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1 저작자표시 - 비영리 - 변경금지 2.0 대한민국 이용자는아래의조건을따르는경우에한하여자유롭게 이저작물을복제, 배포, 전송, 전시, 공연및방송할수있습니다. 다음과같은조건을따라야합니다 : 저작자표시. 귀하는원저작자를표시하여야합니다. 비영리. 귀하는이저작물을영리목적으로이용할수없습니다. 변경금지. 귀하는이저작물을개작, 변형또는가공할수없습니다. 귀하는, 이저작물의재이용이나배포의경우, 이저작물에적용된이용허락조건을명확하게나타내어야합니다. 저작권자로부터별도의허가를받으면이러한조건들은적용되지않습니다. 저작권법에따른이용자의권리는위의내용에의하여영향을받지않습니다. 이것은이용허락규약 (Legal Code) 을이해하기쉽게요약한것입니다. Disclaimer

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4 Multiple Wavelet Shrinkage by Changha Shin A Thesis submitted in fulfillment of the requirement for the degree of Master of Science in Statistics The Department of Statistics College of Natural Sciences Seoul National University February, 2016

5 Abstract Wavelets are used as basis functions in representing other functions. In this paper, we focus on multiple wavelets, which means number of mother and father wavelet is more than one. Meanwhile, noisy data is hard to find hidden inner side of the function. Solution is wavelet shrinkage method. It is effective method to eliminate the noise and see the original form of function. In this paper, we suggest conditions for multiple wavelet shrinkage s adoption. And we find useful properties of multiple wavelet shrinkage in time-series data. Keywords : Discrete wavelet transform, Multiple wavelet, Multiple wavelet shrinkage, Wavelet, Wavelet shrinkage Student Number :

6 Contents 1 Introduction 1 2 Multiple Wavelet and Shrinkage Multiple wavelet Properties Multiple wavelet shrinkage Simulation Study and Data Analysis Example function BabyECG data analysis Conclusion 19 References 20

7 List of Figures 1 Wavelet decomposition coefficients Geronimo multiple wavelets Wavelet from the daubechies family Multiple wavelet shrinkage process Example function Ynoise (added noise 0.1) Single shrinkage, MSE= Multiple shrinkage, MSE= Ynoise (added noise is 0.3) Single shrinkage, MSE= Multiple shrinkage, MSE=

8 List of Figures 12 Ynoise (added noise is 0.5) Single shrinkage, MSE= Multiple shrinkage, MSE= BabyECG Reconstructed function by multiple wavelet shrinkage Reconstructed function by single wavelet shrinkage

9 Chapter 1 Introduction Before, we say multiple wavelet shrinkage, let s briefly cover the concepts and importance of wavelet and wavelet shrinkage. Wavelets mean small waves. It is compound of wave and let (small). Small waves mean they are compactly supported, that is they should be integrated to zero. And wavelets are used as basis functions in representing other functions. The main application and usage of wavelet is by shrinkage. In traditional statistical point of view in data, is all data are regarded as true signal and noise (called aggregation). On the contrary, in wavelet area in statistics, we regard not all data have signal and noise (called sparsity). This is the main point why we use shrinkage. So, if one observes noisy data, it can be transformed by the discrete wavelet transform. And then thresholding the small coefficients, then operate inverse transform to reconstruct the function. That is called wavelet shrinkage. When we use wavelet shrinkage method to noisy function, we can find specific properties of hidden inner side of the function. So it is effective method to eliminate the noise and see the original form of function. It is like restoring contaminated ancient works of art contained traces of the years. Multiple wavelet is special case of above ordinary wavelet, number of 1

10 mother and father wavelet is more than one. In this paper, our main purpose is brighten the advantages of multiple wavelet by comparing with single wavelet. Previously, relatively a little literatures only cover multiple wavelet or multiple wavelet shrinkage. Of course, multiple wavelet has lack of some properties and not appropriate to certain conditions, but it is useful in certain situations. By comparing multiple wavelet with ordinary single wavelet, we can derive specific characteristics of multiple wavelet and multiple wavelet shrinkage. In Chapter 2, we cover more detail concepts of multiple wavelet and multiple wavelet shrinkage. Then in Chapter 3, we cover multiple wavelet shrinkage in various noise level by simulation study and try to find out other discoveries by analysis of time-series data. 2

