Statistical Analysis of Synchrosqueezed Transforms

Transcription

1 Statistical Analysis of Synchrosqueezed Transforms Haizhao Yang Department of Mathematics, Due University October 4, Revised: August 6 Abstract Synchrosqueezed transforms are non-linear processes for a sharpened -uency representation of wave-lie components. They are efficient tools for identifying and analyzing wave-lie components from their superposition. This paper is concerned with the statistical properties of compactly supported synchrosqueezed transforms for wave-lie components embedded in a generalized Gaussian random process in multidimensional spaces. Guided by the theoretical analysis of these properties, new numerical implementations are proposed to reduce the noise fluctuations of these transforms on noisy data. A MATLAB pacage SynLab together with several heavily noisy examples is provided to support these theoretical claims. Keywords. Wave-lie components, instantaneous local properties, synchrosqueezed transforms, noise robustness, generalized Gaussian random process. AMS subject classifications: 4A99 and 65T99. Introduction Non-linear and non-stationary wave-lie signals also termed as chirp signals in D are ubiquitous in science and engineering, e.g., clinical data [43, 44], seismic data [9,, 34], climate data [36, 45], astronomical data [7, ], materials science [6, 33, 5], and art investigation [49]. Analyzing instantaneous properties e.g., instantaneous uencies, instantaneous amplitudes and instantaneous phases [6, 3] or local properties concepts for D signals similar to instantaneous in D of signals has been an important topic for over two decades. In many applications [5, 3, 4, 4, 5, 5], these signals can be modeled as a superposition of several wave-lie components with slowly varying amplitudes, uencies or wave vectors, contaminated by noise. For example, a complex signal fx = K α xe πin φ x + ex, = where α x is the instantaneous local amplitude, N φ x is the instantaneous local phase, N φ x is the instantaneous uency or N φ x as the local wave vector, and ex is a noisy perturbation term. A powerful tool for analyzing signal is the synchrosqueezed transform SST consisting of a linear -uency analysis tool and a synchrosqueezing technique. It belongs to more generally -uency reassignment techniques [3, 8, 9, 6] see also [4] for a

2 recent review. The SST was initialized in [6] and further analyzed in [5] for the D wavelet transform. Suppose W f ξ, x is the wavelet transform of a D wave-lie component fx = αxe πinφx. The wavelet -uency representation has a wide support spreading around the instantaneous uency curve Nφ x, x. It was proved that the instantaneous uency information function v f ξ, x = xw f ξ,x πiw f ξ,x is able to approximate Nφ x. Hence, the synchrosqueezing technique shifts the value of W f ξ, x from ξ, x to v f ξ, x, x, generating a sharpened -uency representation T f v, x with a support concentrating around the curve Nφ x, x. The localization of the new representation not only improves the resolution of the original spectral analysis due to the uncertainty principle but also mae it easier to decompose the superposition in into individual components. A variety of SSTs have been proposed after the D synchrosqueezed wavelet transform SSWT in [6], e.g., the synchrosqueezed short Fourier transform SSSTFT in [37], the synchrosqueezed wave pacet transform SSWPT in [48, 5], the synchrosqueezed curvelet transform SSCT in [5] and the D monogenic synchrosqueezed wavelet transform in [4]. Rigorous analysis has proved that these transforms can accurately decompose a class of superpositions of wave-lie components and estimate their instantaneous local properties if the given signal is noiseless. To improve the synchrosqueezing operator in the presence of strongly non-linear instantaneous uencies, some further methods have been proposed in [4, 5, 8] based on an extra investigation of the higher order derivatives of the phase of a wave-lie component. All these SSTs are compactly supported in the uency domain to ensure accurate estimations. To better analyze signals with a trend or big data sets that need to be handled in real- computation, a recent paper [3] proposes a new synchrosqueezing method based on carefully designed wavelets with sufficiently many vanishing moments and a minimum support in the domain. Previous synchrosqueezed transforms need to access every data sample of fx to compute the synchrosqueezed transform at a specific or location x. However, compactly supported synchrosqueezed transforms only need a small portion of the data samples in a neighborhood of x. Hence, compactly supported synchrosqueezed transforms are computationally more efficient and are better tools for modern big data analysis. However, mathematical analysis on the accuracy of this compactly supported SSWT is still under development. This paper addresses this problem in the framewor of the SSWPT that includes the SSWT as a special case. Another important topic in the study of SSTs is the statistical analysis of the synchrosqueezing operator, since noise is ubiquitous in real applications. A pioneer paper [] in this direction studied the statistical properties of the D spectrogram reassignment method by calculating the probability density function of white Gaussian noise after reassignment. A recent paper [36] focused on the statistical analysis of the D SSWT for white Gaussian noise. It estimated the probability of a good estimation of the instantaneous uency provided by the instantaneous uency information function v f ξ, x. A following paper [] generalized its results to generalized stationary Gaussian process. To support the application of SSTs to real- problems and multidimensional problems, this paper analyzes the statistical properties of multidimensional SSTs that can be compactly supported in the domain. Turning to the robustness issue in a numerical sense, it is of interest to design an efficient implementation of a sharpened -uency representation with reduced noise fluctuations. The idea of multitapering, first proposed in [38] for stationary signals and further extended in [5, ] for non-stationary signals, attempted to improve the statistical stability of spectral analysis by generating multiple windowed trials of the noisy signal and averaging

