM: Philadelphia. [5] Donoho, D.L. & Johnstone, I.M. (994). Ideal Spatial Adaptation via Wavelet Shrinage. Biometria, 8, 425{455. 9
10 [6] Donoho, D.L., Johnstone, I.M., Keryacharian, G. & Picard, D. (995). Wavelet Shrinage: Asymptopia?, J. Roy. Stat. Soc. Ser. B, 57, 30{369. [7] Hall, P. & Turlach, B.A. (997). Interpolation Methods for Nonlinear Wavelet Regression with Irregularly Spaced Design. Ann. Statist [8] Meyer, Y. (990). Ondelettes et Operateurs: I. Ondelettes. Hermann et Cie, Paris. 6 Appendix The four test functions represent dierent degrees of spatial variability. The functions are normalized so that every function has standard deviation 0. Formulae of the test functions can be found in Donoho and Johnstone (994). Doppler HeaviSine Bumps Blocs Figure 2: Test Functions 0
11 SNR = 5 SNR = 7 n Uniform-x Equispaced-x % di. Uniform-x Equispaced-x % di. Doppler % % % % % % % % % % HeaviSine % % % % % * % % % * Bumps % % * % % % % % % % Blocs * * % % % % % % % % Table : Mean Squared Errors From 200 Replications. The \% di." columns are the percentage dierences between the MSEs of random-x and of equispaced-x. The percentage dierences are set to 0 when the dierences are insignicant at 95% level according to the conventional t-test.
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