11 Chapter 2 Multiple Wavelet and Shrinkage 2.1 Multiple wavelet Multiple wavelets are bases with more than one mother and father wavelet. But for simplicity of explanation, we cover only 2 dimension. It is not a complex problem. Multiple wavelet is basically same logic with ordinary single wavelet. So, multiple wavelet reconstruction function can be defined by below equation. f(x) = CJ,k T Φ J,k(x) + k Z where J Dj,k T Ψ j,k(x), j=1 k Z C J,k = (C J,k,1, C J,k,2 ) T, D j,k = (D j,k,1, D j,k,2 ) T, Φ(x) = (ϕ 1 (x), ϕ 2 (x)) T, Ψ(x) = (ψ 1 (x), ψ 2 (x)) T. C J,k and D j,k are vector coefficients of dimension L=2. And Φ(x) and Ψ(x) is respectively father and mother wavelet vector function. The basis functions are orthonormal and the vector Φ(x) and Ψ(x) satisfy the following dilation 3

12 equation, Φ(x) = k Z H k Φ(2x k), Ψ(x) = k Z G k Φ(2x k), where H k and G k are 2 by 2 given matrices (called filter). So we can easily get dilation equation. Discrete multiple wavelet transform is multiple version of single wavelet transform. So the fundamental difference is that each wavelet coefficients are L-dimensional vectors. C j,k = 2 n H n C j+1,n+2k, D j,k = 2 n G n C j+1,n+2k for j = 0,..., J 1. We can check if the finer coefficient (C j+1,n+2k ) has L-dimension input, then also the coarser one has L-dimension output. Figure 1 shows that each coefficient has 2 different values and distinguished by different two colors (green and red). Figure 1: Wavelet decomposition coefficients 4

13 2.2 Properties Multiple wavelet has many advantages. First, multiple wavelets can be symmetric. And they can possess short support and have higher accuracy. Also they can be orthogonal. Multiple wavelets can possess these four properties simultaneously. However, single wavelet cannot guarantee theses properties. By having above properties simultaneously, multiple wavelet transformation is much more effective. Figure 2: Geronimo multiple wavelets Figure 3: Wavelet from the daubechies family Compare above two wavelet function. Figure 2 is multiple wavelet and Figure 3 is single wavelet. As we can see, Figure 1 has shorter support than Figure 2. And Figure 1 has symmetric or antisymmetric structure. 5

14 2.3 Multiple Wavelet Shrinkage Overall process of multiple wavelet shrinkage is equivalent to single wavelet shrinkage. First, mapping a sequence of univariate noisy observations, f(x), to the L-dimensional input father wavelet coefficients (prefiltering). Then perform discrete multiple wavelet transform, and then thresholding small coefficients (shrinkage). Finally, inverts the thresholded wavelet coefficients and we can get reconstructed univariate sequence (postfiltering). Only important difference between single wavelet shrinkage and multiple wavelet shrinkage is prefiltering and postfiltering. Figure 4: Multiple wavelet shrinkage process Multiple wavelet transform has been proposed for denoising univariate signals. But the starting coefficients of the discrete multiple wavelet transform, C 0,k (In this chapter, let j = 0 is the finest), are 2D vectors. So we 6

15 have to find a way to transform a univariate input sequence into a sequence of 2D vectors, which is called prefilter. Prefilter is given by matrix Q n ( L L) and by below equation, we can get starting coefficients (C 0,k ). C 0,k = n Q n f 2(n+k) f 2(n+k)+1. So, if prefilter (Q n ) is given like below example (Interpolation prefilter), starting coefficients are easily computed. However, even if the noisy sequence f(x) is comprised of independent observations, multiple wavelet coefficients are correlated after prefiltering. It is shown in below example. C 0,k = n Q n = Q 0 f 2k = = f 2(n+k) f 2k+1 f 2(n+k) Q 1 f 2k+2 f 2k+3 f 2k f 2k f 6 2k f 6 2k f 2k f 2k+2 f 2k+2. f 2k+3 where observation data are, f = (f 0, f 1,..., f k,...). When we compute C 0,0, we have to use f o, f 1, f 2. Also when computing C 0,1, we have to use f 2, f 3, f 4. And general C 0,k are also computed by this pattern. 7