3 the spectral analysis of these trials. By combining the multitapering and -uency reassignment techniques, [46] proposed the D multitapering -uency reassignment for a sharpened -uency representation with reduced noise fluctuations. Since its implementation is based on Hermite functions, its efficient generalization in multidimensional spaces is not straightforward. Guided by the theoretical analysis of the statistical properties of SSTs, this paper proposes efficient numerical implementations of multidimensional SSTs based on highly redundant frames. Since the SST of a noiseless signal is frame-independent, averaging the SSTs from multiple -uency frames reduces the noise fluctuation while eeping the localization of the synchrosqueezed -uency representation. The rest of this paper is organized as follows. In Section, the main theorems for the compactly supported SSTs in multidimensional spaces and their statistical properties are presented. In Section 3, a few algorithms and their implementations are introduced in detail to improve the statistical stability of SSTs. Several numerical examples with heavy noise are provided to demonstrate the proposed properties. We conclude this paper in Section 4. Theory for synchrosqueezed transforms SSTs Let us briefly introduce the basics and assumptions of compactly supported SSTs in Section. before discussing their statistical properties in Section.. While the synchrosqueezing technique can be applied to a wide range of -uency transforms, the discussion here is restricted to the framewor of multidimensional wave pacet transforms to save space. It is easy to extend these results to other transforms see [47] for the example of the D synchrosqueezed curvelet transform. Due to the space limitation, only the main ideas of the proofs are presented. Readers are referred to [47] for more details.. Compactly supported SSTs Previously, the synchrosqueezed wave pacet transform SSWPT was built using mother wave pacets compactly supported in the uency domain. This paper studies a wider class of mother wave pacets defined below. Definition.. An n-dimensional mother wave pacet wx C m R n is of type, m for some, and some non-negative integer m, if ŵξ is a real-valued smooth function with a support that covers the ball B centered at the origin with a radius satisfying that: ŵξ + ξ m, for ξ > and ŵξ >, for ξ <. Since w C m R n, the above decaying requirement is easy to satisfy. Actually, we can further assume ŵξ is essentially supported in a ball B d with d, ] to adapt signals with close instantaneous uencies, i.e., ŵξ is approximately zero outside this support up to an truncation error. However, d is just a constant in later asymptotic analysis. Hence, we omit this discussion and consider it as in the analysis but implement it in the numerical tool. Similarly to the discussion in [48, 5], we can use this mother wave pacet wx to define a family of n-dimensional compactly supported wave pacets w ab x = a ns/ w a s x b e πix b a, 3

