One of the basic techniques in the vast wavelet literature is the wavelet transform. Advantages provided by such a transform have been demonstrated in

Transcription

1 K-Stationarity and Wavelets Bing Cheng and Howell Tong Institute of Mathematics and Statistics University of Kent at Canterbury Kent CT2 7NF, UK April 29, Introduction Wavelets oer exciting possibilities for statistical problems. For example, Donoho et al. (1995) aroused extensive discussions and gave a very substantial list of references. One of the most recent papers on the subject is Hall and Patil (1996). Numerous major statistical society meetings have featured wavelets in statistics. For example, in the Royal Statistical Society Conference in September 1994, the talks by Nason and Silverman and by McCoy and Walden gave many references and highlighted some of the possibilities in a statistical context. The Institute of Mathematical Statistics, the International Statistical Institute, the Bernoulli Society World Congress, the Franco-Belgian meeting of statisticianss and many others have all featured similar papers in their international meetings. Numerous interesting applications in signal processing, computer vision, numerical analysis, applied mathematics, physics and others have been reported in the mathematical, engineering and physical literature. Some of these also discuss statistically related problems. (See for example, Chui, 1992, Mallat, 1989b, 1989c, 1991, Segmen and Zeevi, 1993 and many others) 1

2 One of the basic techniques in the vast wavelet literature is the wavelet transform. Advantages provided by such a transform have been demonstrated in terms of the ability to handle long-memory stochastic processes, edge detection with noisy data, time-frequency resolution of random signals (See for example, Flandrin, 1992, Cheng and Kay, 1993, and Basseville et al 1992). Behind all the statistically related applications is a stochastic process and this suggests that there is a common theoretical basis, namely a common generic wavelet representation of a stochastic process. Needless to say, such a representation theory is fundamental. The situation is not unlike the spectral representation of a stationary time series. Cambanis and Masry (1994) used wavelets as basis functions for approximations of stochastic processes. Cheng and Tong (1996) have developed an alternative approach. In fact, there are important dierences between the two approaches. (1) The more fundamental feature behind the wavelet methodology is its ability to construct a uniform multiresolution representation and decomposition (See Mallat, 1989b, for example) rather than its ability to approximate any shaped-signals, which can be equally well provided by such techniques as the radial basis function, the spline, the kernel and others. In particular, the approximation provided by the wavelets for deterministic signals is uniformly functionindependent (See Daubechies, 1992, for example) and the same result should be expected for random signals. However, the approximation obtained by Cambanis and Masry (op. cit.) is apparently function-dependent and has thus under-utilised the power of wavelets. (2) The rate of convergence they have obtained is function-dependent in that they assume that the function to be approximated has the same degree of smoothness as the wavelet functions. However, in practice such a rate of convergence is of very limited use since we rarely know the form of the true function and even if we assume that it is known the functions the degrees of smoothness could still vary. (3) The assumptions they have made for the approximation are somewhat obscure and not easy to verify. This paper is a continuation of the development of Cheng and Tong (op. cit.). The plan of this paper is as follows. In section 2, we quote our representation thoerem for a general stochastic process (Theorem 2.1) to be followed by a decomposition theorem (Theorem 2

3 2.2). The main results of this paper are in section 3. In section 3.1, we introduce the notion of k-stationarity, i.e. resolution- dependent stationarity. In section 3.2, we run some simulations for k-stationary processes. In section 3.3, we introduce the notion of k-weak stationarity. In section 4, we give a brief concluding discussion. 2 Wavelet representation of a general stochastic process First, we quote the representation theorem developed in Cheng and Tong (1996). Let (; F; P ) denote a probability space. Let L 2 ( T ) = fx(t)je Z T X 2 (t)dt < 1g: Recall that L 2 ( T ) is a Hilbert space. Now, dene a sequence of subspaces, V k ( T ) say, of L 2 ( T ) for each k 2 Z = f0; 1; : : :g by V k = V k ( T ) = ( X 2 L 2 ( T ) : X(t) = X l kl kl (t); X l where f kl g l2z is a squence of random variables and kl (t) is dened by In the following we use a continuous scaling function. ) E 2 kl < +1 ; (3:1) kl (t) = p 2 k (2 k t? l) : (3:2) Theorem 2.1: (Representation Theorem) fv k ( T )g k2z constitute a multiresolution approximation to the space L 2 ( T ). Specically, (M0) for each k 2 Z, V k = V k ( T ) is a closed subspace of L 2 ( T ); (M1) V k V k+1 ; 8 k 2 Z; (M2) +1[ V k is dense in L 2 ( T ) and +1\ k=?1 k=?1 (M3) X(t) 2 V k, X(2t) 2 V k+1 ; 8k 2 Z; (M4) X(t) 2 V k, X(t? 2?k l) 2 V k ; 8l 2 Z. 3 V k = f0g;