16 So, we can see starting coefficients (C 0,0, C 0,1,..., C 0,k,...) are correlated. If components of each vector coefficients are correlated, they interrupt denoising. So, we have to find a way to decorrelate. There are two main ways to decorrelate. First method is weighting universal threshold (by Strela and Walden). This method gives weight to universal threshold and denoising while multiple wavelet transformation is computed. Other method is normalize wavelet coefficients (by Jansen). This method divides correlated coefficients by their standard deviation, so get normalized coefficients. Now we cover the way to shrink multiple wavelet coefficients Dj,k, D j,k = D j,k + E j,k, where Dj,k is L-dimensional vector. If f(x) is distributed as normal with mean zero, then E j,k is distributed as multivariate normal distribution N L (0, V j ). Now then, we have to define the new quantity θ j,k, to determine whether it is large or small. θ j,k = D T j,k V 1 j D j,k χ2 L. In the null situation (D j,k = 0), the above θ j,k is distributed as a χ 2 L. So, the θ j,k can be compared to an appropriate critical threshold. In hard thresholding, if θ j,k is larger than the threshold, then D j,k is estimated by Dj,k, and if θ j,k is smaller than the threshold, then the D j,k is estimated by zero. Then now define the threshold. In this paper, we cover universal-type threshold. For multiple wavelet case, multivariate universal threshold is defined by the maximum of χ 2 L random variables. As a result, multivariate universal threshold is defined by λ 2 n = 2logn + (L 2)loglogn. 8

17 So far, we cover the broad concepts of multiple wavelets and multiple wavelet shrinkage. In the next chapter, by using these theoretical background, we cover simulation study and data analysis. 9

18 Chapter 3 Simulation Study and Data Analysis In this chapter, we compare multiple wavelet shrinkage with single wavelet shrinkage at various situations. In wavethresh packages in R, only 2-dimensional case (L=2) is available. So in this chapter, we cover 2-dimensional case. Before starting, we try to set up same condition for single and multiple case. First, for multiple wavelet transformation, we use interpolation filter. Second, SURE shrink is unavailable in wavethresh packages in R, so we use hard thresholding and universal threshold method in both single and multiple cases. 3.1 Example function First, we consider example function in wavethresh packages. True function of example is shown in Figure 5. This function has high jump. Now, we add gradually increasing noise step by step. 10

19 Figure 5: Example function Added standard deviation (added noise) is 0.1 Added noise function is shown in Figure 6. And Figure 7 shows reconstruction function with single wavelet shrinkage, and Figure 8 shows reconstruction function with multiple wavelet shrinkage. And we use mean squared error (MSE) to check the fitness. Then we conclude that the single wavelet shrinkage has better fit. Figure 6: Noisy data with noise level

20 Figure 7: Single shrinkage, MSE= Figure 8: Multiple shrinkage, MSE= Added standard deviation (added noise) is 0.3 Figure 9: Noisy data with noise level

21 Figure 10: Single shrinkage, MSE= Figure 11: Multiple shrinkage, MSE= Added noise function and reconstructed function is shown in Figure 9. If added standard deviation is 0.3, then we can see the multiple wavelet shrinkage has better reconstruction function. Moreover, multiple wavelet shrinkage can catch the peak. On the contrary, single wavelet shrinkage reconstructed function is too smooth. 13

22 3.2.3 Added standard deviation (added noise) is 0.5 Figure 12: Noisy data with noise level 0.5 Figure 13: Single shrinkage, MSE= Figure 14: Multiple shrinkage, MSE=

23 If added standard deviation is 0.5, then the multiple reconstructed function is extremely distracted. So we conclude that the single wavelet shrinkage has better fit. But actually, if added standard deviation is too large then the signal analysis is meaningless. So by above steps, our observations are as follows. (Differences between single and multiple) Single wavelet shrinkage is smooth whether the added noise is large or small (underfit). Single wavelet shrinkage has better fit in relatively small standard deviation. Multiple wavelet shrinkage has better fit in appropriately large standard deviation, but if added standard deviation too large, then it does not fit very well. To sum up, if the noise is small then the single wavelet shrinkage has better reconstruction, but if the noise is somewhat large then the multiple wavelet shrinkage is better. 15