4 through scaling, modulation, and translation, controlled by a geometric parameter s, for a, b R n. With this family of wave pacets ready, we define the wave pacet transform via W f a, b = f, w ab = fxw ab xdx. Previously in [48, 5], the synchrosqueezed wave pacet transform SSWPT was proposed to analyze a class of intrinsic mode type functions as defined below. Definition.. A function fx = αxe πinφx in R n is an intrinsic mode type function IMT of type M, N if αx and φx satisfy αx C, αx M, /M αx M, φx C, /M φx M, φx M. It has been proved that the instantaneous uency or local wave vector when n > information function v f a, b = bw f a,b πiw f a,b of an IMT fx can approximate N φb if the mother wave pacet wx is of type, and N is sufficiently large. A careful inspection of previous proofs shows that the approximation v f a, b N φb is still valid up to an relative error if the mother wave pacet is of type, m for any positive integer. See Theorem..7 in [47] for a detailed proof. Hence, if we squeeze the coefficients W f a, b together based upon the same information function v f a, b, then we would obtain a sharpened -uency representation of fx. This motivates the definition of the synchrosqueezed energy distribution T f v, b = W f a, b δ Rv f a, b v da R n for v, b R n. Here δ denotes the Dirac delta function and Rv f a, b means the real part of v f a, b. For a multi-component signal fx = K = α xe πin φ x, the synchrosqueezed energy of each component will also concentrate around each N φ x if these components satisfy the well-separation condition defined below. Definition.3. A function fx is a well-separated superposition of type M, N, K, s if fx = K f x, = where each f x = α xe πin φ x is an IMT of type M, N with N N and the phase functions satisfy the separation condition: for any a, b R n, there exists at most one f satisfying that a s a N φ b. We denote by F M, N, K, s the set of all such functions. In real applications, this well-separation condition might not be valid for a multicomponent signal at every x. However, the SST will wor wherever the well-separation condition is satisfied locally. The ey analysis of the SSWPT is how well the information function v f a, b approximates the instantaneous uencies or local wave vectors. If the approximation is accurate 4

5 enough, the synchrosqueezed energy distribution T f a, b gives a sharpened -uency representation of fx. We close this section with the following theorem that summarizes the main analysis of the n-dimensional SSWPT for a superposition of IMTs without noise or perturbation. In what follows, when we write O,, or, the implicit constants may depend on M, m and K. Readers are referred to Theorem..7 in [47] for the proof of Theorem.4 here. Theorem.4. Suppose the n-dimensional mother wave pacet is of type, m, for any fixed, and any fixed integer m. For a function fx, we define and R = {a, b : W f a, b a ns/ }, S = {a, b : W f a, b }, Z = {a, b : a N φ b a s } for K. } For fixed M, m, K, s, and, there exists a constant N M, m, K, s, max { s, s such that for any N > N and fx F M, N, K, s the following statements hold. i {Z : K} are disjoint and S R K Z ; ii For any a, b R Z, v f a, b N φ b N φ b ; iii For any a, b S Z, v f a, b N φ b N φ b N ns/.. Statistical Properties of SSTs Similarly to the noiseless case, we will analyze how well the information function v f a, b approximates instantaneous uencies or local wave vectors in the case when a superposition of IMTs is contaminated by random noise. To simplify the discussion, we will setch out the proofs in the one-dimensional case and refer the readers to [47] for higher dimensional cases. Let us start with a simple case in which the superposition is perturbed slightly by a contaminant, Theorem.5 below shows that the information function v f a, b can approximate instantaneous uencies with a reasonable error determined by the magnitude of the perturbation. Theorem.5. Suppose the mother wave pacet is of type, m, for any fixed, and any fixed integer m. Suppose gx = fx + ex, where ex L is a small error term that satisfies e L for some >. For any p, ], let δ = + p. Define R δ = {a, b : W g a, b a s/ δ}, S δ = {a, b : W g a, b δ}, 5

6 and Z = {a, b : a N φ b a s } for K. } For fixed M, m, K, s, and, there exists a constant N M, m, K, s, max { s, s such that for any N > N M, m, K, s, and fx F M, N, K, s the following statements hold. i {Z : K} are disjoint and S δ R δ K Z ; ii For any a, b R δ Z, v g a, b N φ b N φ b + p ; iii For any a, b S δ Z, v g a, b N φ b N φ b + p. N s/ We introduce the parameter p to clarify the relation among the perturbation, the threshold and the accuracy for better understanding the influence of perturbation or noise. For the same purpose, a parameter q will be introduced in the coming theorems. If the threshold δ is larger, e.g., δ, the relative estimate errors in ii and iii are bounded by and N s/, respectively. Similarly, one can show that the information function computed from the wave pacet coefficient with a larger magnitude can better approximate the instantaneous uency. Below is a setch of the proof of Theorem.5. See the proof of Theorem 3.. in [47] for a detailed proof. Proof. We only need to discuss the case when a >. By the definition of the wave pacet transform, we have W e a, b a s/ and b W e a, b a s/ + a s/. If a, b R δ, then W g a, b a s/ δ. Together with Equation, it holds that W f a, b W g a, b W e a, b a s/ δ a s/. 3 Hence, S δ R δ R, where R is defined in Theorem.4 and is a subset of K Z. So, i is true by Theorem.4. Since R δ R, a, b R δ Z implies a, b R Z. Hence, by Theorem.4, it holds that v f a, b N φ b N φ b, 4 } when N is larger than a constant N M, m, K, s, max { s, s. Notice that a, b Z implies a N. Hence, by Equation to 4, v g a, b N φ b N φ b v f a, b N φ b N φ b + b W f a,b πiw f a,b bw ga,b πiw ga,b when N > N. Hence, ii is proved. The proof of iii is similar. N φ b + p, 6