4 Note that the assumption of the nitteness of E R T X(t) 2 dt, which will exclude the conventional stationary processes over the innite time interval, can be weakened by introducing a weighting function, say s, such that E R T X(t) 2 s(t)dt < 1. Cheng and Tong (op. cit.) have also argued that from a practical point of view a stationary time series over an inntie time interval is, in any case, an unrealistic assumption, which is often made for mathematical convenience. Cheng and Tong (op. cit.) have also obtained the following Decompositon Theorem. Theorem 2.2: (Time-Frequency Decomposition Theorem) L 2 ( T ) = X k W k ; that is 8X 2 L 2 ( T ), we have X(t) = P k Pl kl W kl (t); where kl = R +1?1 X(t)W kl (t)dt: Denition 2.1: For X 2 L 2 ( T ), and real numbers a (a > 0) and b, the wavelet trans- dt: form S a;b X, say, for the stochastic process X is dened by S a;b X = R +1?1 X(t)W t?b a Clearly, if we use compactly supported wavelets such as Daubechies wavelet, then the wavelet transform S a;b X localizes X in both the time-domain and the frequency-domain. Theorem 2.2 indicates that we can reconstruct X from wavelet transform over a discrete grid by X(t) = P P p k l 1 2 S2 k?k ;2?k l XW kl (t): 3 Stationarity, Resolution and k{stationarity Resolution is actually a fundamental feature when viewing any signal or image. We pursue the notion of stationarity from this perspective. 3.1 Stationarity in distribution Recall that 8X 2 V k ( T ), X(t) = X l kl kl (t): (1) 4

5 Therefore, X X(t + s) = X l = X l = X l = l kl kl (t + s) p kl 2k (2 k (t + s)? l) p kl 2k (2 k t? (l? 2 k s)) k;l+2 k s kl (t): (2) Denition 3.1: fx(t)g is said to be stationary at the kth level of resolution (or simply k-stationary) if 8n 1, t 1 ; : : : ; t n 2 T and l 2 Z, fx(t)g satises fx(t 1 + 2?k l); : : : ; X(t n + 2?k l)g d = fx(t 1 ); : : : ; X(t n )g where \ d =" denotes that the two sides have the same distribution. Now, dene a random vector by k () = f kl proposition. : l 2 Zg: Then, we have the following Proposition 3.1: fx(t)g is a stationary process at the kth level of resolution if and only if k () is a stationary sequence, that is, k () d = k ( + l); 8l 2 Z. Proof: X(t i + 2?k l) X = k;l+l 0 k;l 0(t i ) X l 0 = k;l+l 0p 2k (2 k t i? l 0 ) X l 0 = k;l+l 0 +2 k ti kl 0(0) l 0 and X(t i ) = X l 0 k;l 0 +2 k ti kl 0(0): 5

6 Therefore, fx(t)g is k{stationary if and only if k () d = k ( + l) 8l 2 Z: Proposition 3.2: Suppose X 2 V ~ k ( T ). If fx(t)g is k{stationary, then for ~ k < k, fx(t)g is also ~ k{stationary. Proof: We only need to prove the case when ~ k = k?1. Since k?1;i (t) = P j h(j?2i) k;j (t); we have k?1;i = X = X j = j Z +1?1 X(t) k?1;i (t)dt h(j? 2i) Z +1?1 h(j? 2i) k;j : X(t) k;j (t)dt Therefore, it follows that k?1;i+l = X j h(j? 2(i + l)) k;j = X j = X j h(j? 2l)? 2i) k;j h(j? 2i) k;j+2l : Thus, k?1 () d = k?1 ( + l): By Proposition 3.1, X is (k? 1)-stationary and the proof is completed. (Actually, we may have a more general form for k{stationarity.) Proposition 3.3: If k 0 k, then k{stationarity implies k 0 {stationarity. Proof: 8n 1, t 1 ; : : : ; t n 2 T and 8l 2 Z, fx(t 1 + 2?k0 l); : : : ; X(t n + 2?k0 l)g = fx(t 1 + 2?k l 1 ); : : : ; X(t n + 2?k l 1 )g; where l 1 = 2 k?k0 l. Since k 0 k, l 1 2 Z. Therefore, the above random vector has the same joint distribution as that of fx(t 1 ); : : : ; X(t n )g since fx(t)g is k{stationary. 6