24 3.2 BabyECG Data Analysis Figure 15: BabyECG We perform multiple wavelet shrinkage and single wavelet shrinkage, and compare reconstruction functions (Figure 16 and Figure 17). In terms of model fit, it is hard to determine which one has better fit. Of course, we can check both cases do not thresholding high jump and there is little differences between two cases. But still it is hard to determine which one is better. Then think of other point of view. BabyECG data is known as not a stationary series but a locally stationary series. If the time series data is locally stationary series, then the variance of a time series changes slowly as a function of time. So if the structure of variance change slowly, then the time series is locally stationary. But people cannot directly know the original BabyECG data is whether locally stationary or not. So if we perform multiple wavelet shrinkage, we can easily find the structure of variance. Below reconstructed figures show that both are definitely not a stationary 16

25 series. But Figure 16 shows that the variance is changing slowly between time and or other intervals. It is easy to find Figure 16 is much more slowly changes than Figure 17. As a result, we conclude that multiple wavelet shrinkage can be a method to check whether the series is locally stationary or not. Figure 16: Reconstructed function by multiple wavelet shrinkage 17

26 Figure 17: Reconstructed function by single wavelet shrinkage 18

27 Chapter 4 Conclusion In this paper, we broadly cover multiple wavelet shrinkage and find some discoveries by simulation study and data analysis. To sum up, we conclude that multiple wavelet shrinkage has better reconstruction function in appropriately large noisy data. Also we can find if the noise is too big or small, single wavelet shrinkage has better reconstruction function. Furthermore, in time series data, multiple wavelet shrinkage can be a method to check whether the series is locally stationary or not. I hope this brief and simple discoveries would be widely used for denoising and checking the stationarity, and also hope that multiple wavelet shrinkage would be more adopted in various fields such as physics and statistics. 19

28 References [1] T. R. Downie and B.W. Silverman (1996). The discrete multiple wavelet transform and thresholding methods, IEEE Trans. SP, Vol. 44, No. 1, [2] G.P.Nason (2006). Wavelet Methods in Statistics with R, Springer. [3] V. Strela and A. Walden (1998). Signal image denoising via wavelet thresholding : Orthogonal and biorthogonal scalar and multiple wavelet transform, Technical Report TR-98-01, Statistics Section, Imperial College, London. [4] Maarten Jansen (2001). Noise Reduction by Wavelet Thresholding, Springer. 20

29 주요어 학번

30 저작자표시 - 비영리 - 변경금지 2.0 대한민국 이용자는아래의조건을따르는경우에한하여자유롭게 이저작물을복제, 배포, 전송, 전시, 공연및방송할수있습니다. 다음과같은조건을따라야합니다 : 저작자표시. 귀하는원저작자를표시하여야합니다. 비영리. 귀하는이저작물을영리목적으로이용할수없습니다. 변경금지. 귀하는이저작물을개작, 변형또는가공할수없습니다. 귀하는, 이저작물의재이용이나배포의경우, 이저작물에적용된이용허락조건을명확하게나타내어야합니다. 저작권자로부터별도의허가를받으면이러한조건들은적용되지않습니다. 저작권법에따른이용자의권리는위의내용에의하여영향을받지않습니다. 이것은이용허락규약 (Legal Code) 을이해하기쉽게요약한것입니다. Disclaimer

31

32

33 Multiple Wavelet Shrinkage by Changha Shin A Thesis submitted in fulfillment of the requirement for the degree of Master of Science in Statistics The Department of Statistics College of Natural Sciences Seoul National University February, 2016

34 Abstract Wavelets are used as basis functions in representing other functions. In this paper, we focus on multiple wavelets, which means number of mother and father wavelet is more than one. Meanwhile, noisy data is hard to find hidden inner side of the function. Solution is wavelet shrinkage method. It is effective method to eliminate the noise and see the original form of function. In this paper, we suggest conditions for multiple wavelet shrinkage s adoption. And we find useful properties of multiple wavelet shrinkage in time-series data. Keywords : Discrete wavelet transform, Multiple wavelet, Multiple wavelet shrinkage, Wavelet, Wavelet shrinkage Student Number :

35 Contents 1 Introduction 1 2 Multiple Wavelet and Shrinkage Multiple wavelet Properties Multiple wavelet shrinkage Simulation Study and Data Analysis Example function BabyECG data analysis Conclusion 19 References 20

36 List of Figures 1 Wavelet decomposition coefficients Geronimo multiple wavelets Wavelet from the daubechies family Multiple wavelet shrinkage process Example function Ynoise (added noise 0.1) Single shrinkage, MSE= Multiple shrinkage, MSE= Ynoise (added noise is 0.3) Single shrinkage, MSE= Multiple shrinkage, MSE=