7 Next, we will discuss the case when the contamination e is a random perturbation. [, 4, 7, 35, 39] are referred to for basic facts about generalized random fields and complex Gaussian processes. To warm up, we start with additive white Gaussian process in Theorem.6 and extend it to a generalized zero mean stationary Gaussian process in Theorem.7. We assume that e has an explicit power spectral function denoted by êξ. represents the L norm and, is the standard inner product. Theorem.6. Suppose the mother wave pacet is of type, m, for any fixed, and any fixed integer m s + 4. Suppose gx = fx + e, where e is zero mean white Gaussian process with a variance +q for some q > and some >. For any p, ], let δ = + p. Define and R δ = {a, b : W g a, b a s/ δ}, S δ = {a, b : W g a, b δ}, Z = {a, b : a N φ b as } for K. } For fixed M, m, K, s, and, there exists a constant N M, m, K, s, max { s, s such that for any N > N M, m, K, s, and fx F M, N, K, s the following statements hold. i {Z : K} are disjoint. ii If a, b R δ, then a, b K Z with a probability at least s ON e q + O. N m s iii If a, b S δ, then a, b K Z with a probability at least e q w + O. iv If a, b R δ Z for some, then is true with a probability at least e ON 3s v If a, b S δ Z, then q N m s v g a, b N φ b N φ b + p e is true with a probability at least e ON s q s ON q + O v g a, b N φ b + p N φ b N s/ e ON q 7 + O N m 4 s N m 4 s..

8 We only setch the proof of the above theorem. See the proof of Theorem 3.. in [47] for a long proof. Proof. Step : we prove this theorem when the mother wave pacet is of type, m first, i.e., compactly supported in the uency domain. Since w ab and b w ab are in L C m, W e a, b and b W e a, b are Gaussian variables. Hence, W g a, b = W f a, b + W e a, b and b W g a, b = b W f a, b + b W e a, b can be understood as Gaussian variables. Furthermore, W e a, b and W e a, b, b W e a, b are circularly symmetric Gaussian variables by checing that their pseudo-covariance matrices are zero. Therefore, the distribution of W e a, b is determined by its variance If we define e +q z w π +q w. ŵ V = ŵ ab, πiξŵ ab, πiξŵ ab, ŵ ab πiξŵ ab, πiξŵ ab then +q V is the covariance matrix of W e a, b, b W e a, b and its distribution is described by the joint probability density e +q z V z π +q det V, where z = z, z T, T and denote the transpose operator and conjugate transpose operator. V is an invertible and self-adjoint matrix, since W e a, b and b W e a, b are linearly independent. Hence, there exist a diagonal matrix D and a unitary matrix U such that V = U DU. Part i is true by previous theorems. Define the following events G = { W e a, b < a s/ }, and G = { W e a, b < }, G 3 = { b W e a, b < a s/ + a s/}, { vg a, b N φ H = b N φ b } + p, J = { v g a, b N φ b N φ b } + p N s/, for K. To conclude Part ii to v, we need to estimate the probability P G, P G, P G G 3, P G G 3, P H and P J. Algebraic calculations show that P G = e a s q w and P G = e q w. We are ready to summarize and conclude ii and iii. If a, b R δ, then W e a, b + W f a, b a s/ / p