7 Proposition 3.4: Suppose that every nitely dimensional distribution of fx(t)g is continuous. Then fx(t)g is a stationary process if and only if fx(t)g is stationary at all levels of resolution. Proof: \)": Obvious. \(": Consider the distribution of fx(t 1 + r); : : : ; X(t n + r)g. Pick any t 1 ; : : : ; t n 2 T and any r 2 T. There is a sequence fl k jk 2 Zg Z such that lim k!+1 2?k l k = r: Since fx(t)g is stationary at all levels of resolution, fx(t 1 + 2?k l k ); : : : ; X(t n + 2?k l k )g d = fx(t 1 ); : : : ; X(t n )g: Let k! 1. We have fx(t 1 + r); : : : ; X(t n + r)g Therefore fx(t)g is a stationary process. d = fx(t 1 ); : : : ; X(t n )g: 3.2 Generating k-stationarity and modelling non-stationarity Let integers ~ k = k nest 1 and k = k stat 1 with ~ k k. Suppose that fx(t)g is k?stationary and X 2 V k ~ ( R). So we have a representation for fx(t)g by X(t) = 1X l=?1 ~ k;l ~ k;l (t) Before we prove Proposition 3.5 below, we rst prove a lemma. Let integer k free = k nest? k stat = ~ k? k and integer ws = ws free = 2 k free. Lemma: Suppose X 2 V ~ k ( R). Then fx(t)g is k?stationary (k ~ k) if and only if 8j 2 Z, ~ k;wsj+ has the same distribution as ~ k;, where ~ k; = (: : : ; ~ k;?1 ; ~ k;0 ; ~ k;+1 ; : : :) ; 7

8 where refers to the transpose of a vector in l 2 (Z). Proof: Dene Then we have k; (t) = (: : : ; ~ ~ (t); k;?1 ~ (t); k;0 ~ (t); : : :) k;+1 X(t) = ( ~ k; (t)) ~ k; where refers to the inner product in l 2 (Z), that is 8a; b 2 l 2 (Z), a b = P l a l b l. For n 1, t 1 < t 2 ; ; t n, and l 2 Z, X(t i + 2?k j) = X l ~ k;2 ~ k?k j+l ~ k;l (t i) = ( ~ k; (t i)) ~ k;2 ~ k?k j+ : So we have (X(t 1 + 2?k j); : : : ; X(t n + 2?k j)) = ( k; (t ~ 1) k;2 ~ k?k ~ j+ ; : : : ; k; (t ~ n) k;2 ~ k?k ~ j+ ) and similarly (X(t 1 ); : : : ; X(t n )) = ( k; (t ~ 1) k; ; : : : ; ~ ~ k; (t n) k; ): ~ Since ws = ws free = 2 ~ k?k, fx(t)g is k?stationary (k ~ k) if and only if 8j 2 Z, ~ k;wsj+ has the same distribution as ~ k; Now, dene a ws-dimensional vector, Vi ws, by V ws i = ( ~ k;wsi+1 ; : : : ; ~ k;wsi+ws ) Proposition 3.5: Suppose X 2 V ~ k ( R), then fx(t)g is k?stationary (k ~ k) if and only if fv ws i g i2z is a stationary sequence of ws-dimensional random vectors, with ws = ws free ; Proof: "=)": For M 1, let integers i 1 < : : : < i M. For n 2 Z, denote l(n; m) = ws l + m for 1 m ws. Then (V ws i 1 +j; : : : ; V ws im +j) = ( ~ k;l(i 1 ;1)+j ; : : : ; ~ k;l(i 1 ;ws)+j ; : : : : : : ; ~ k;l(im ;1)+j ; : : : ; ~ k;l(im ;ws)+j ) 8