37 List of Figures 12 Ynoise (added noise is 0.5) Single shrinkage, MSE= Multiple shrinkage, MSE= BabyECG Reconstructed function by multiple wavelet shrinkage Reconstructed function by single wavelet shrinkage

38 Chapter 1 Introduction Before, we say multiple wavelet shrinkage, let s briefly cover the concepts and importance of wavelet and wavelet shrinkage. Wavelets mean small waves. It is compound of wave and let (small). Small waves mean they are compactly supported, that is they should be integrated to zero. And wavelets are used as basis functions in representing other functions. The main application and usage of wavelet is by shrinkage. In traditional statistical point of view in data, is all data are regarded as true signal and noise (called aggregation). On the contrary, in wavelet area in statistics, we regard not all data have signal and noise (called sparsity). This is the main point why we use shrinkage. So, if one observes noisy data, it can be transformed by the discrete wavelet transform. And then thresholding the small coefficients, then operate inverse transform to reconstruct the function. That is called wavelet shrinkage. When we use wavelet shrinkage method to noisy function, we can find specific properties of hidden inner side of the function. So it is effective method to eliminate the noise and see the original form of function. It is like restoring contaminated ancient works of art contained traces of the years. Multiple wavelet is special case of above ordinary wavelet, number of 1

39 mother and father wavelet is more than one. In this paper, our main purpose is brighten the advantages of multiple wavelet by comparing with single wavelet. Previously, relatively a little literatures only cover multiple wavelet or multiple wavelet shrinkage. Of course, multiple wavelet has lack of some properties and not appropriate to certain conditions, but it is useful in certain situations. By comparing multiple wavelet with ordinary single wavelet, we can derive specific characteristics of multiple wavelet and multiple wavelet shrinkage. In Chapter 2, we cover more detail concepts of multiple wavelet and multiple wavelet shrinkage. Then in Chapter 3, we cover multiple wavelet shrinkage in various noise level by simulation study and try to find out other discoveries by analysis of time-series data. 2

40 Chapter 2 Multiple Wavelet and Shrinkage 2.1 Multiple wavelet Multiple wavelets are bases with more than one mother and father wavelet. But for simplicity of explanation, we cover only 2 dimension. It is not a complex problem. Multiple wavelet is basically same logic with ordinary single wavelet. So, multiple wavelet reconstruction function can be defined by below equation. f(x) = CJ,k T Φ J,k(x) + k Z where J Dj,k T Ψ j,k(x), j=1 k Z C J,k = (C J,k,1, C J,k,2 ) T, D j,k = (D j,k,1, D j,k,2 ) T, Φ(x) = (ϕ 1 (x), ϕ 2 (x)) T, Ψ(x) = (ψ 1 (x), ψ 2 (x)) T. C J,k and D j,k are vector coefficients of dimension L=2. And Φ(x) and Ψ(x) is respectively father and mother wavelet vector function. The basis functions are orthonormal and the vector Φ(x) and Ψ(x) satisfy the following dilation 3

41 equation, Φ(x) = k Z H k Φ(2x k), Ψ(x) = k Z G k Φ(2x k), where H k and G k are 2 by 2 given matrices (called filter). So we can easily get dilation equation. Discrete multiple wavelet transform is multiple version of single wavelet transform. So the fundamental difference is that each wavelet coefficients are L-dimensional vectors. C j,k = 2 n H n C j+1,n+2k, D j,k = 2 n G n C j+1,n+2k for j = 0,..., J 1. We can check if the finer coefficient (C j+1,n+2k ) has L-dimension input, then also the coarser one has L-dimension output. Figure 1 shows that each coefficient has 2 different values and distinguished by different two colors (green and red). Figure 1: Wavelet decomposition coefficients 4

42 2.2 Properties Multiple wavelet has many advantages. First, multiple wavelets can be symmetric. And they can possess short support and have higher accuracy. Also they can be orthogonal. Multiple wavelets can possess these four properties simultaneously. However, single wavelet cannot guarantee theses properties. By having above properties simultaneously, multiple wavelet transformation is much more effective. Figure 2: Geronimo multiple wavelets Figure 3: Wavelet from the daubechies family Compare above two wavelet function. Figure 2 is multiple wavelet and Figure 3 is single wavelet. As we can see, Figure 1 has shorter support than Figure 2. And Figure 1 has symmetric or antisymmetric structure. 5