9 If a, b / K Z, then W f a, b a s/. 6 Equation 5 and 6 lead to W e a, b a s/. Hence, P a, b / P W e a, b a s/ = P G. K Z This means that if a, b R δ, then a, b K Z with a probability at least P G = ON s e a s q w = e q, since a N if a, b Z. So, ii is true. A similar argument applied to a, b S δ shows that a, b K Z with a probability at least P G = e q w. Hence, iii is proved. Recall that V = U DU. By the change of variables z = Uz, we can show that P G G 3 e Dd +q e Dd +q, and P G G 3 e Dd +q e Dd +q. We can further estimate that D a s and D a. Therefore, P G G 3 e Oa 3s q e Oa s q, and P G G 3 e Oa s q e Oa q. By Theorem.5, if a, b R δ Z for some, then P H P H G G 3 P G G 3 = P G G 3 Note that a N when a Z, then P H e ON 3s q e Similarly, if a, b S δ Z for some, then P J P J G G 3 P G G 3 = P G G 3 e e Oa 3s q e Oa s q. s ON q. ON s q e ON q These arguments prove iv and v. Step : we go on to prove this theorem when the mother wave pacet is of type, m with m s + 4. We would lie to emphasize that the requirement is crucial to the following asymptotic analysis and it eeps the error caused by the non-compact support of ŵ reasonably small. The setch of the proof is similar to the first step. W e a, b and W e a, b, b W e a, b are still Gaussian variables but in general not circularly symmetric, because they would 9.

10 not have zero pseudo-covariance matrices. Suppose they have covariance matrices C and C, pseudo-covariance matrices P and P, respectively. We can still chec that they have zero mean, C = +q w and C = +q V, where V is defined in the first step. By the definition of the mother wave pacet of type, m, the magnitude of every entry in P and P is bounded by O +q. Notice that the covariance matrix of W a m s e a, b, We a, b is C P V = P C. By Equation 7 in [7] and the Taylor expansion, the distribution of W e a, b is described by the following distribution e z,z V z,z T +q π = e z w z det V π +q + O w +q. a m s By the same argument, the covariance matrix of W e a, b, b W e a, b, We a, b, b We a, b is C P V = P C. Let z = z, z T. Then the distribution of W e a, b, b W e a, b is described by the joint probability density e z,z,z,z V z,z,z,z T π. 7 det V Notice that C = +p V and V has eigenvalues of order a and a s determined by estimating the diagonal entries of the matrix D in the diagonalization V = U DU. Hence, C has eigenvalues of order +p a and +p a s. Recall that the magnitude of every entry in P is bounded by O +q. This means that V is nearly dominated by a m s diagonal blocs C and C. Basic spectral theory for linear transforms shows that V C = + P, C +q a m 4s where P is a matrix with -norm bounded by O to the above spectral analysis. Since every entry of P is bounded by O det V = det C 4+q + O a m m+s,. m 6 m 4 s is crucial +q, a m s where the residual comes from the entry bound and the eigenvalues of C. Hence 7 is actually e +q z V z e z,z,z,z P z,z,z,z T. π +q det V + O a m m+s By the same argument as in the first step, we can show that there exist a diagonal matrix D = diag{a s, a } and a unitary matrix U such that V = U DU. Part i is

11 still true by previous theorems. To conclude Part ii to v, we still need to estimate the probability of those events defined in the first step, i.e., P G, P G, P G G 3, P G G 3, P H and P J. By the estimations above, one can show that P G = e a s q w + O a m s, and P G = e q w + O a m s. Hence, we can conclude ii and iii follows the same proof in the first step. Next, we loo at the last two part of this theorem. Let us introduce notations δ = a s/, δ =, δ 3 = a s/ + a s/, d = min{ δ, δ 3 }, and d = min{ δ, δ 3 }. By previous estimations, we have = = P G G 3 { z <δ, z <δ 3 } { z <δ, z <δ 3 } e z,z,z,z V z,z,z,z T π det V dz dz e +q z V z e z,z,z,z P z,z,z,z T π +q det V + O dz dz. 8 a m m+s Since det V det V + O we can drop out the term O no more than O a m s = + O a m m+s a m m+s a m s, 9 in 8, which would generate an absolute error in the estimate of P G G 3. Let gz = z, z, z, z P z, z, z, z T,

12 then by the change of variables we have = = = P G G 3 { z <δ, z <δ 3 } { z <δ, z <δ 3 } e +q z V z e gz π +q det V e +q { z <d, z <d } d d π π π +q det V dz dz D z +D z e gu z π +q dz dz det V e +q D z +D z e gu z π +q dz dz det V d d π π r r e Dr +q e Dr +q e gr,θ,r,θ dθ dθ dr dr π +q r r e Dr +q e Dr +q e gr,θ,r,θ dθ dθ dr dr det V + e Dd +q e Dd +q, where gr, θ, r, θ = gu z. Recall that the -norm of P is bounded by O +q a. m 4s Hence, gr, θ, r, θ O +q a m 4s z + z = O +q a m 4s r + r. Therefore, the first term in is bounded by d d O a m 4s 3+q r r e Dr +q e Dr +q r + r dr dr det V O D a m 4 s r r r + r e r e r dr dr = O a m 4 s. The analysis in 9 and implies that P G G 3 e Dd +q e Dd +q + O a m 4 s. and similarly P G G 3 e Dd +q e Dd +q + O a m 4 s. The rest of the proof is exactly the same as the one in the first step and consequently we now this theorem is also true for a mother wave pacets of type, m with m satisfying m s + 4.