9 and similarly (V ws i 1 : : : V ws ) = im (~ ; : : : ; k;l(i 1 ;1) ~ ; : : : : : : ; k;l(i 1 ;ws) ~ ; : : : ; k;l(im ;1) ~ ): k;l(im ;ws) Therefore (V ws i 1 +j; : : : ; V i M +jws) =d (V ws i 1 + : : : + V ws im ): So fv ws i g i2z is a stationary sequence of ws-dimensional random vectors with ws = ws free. "(=": For all M 1, and l 1 < < l M, we wish to show that ( ~ k;l 1 +wsj ; : : : ; ~ k;lm +wsj ) =d ( ~ k;l 1 ; : : : ; ~ k;wsj ) for 8j 2 Z. Now, group the indices l 1 ; : : : ; l M into patches by 0 i(p? 1) i(p) M, p = 1; : : : M? 1 satisfying l(i(p); 1) l i(p?1)+1 : : : l i(p) l(i(p); ws), p = 1; : : : ; M? 1. (Some patches may be empty.) Therefore, ( ; : : : ; k;l ~ i(p?1) +1+wsj ~ k;l i(p) +wsj ) is part of the vector V ws ws i(p)+j. Since fvi g i2z is stationary, we have (V ws i(1)+j; : : : ; V ws i(m?1)+j) = d (V ws i(1); : : : ; V ws i(m?1)): Further, the distribution of ( ; : : : ; k;l ~ 1 +wsj ~ k;lm +wsj ) is a marginal distribution of (V ws ; : : : ; V ws ), and i(1)+j i(m?1)+j (~ ws k;l 1 ; : : : ; ~ ) is a marginal distribution of (V ; : : : ; V ws ), k;lm i(1) i(m?1) correspondingly. Therefore, we have ( ~ k;l 1 +wsj ; : : : ; ~ k;lm +wsj ) =d ( ~ k;l 1 ; : : : ; ~ k;lm ): By the lemma, fx(t)g is k-stationary. Now, suppose that X 2 V k ~ ( <) and X is k-stationary but not (k+1)-stationary. First, we introduce some notations. Recall that we have denoted k nest by ~ k, k stat by k, k free = k nest?k stat and ws free = 2 k free. Given data, usually k stat is unknown and needs to be determined. In the following, we will show that k-stationarity is related to "window-based stationarity". That is, it depends on how we view the data from a window with a certain window size. Let k view be an integer, 9

10 and k arbi = k nest? k view, ws view = 2 k view and ws arbi = 2 k arbi. Let t 0 be a real number, for j 2 Z and 0 j 0 ws arbi? 1. We generate data Y 1 ; : : : ; Y n by Y i = Y (t 0 ; j 0 ; j) = X(t 0 + 2?knest (j 0 + j ws arbi )) where i = j ws arbi + j 0. Since 2?knest ws arbi = 2?k view, we have Y i = X(t 0 + 2?knest j 0 + 2?k view j) = X(t + 2?k view j) where t = t 0 + 2?knest j 0. For 1 i n, based on ws arbi, we can always decompose i by i = j ws arbi + j 0 with 0 j 0 ws arbi and 0 j J. Equivalently, such a decomposition allows us to look at the data fy i g through a window with size ws arbi. Specically, given each 0 j 0 ws arbi, we screen j for 0 j J, where we call J the screening length. For the same data, by varying the window size, we should be able to visualize stationarity and non-stationarity. Now, Y i = X(t 0 + 2?knest j 0 + 2?k view j) = X l ~ k;l ~ k;l (t 0 + 2?knest j 0 + 2?k view j) = X l ~ k;l+j 0 +ws arbi j ~ k;l (t 0): Thus, by Proposition 3.5, when k view k stat, for each 0 j 0 ws arbi?1, fy (t 0 ; j 0 ; j)g j2z is a stationary sequence. When k view > k stat, for each 0 j 0 ws arbi, fy (t 0 ; j 0 ; j)g j2z could be a non-stationary sequence. Hence by viewing the data fy i g through a window with size ws arbi, we will see k stat -stationarity and (k stat + 1)-non-stationarity, by varying k view and j 0. In all the following gures, we only plot j0 = 0 for 0 < k view < k nest? 1. Example 1: Let k stat = 2 and k nest = 7. So k free = k nest? k stat = 5 and ws free = 2 k f ree = 2 5 = 32. By Proposition 3.5, we can rst generate a 32-dimensional random vector V 32 = ( 7;1 ; : : : 7;32 ) and then generate other random coecients f 7;l g by independent and 10