43 2.3 Multiple Wavelet Shrinkage Overall process of multiple wavelet shrinkage is equivalent to single wavelet shrinkage. First, mapping a sequence of univariate noisy observations, f(x), to the L-dimensional input father wavelet coefficients (prefiltering). Then perform discrete multiple wavelet transform, and then thresholding small coefficients (shrinkage). Finally, inverts the thresholded wavelet coefficients and we can get reconstructed univariate sequence (postfiltering). Only important difference between single wavelet shrinkage and multiple wavelet shrinkage is prefiltering and postfiltering. Figure 4: Multiple wavelet shrinkage process Multiple wavelet transform has been proposed for denoising univariate signals. But the starting coefficients of the discrete multiple wavelet transform, C 0,k (In this chapter, let j = 0 is the finest), are 2D vectors. So we 6

44 have to find a way to transform a univariate input sequence into a sequence of 2D vectors, which is called prefilter. Prefilter is given by matrix Q n ( L L) and by below equation, we can get starting coefficients (C 0,k ). C 0,k = n Q n f 2(n+k) f 2(n+k)+1. So, if prefilter (Q n ) is given like below example (Interpolation prefilter), starting coefficients are easily computed. However, even if the noisy sequence f(x) is comprised of independent observations, multiple wavelet coefficients are correlated after prefiltering. It is shown in below example. C 0,k = n Q n = Q 0 f 2k = = f 2(n+k) f 2k+1 f 2(n+k) Q 1 f 2k+2 f 2k+3 f 2k f 2k f 6 2k f 6 2k f 2k f 2k+2 f 2k+2. f 2k+3 where observation data are, f = (f 0, f 1,..., f k,...). When we compute C 0,0, we have to use f o, f 1, f 2. Also when computing C 0,1, we have to use f 2, f 3, f 4. And general C 0,k are also computed by this pattern. 7

45 So, we can see starting coefficients (C 0,0, C 0,1,..., C 0,k,...) are correlated. If components of each vector coefficients are correlated, they interrupt denoising. So, we have to find a way to decorrelate. There are two main ways to decorrelate. First method is weighting universal threshold (by Strela and Walden). This method gives weight to universal threshold and denoising while multiple wavelet transformation is computed. Other method is normalize wavelet coefficients (by Jansen). This method divides correlated coefficients by their standard deviation, so get normalized coefficients. Now we cover the way to shrink multiple wavelet coefficients Dj,k, D j,k = D j,k + E j,k, where Dj,k is L-dimensional vector. If f(x) is distributed as normal with mean zero, then E j,k is distributed as multivariate normal distribution N L (0, V j ). Now then, we have to define the new quantity θ j,k, to determine whether it is large or small. θ j,k = D T j,k V 1 j D j,k χ2 L. In the null situation (D j,k = 0), the above θ j,k is distributed as a χ 2 L. So, the θ j,k can be compared to an appropriate critical threshold. In hard thresholding, if θ j,k is larger than the threshold, then D j,k is estimated by Dj,k, and if θ j,k is smaller than the threshold, then the D j,k is estimated by zero. Then now define the threshold. In this paper, we cover universal-type threshold. For multiple wavelet case, multivariate universal threshold is defined by the maximum of χ 2 L random variables. As a result, multivariate universal threshold is defined by λ 2 n = 2logn + (L 2)loglogn. 8

46 So far, we cover the broad concepts of multiple wavelets and multiple wavelet shrinkage. In the next chapter, by using these theoretical background, we cover simulation study and data analysis. 9

47 Chapter 3 Simulation Study and Data Analysis In this chapter, we compare multiple wavelet shrinkage with single wavelet shrinkage at various situations. In wavethresh packages in R, only 2-dimensional case (L=2) is available. So in this chapter, we cover 2-dimensional case. Before starting, we try to set up same condition for single and multiple case. First, for multiple wavelet transformation, we use interpolation filter. Second, SURE shrink is unavailable in wavethresh packages in R, so we use hard thresholding and universal threshold method in both single and multiple cases. 3.1 Example function First, we consider example function in wavethresh packages. True function of example is shown in Figure 5. This function has high jump. Now, we add gradually increasing noise step by step. 10