13 Thus far, we have considered the analysis for small perturbation and white Gaussian process. Next, Theorem.6 is extended to a broader class of colored random processes. Theorem.7. Suppose the mother wave pacet is of type, m, for any fixed, and any fixed integer m s + 4. Suppose gx = fx + e, where e is a zero mean stationary Gaussian process. Let êξ denote the spectrum of e, max ξ êξ and M a = max ξ < êa s ξ + a. For any p, ] and q >, let δ a = M p/+q a +, and R δa = {a, b : W g a, b a s/ δ a }, S δa = {a, b : W g a, b δ a }, Z = {a, b : a N φ b as } for K. } For fixed M, m, K, s, and, there exists a constant N M, m, K, s, max { s, s such that for any N > N M, m, K, s, and fx F M, N, K, s the following statements hold. i {Z : K} are disjoint. ii If a, b R δa, then a, b K Z with a probability at least e O N s M a q/+q + O N m s iii If a, b S δa, then a, b K Z with a probability at least e q/+q OMa iv If a, b R δa Z for some, then is true with a probability at least e O N 3s Ma q/+q v If a, b S δa Z, then is true with a probability at least e O N s + O N m s v g a, b N φ b N φ b + Ma p/+q e O N s v g a, b N φ b N φ b N s/ Ma q/+q e O 3 N Ma q/+q.. + O + M p/+q a M a q/+q + O N m 4 s N m 4 s..

14 Proof. The proof of this theorem is nearly identical to Theorem.6. Although calculations are more cumbersome, the derivation proceeds along the same line see the proof of Theorem 3..3 in [47] for details. Theorem.6 and.7 provide a new insight that a smaller s yields a synchrosqueezed transform that can provide a good estimation with higher probability. The parameter m in the mother wave pacet is also important, e.g., satisfying m s + 4. In a special case, if a compactly supported synchrosqueezed wavelet transform corresponding to s = is preferable, then we require that m s + 4 =. Hence, the mother wavelet is better to be C. We have not optimized the requirement of the variance of the Gaussian process e and the probability bound in Theorem.6 and Theorem.7. According to the numerical performance of the SSTs, the requirement of the variance could be weaened and the probability estimation could be improved. A ey step is to improve the estimate of P G G 3 and P G G 3 in the above proofs. This is left as future wor. The above statistical property of the D SSWPT can be extended to higher dimensional cases, e.g. the D SSWPT and SSCT. However, the notations are much heavier and the calculations are more tedious. We close this section with the theorem for the D SSWPT. See the proof of Theorem and a similar theorem for the SSCT in [47]. Theorem.8. Suppose the D{ mother wave pacet is of type, m, for any fixed, and any fixed integer m max +s s, s }. + 4 Suppose gx = fx+e, where e is a zero mean stationary Gaussian process with a spectrum denoted by êξ and max ξ êξ. Define M a = max ξ < ê a s ξ+a. For any p, ] and q >, let δ a = M p/+q a +, and R δa = {a, b : W g a, b a s δ a }, S δa = {a, b : W g a, b δ a }, Z = {a, b : a N b φ b a s } for K. } For fixed M, m, s, and K, there exists a constant N M, m, K, s, max { s, s such that for any N > N and fx F M, N, K, s the following statements hold. i {Z : K} are disjoint. ii If a, b R δa, then a, b K Z with a probability at least e O N s Ma q/+q + O N m s iii If a, b S δa, then a, b K Z with a probability at least e q/+q OMa + O N m s.. 4