11 Figure 1: Data window 35 data.g X_t t identically distributed samples of V 32. For V 32 itself, we assume it is a multivariate normal random vector i.e. V 32 d N(M; ) with M = (m 1 ; m 2 ; : : : ; m wsfree ) and = diag(1; 2 : : : ; ws 2 free ), with ws free = 32. In this example, we use a screening length of 64 for varying j and t 0 = 0. The mean values are m l Figure 1 depicts the noise-free data (i.e. l = 0). Figure 2 depicts the result of viewing Y (0; 0; j) = X(2?k view j) for k view = 1; : : : ; 5 (i.e. bottom-up) and correspondingly ws arbi varies from 64 to 4 and j = 1; : : : ; 64. (In order to have a good viewing, we have presented the gures in a bottom-up direction in accordance with k view ). From this view window, we can clearly see that, when k view k stat = 2, i.e. k view = 1 and 2, we can see two straight lines which correspond to X(j=2) and X(j=4) (one of them merging into the horizontal axis). However, if we zoom in on fx(t)g further by = l 11

12 Figure 2: View window View window for stationarity and non-stationarity 120 X_t t letting k view = 3, which corresponds to X(j=8). We can see that the path of fx(j=8)g 64 j=1 jumps between two points and this is due to the fact that these two points have two dierent mean values. Thus, this is an example of a (noise-free) 2-stationary but 3-non-stationary signal. The non-stationarity prevails when k view = 4 and 5, i.e. when we zoom in fx(t)g further to X(j=16) and X(j=32), correspondingly. Example 2: We continue to use k stat = 2, k nest = 7, mean values m l = l for l = 1; : : : ; 32 and screening length = 64, but variance l 2 = 4. Figure 3 depicts the data fy i g. Figure 4 depicts the results of viewing Y (0; 0; j) = X(2?k view j) for k view = 1; : : : ; 5 (bottom-up direction) and j = 1; : : : ; 64. From Figure 4, we can see that, when k view = 1 and 2, the sampled data X(j=2) and X(j=4) have some random uctuations which may be attributed to the random coecients kkest ; l d N(m l ; l 2 ) with m l = l mod(32); and 2 = 4: When k view moves from level 2 to 3, i.e. we zoom in fx(t)g to X(j=8), we can see clearly 12

13 Figure 3: Data window 35 data.g X_t t non-stationary uctuations of the data and such non-stationary uctuations continue with k view increased to 4 and 5. Example 3: Let k stat = 3 and k nest = 8. So k free = k nest? k stat = 5 and ws free = 2 k f ree = 2 5 = 32. Let the mean value of 8;l be m l = cos(2 l=ws free ) mod(32); and 2 l = 0:1. Again the screening length is set at 64. Figure 5 depicts the time plot and Figure 6 depicts the results of varying k view from 1 to 6, (correspondingly, varying ws arbi from 128 to 4). From Figure 6, we can see, when k view is increased from 3 to 4, there is a shift from a stationary sequence to a non-stationary one. Example 4: (Example 3 continued.) Still choose k stat = 3, k nest = 8 and the screening length to be 64. If we increase the variances of the random coecients - f 8;l g, we could kill o 3-stationarity and the distinction between stationarity and non-stationarity. Let 13

14 Figure 4: View window View window for stationarity and non-stationarity X_t t Figure 5: Data window 2 data.g X_t t 14

15 Figure 6: View window View window for stationarity and non-stationarity 25 X_t t 2 l = 1:0. Figure 7 depicts the time plot and Figure 8 depicts the "zoom-in" sampling of fx(t)g. Example 5: Let k stat = 3, k nest = 8 and screening length be 64. So ws free = 32. Let m l 1; and 2 l = l 2 mod(ws free ): Figure 9 depicts the time plot and Figure 10 the results of varying k view from 1 to 6, (correspondingly varying ws arbi from 128 to 4). From Figure 10, we can see, as k view increases from 3 to 4, there is a clear switch from a stationary sequence to a non-stationary one. 3.3 k-weak stationarity From X(t) = X l kl kl (t), we have X(t + s) = X l k;l+2 k t kl (s). Now, R(t; t + s) = Ef[X(t)? EX(t)][X(t + s)? EX(t + s)]g: 15

16 Figure 7: Data window 4 data.g X_t t Figure 8: View window View window for stationarity and non-stationarity 25 X_t t 16