48 Figure 5: Example function Added standard deviation (added noise) is 0.1 Added noise function is shown in Figure 6. And Figure 7 shows reconstruction function with single wavelet shrinkage, and Figure 8 shows reconstruction function with multiple wavelet shrinkage. And we use mean squared error (MSE) to check the fitness. Then we conclude that the single wavelet shrinkage has better fit. Figure 6: Noisy data with noise level

49 Figure 7: Single shrinkage, MSE= Figure 8: Multiple shrinkage, MSE= Added standard deviation (added noise) is 0.3 Figure 9: Noisy data with noise level

50 Figure 10: Single shrinkage, MSE= Figure 11: Multiple shrinkage, MSE= Added noise function and reconstructed function is shown in Figure 9. If added standard deviation is 0.3, then we can see the multiple wavelet shrinkage has better reconstruction function. Moreover, multiple wavelet shrinkage can catch the peak. On the contrary, single wavelet shrinkage reconstructed function is too smooth. 13

51 3.2.3 Added standard deviation (added noise) is 0.5 Figure 12: Noisy data with noise level 0.5 Figure 13: Single shrinkage, MSE= Figure 14: Multiple shrinkage, MSE=

52 If added standard deviation is 0.5, then the multiple reconstructed function is extremely distracted. So we conclude that the single wavelet shrinkage has better fit. But actually, if added standard deviation is too large then the signal analysis is meaningless. So by above steps, our observations are as follows. (Differences between single and multiple) Single wavelet shrinkage is smooth whether the added noise is large or small (underfit). Single wavelet shrinkage has better fit in relatively small standard deviation. Multiple wavelet shrinkage has better fit in appropriately large standard deviation, but if added standard deviation too large, then it does not fit very well. To sum up, if the noise is small then the single wavelet shrinkage has better reconstruction, but if the noise is somewhat large then the multiple wavelet shrinkage is better. 15

53 3.2 BabyECG Data Analysis Figure 15: BabyECG We perform multiple wavelet shrinkage and single wavelet shrinkage, and compare reconstruction functions (Figure 16 and Figure 17). In terms of model fit, it is hard to determine which one has better fit. Of course, we can check both cases do not thresholding high jump and there is little differences between two cases. But still it is hard to determine which one is better. Then think of other point of view. BabyECG data is known as not a stationary series but a locally stationary series. If the time series data is locally stationary series, then the variance of a time series changes slowly as a function of time. So if the structure of variance change slowly, then the time series is locally stationary. But people cannot directly know the original BabyECG data is whether locally stationary or not. So if we perform multiple wavelet shrinkage, we can easily find the structure of variance. Below reconstructed figures show that both are definitely not a stationary 16

54 series. But Figure 16 shows that the variance is changing slowly between time and or other intervals. It is easy to find Figure 16 is much more slowly changes than Figure 17. As a result, we conclude that multiple wavelet shrinkage can be a method to check whether the series is locally stationary or not. Figure 16: Reconstructed function by multiple wavelet shrinkage 17

55 Figure 17: Reconstructed function by single wavelet shrinkage 18

56 Chapter 4 Conclusion In this paper, we broadly cover multiple wavelet shrinkage and find some discoveries by simulation study and data analysis. To sum up, we conclude that multiple wavelet shrinkage has better reconstruction function in appropriately large noisy data. Also we can find if the noise is too big or small, single wavelet shrinkage has better reconstruction function. Furthermore, in time series data, multiple wavelet shrinkage can be a method to check whether the series is locally stationary or not. I hope this brief and simple discoveries would be widely used for denoising and checking the stationarity, and also hope that multiple wavelet shrinkage would be more adopted in various fields such as physics and statistics. 19

57 References [1] T. R. Downie and B.W. Silverman (1996). The discrete multiple wavelet transform and thresholding methods, IEEE Trans. SP, Vol. 44, No. 1, [2] G.P.Nason (2006). Wavelet Methods in Statistics with R, Springer. [3] V. Strela and A. Walden (1998). Signal image denoising via wavelet thresholding : Orthogonal and biorthogonal scalar and multiple wavelet transform, Technical Report TR-98-01, Statistics Section, Imperial College, London. [4] Maarten Jansen (2001). Noise Reduction by Wavelet Thresholding, Springer. 20

58 주요어 학번