15 iv If a, b R δa Z for some, then v g a, b N φ b N φ b + M p/+q a is true with a probability at least e O e O N 4s Ma q/+q +O N m 4 s N 4s Ma q/+q + O e O N m m+s. N s Ma q/+q v If a, b S δa Z for some, then v g a, b N φ b N φ b N s + M p/+q a is true with a probability at least e O e O N s Ma q/+q +O N m 4 s N s Ma q/+q + O e O N m m+s 3 Implementation and numerical results. N M a q/+q In this section, we provide numerical examples to demonstrate some statistical properties discussed in Section. Guided by these properties, we explore several new ideas to improve the statistical stability of discrete SSTs in the presence of heavy noise. We have developed SynLab, a collection of MATLAB implementation for various SSTs that has been publicly available at: Most numerical examples presented in this paper can be found in this toolbox. 3. Implementation for better statistical stability From the discussion in Section, we can summarize several observations for designing better implementation of SSTs to reduce the fluctuation of noise.. Smaller geometric parameter s As we can see in the theorems for noisy data, a smaller geometric parameter s results in a higher probability of a good estimation of the instantaneous uency or the local wave vector via the information function v f a, b. Hence, in the case when noise is heavy, it is better to apply an SST with a smaller geometric parameter s.. Highly redundant frames Previously in the synchrosqueezed wavelet transform SSWT [5], the synchrosqueezed short- Fourier transform SSSTFT [37], and the synchrosqueezed wave pacet transform SSWPT [48], various SSTs were generated from -uency transforms with low redundancy to obtain efficient forward and inverse transforms. However, the resultant transforms are not reliable when noise is heavy as we can see in the example in Figure. In this example, one single IMT fx = e 6πix+.5 cosπx is 5

16 Figure : Top-left: The real instantaneous uency. Top-right: The standard SSWT available in []. Bottom-left: The standard SSSTFT available in []. Bottom-right: The SSWPT with only one frame, i.e., the redundancy parameter red = in SynLab, which only one wave pacet frame is used to compute the synchrosqueezed transform. Parameters in these transforms have been tuned for reasonably good visualization. embedded in white Gaussian noise with a distribution N,. The signal is sampled in a interval [, ] with a sampling rate 4 Hz. We apply the standard SSWT, SSSTFT in [] and the SSWPT with s =.75 in SynLab and visualize their results using the same discrete grid in the -uency domain. Although we could identify a rough curve in these results to estimate the instantaneous uency of fx, the accuracy is poor in some areas and there are many misleading curves coming from the noise. As we can see in the theory for SSTs, the accuracy of the estimation provided by the information function v f a, b is essentially independent of a and the mother wave pacet wx. This motivates us to synchrosqueeze a highly redundant -uency transform with over-complete samples in a and different mother wave pacets wx. This generates many samples of estimations from a single realization of the noisy data. Averaging these estimation samples leads to a better result. It can be understood in the point of view that the contribution of IMTs to the synchrosqueezed energy distribution T f a, b will remain the same due to the coherent averaging, but the contribution of the noise will be smoothed out because of the incoherent averaging. Applying different mother wave pacets is essentially the same as applying multitapers in the multitaper -uency reassignment in [46]. In this paper, we only focus on 6

17 oversampling the variable a and leave the design of different multidimensional mother wave pacets as future wor. In the numerical examples later, for a fixed geometric parameter s, we synchrosqueeze a union of wave pacet frames generated by uency shifting, dilation and rotation in multidimensional spaces. The number of frames is denoted as red in SynLab. 3. Selective -uency coefficients As we can see in the proofs of previous theorems, a larger -uency coefficient results in a higher probability for a good estimation provided by the information function v f a, b. This inspires two ideas: applying an adaptive -uency transform before synchrosqueezing; only reassigning the largest coefficient in the domain where there is at most one IMT. The first idea aims at generating large coefficients while the second idea avoids incorrect reassignment as much as possible. Selecting the best coefficients to reassign, in some sense, is similar to the idea of sparse matching pursuit in a highly redundant frame, but we avoid its expensive optimization. 3. Numerical examples Here we provide numerical examples to demonstrate the efficiency of the proposed implementation in Section 3.. In all examples, we assume the given data gx = fx + ex is defined in [, ] n, where fx is the target signal, ex is white Gaussian noise with a distribution σ N,, and n is the dimension. As we have seen in the theorems in Section, a proper threshold adaptive to the noise level after the wave pacet transform is important to obtain an accurate instantaneous uency/local wave vector estimate. We refer to [7, 8] for estimating noise level and [36] for designing thresholds for the SSWT. The generalization of these techniques for the transforms here is straightforward. In this paper, we use a small uniform threshold δ = rather than a threshold adaptive to noise level and set σ such that the noise is overwhelming the original signal. We refer the reader to [5, 5] for detailed implementation using these parameters. 3.. Visual illustrations for statistical properties To support the theoretical analysis in Section and the proposals in Section3., we compare the performance of the SSWPT with different redundancy parameter red and s = /+/8, where =, and 3, in both noiseless cases and highly noisy cases. One-dimensional examples: We start with the D SSWPT. In some real applications, e.g., seismic data analysis [34, 5], wave-lie components are only supported in a bounded domain or they have sharp changes in instantaneous uencies. Hence, we would lie to test a benchmar signal fx in which there is a component with a bounded support and an oscillatory instantaneous uency, and a component with an exponential instantaneous uency see Figure. Of a special interest to test the performance of synchrosqueezed transforms for impulsive waves, a wavelet component is added in this signal at x =.. The synthetic benchmar 7