17 Figure 9: Data window data.g X_t t Figure 10: View window View window for stationarity and non-stationarity X_t t 17

18 Therefore = X m;l Ef[ km? E km ][ k ; l + 2 2t? E k;l+2 k t ]g km (t) kl (s) = X m;l Ef[ k;m+2 k t? E k;m+2 k t ][ k;l+2 k t? E k;l+2 k t ]ig km (0) kl (s): Dene C k (m; l) = Ef[ k;m? E k;m ][ k;l? E k;l ]g. Then, we have R(t; t + s) = X m;l C k (m + 2 k t; l + 2 k t) km (0) kl (s): (3) Denition 3.2: fx(t)g is weakly stationary at the k-level of resolution (or simply k{ weakly stationary) if, 8s 2 T and l 2 Z, R(2?k l; 2?k l + s) = R(0; s) = R k (s), say. Proposition 3.6: If ~ k < k and fx(t)g is k{weakly stationary, then fx(t)g is ~ k{weakly stationary. Proof: Simply note that with l 1 = 2 k?~ k l 2 Z, R(2?~k l; 2?~k l + s) = R(2?k l 1 ; 2?k l 1 + s) = R(2?~k l; 2?~k + s) = R(0; s): Proposition 3.7: fx(t)g is weakly stationary if and only if fx(t)g is weakly stationary at all levels of resolution. Proof: \)": Obvious. \(" : 8r 2 T and for each k 2 Z, there is a l k 2 Z such that lim k!1 2?k l k = r: Now, R(2?k l k ; 2?k l k + s) = R(0; s), 8s 2 T. Letting k! 1, we have R(r; r + s) = R(0; s): because, by Lemma 3.1 of Cheng and Tong (1996), R(; ) is a continuous function. Proposition 3.8: If fx(t)g is weakly stationary and R(s) = R(t; t + s) 2 L 2 (T ), then R(s) = X k;l d kl W k;l (s): 18

19 Proof: Simply note that L 2 (T ) = P k L Wk : 4 Discussion In this paper, based on our representation and decomposition theory for a general stochastic process, we have introduced the new notion of k-stationarity. This new notion reects the fact, which is perhaps not commonly realized todate, that stationarity is relative. It is resolution-dependent. We suggest that the new notion of k-stationarity provides us with a deeper understanding of the absolute notion of stationarity and its relation with nonstationarity. Moreover, Proposition 3.5 opens a way to generating k-stationary processes as we have demonstrated. It also opens a way to a number of fascinating new avenues. We list some of these below which we shall report elsewhere: (1) spectral analysis of k-stationary process; (2) detection of k-stationarity from data; (3) the use of a time varying ws free to characterize k-stationary process; (4) parallel development in respect of k-self similar processes. There are many other possibilities. For example, we have found one way to calibrate a non-stationary process by introducing the notion of k-stationarity. In this respect, we can regard the window size ws free as 1 (k nest = k stat ), and k stat = 1 corresponds to the conventional stationary process. On the other hand, when ws free = 1, we have, in a sense, the most non-stationary process. In other words, we can classify non-stationary processes into classes: each non-stationary class has a degree of non-stationarity via ws = ws free. Next, according to Proposition 3.5, fx(t)g is k-stationary process if only if the random coecient vector fv ws i g is a ws-dimensional stationary sequence. This suggests one potential way to bootstrap fx(t)g, namely by bootstraping its coecient vectors. Especially when fv ws i g are independent and identically distributed, we can use the conventional bootstraping method for independent data by simply permuting the former. Finally, by considering the conditional expectation E[Vi ws jfx t g], we can transform such wavelet-based non-stationary 19

20 processes into non-linear processes. In conclusion, the wavelet representation and decomposition theory that we have developed points to very fertile statistical areas, both in terms of statistical theory and statistical practice. References [1] Basseville, M., Benveniste, A., Chou, K.C. Colden, S.A., Nikoukhan, R. and Willsky, A.S. (1992). Modeling and estimation of multiresolution stochastic processes. IEEE Trans. Inf. Th., 38, [2] Burt, P., and Adelson, E. (1983). The Laplacian pyramid as a compact image code. IEEE Trans. Comm., 31, [3] Cambanis, S. and Masry, E. (1994). Wavelet approximation of deterministic and random signals: convergence properties and rates. IEEE Trans on Information Theory 40 pp [4] Cheng, B. and Kay, J. (1993). Multiresolution image reconstruction using the wavelet transform. Submitted to Statistics and Computing. [5] Cheng, B. and Tong, H. (1996). A Theory of Wavelet Representation and Decomposition for a General Stochastic Process. to appear in a Festschrift in Honour of Professor E.J.Hannan, ed. M. Rosenblatt and P.M.Robinson, Springer-Verlag. [6] Chui, C.K. (Ed.) (1992) Wavelet Analysis and its Applications, Vols. 1 and 2, Academic Press, London. [7] Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 41, [8] Daubechies, I. (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Th., 36,

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