18 x x x Figure : Left: A D synthetic benchmar signal. It is normalized using L norm. Middle: A noisy version generated with white Gaussian noise.75n,. Right: A noisy version contaminated by an α stable random noise [] with parameters α =, dispersion=.9, δ =, N = 89. The noise is re-scaled to have a L -norm equal to 5 by dividing a constant factor. signal fx is defined as where fx = χ [,.6] xf x + χ [,.6] xf x + χ [.6,] xf 4 x + χ [.4,.8] xf 3 x + f 5 x, f x =.6 cos7πx, f x =.8 cos3πx, f 3 x =.7 cos3πx + 5 sinπx, 8π 5x/4 f 4 x = sin, ln f 5 x = 3e 5x. cos5x., and χx is the indicator function. fx is sampled in [, ] with a sampling rate 89 Hz and the range of instantaneous uencies is 5 6 Hz. The white Gaussian noise in this example is.75n,. Although we are not aware of the optimal value of the scaling parameter s, it is clear from Theorem.6 and.7 that the synchrosqueezed transform with a smaller s is more suitable for noisy signals. As shown in the second and the third rows in Figure 3, in the noisy cases, the synchrosqueezed energy distribution with s =.65 in the first column is better than the one with s =.75 in the second column, which is better than the one with s =.875 in the last column. This agrees with the conclusion in Theorem.6 and.7 that a smaller s results in a higher probability to obtain a good instantaneous uency estimate. Another ey point is that a wave pacet coefficient with a larger magnitude has a higher probability to give a good instantaneous uency estimate. A highly redundant wave pacet transform would have wave pacets better fitting the local oscillation of wavelie components. In another word, there would be more coefficients with large magnitudes. The resulting synchrosqueezed energy distribution has higher non-zero energy concentrating Prepared by Miro van der Baan and available in [34, 4]. 8

19 around instantaneous uencies. This is also validated in Figure 3. The results in the third row obtained with red = 6 is better than those in the second row obtained with red =. It is also interesting to observe that the SST with a smaller s is better at capturing the component boundaries, e.g. at x =.39,.59 and.77 and is more robust to an impulsive perturbation see Figure and 3 at x =. and an example of α stable noise in Figure and 4. Boundaries and impulse perturbations would produce uency aliasing. The SSWPT with a smaller s has wave pacets with a smaller support in uency and a larger support in space. Hence, it is more robust to uency aliasing in the sense that the influence of impulsive perturbations is smoothed out and the synchrosqueezed energy of the target components might not get dispersed when it meets the uency aliasing, as shown in Figure 4. However, if s is small, the instantaneous uency estimate might be smoothed out and it is difficult to observe detailed information of instantaneous uencies. As shown in the first row of Figure 3, when the input signal is noiseless, the synchrosqueezed transforms with s =.75 and.875 have better accuracy than the one with s =.65. In short, it is important to tune scaling parameters for data-dependent synchrosqueezed transforms, which has been implemented in the SynLab toolbox. Two-dimensional examples: We now explore the performance of the D SSWPT using a single wave-lie component in Figure 5. The function fx = e πi6x +.5 sinπx +6x +.5 sinπx is uniformly sampled in [, ] with a sampling rate 5 Hz and is disturbed by additive white Gaussian noise 5N,. The D SSWPTs with s =.65,.75 and.875 are applied to this noisy example and their results are shown in Figure 6. Since the synchrosqueezed energy distribution T f v, v, x, x of an image is a function in R 4, we fix x =, stac the results in v, and visualize R T f v, v, x, dv. The results in Figure 6 again validate the theoretical conclusion in Theorem.8 that a smaller scaling parameter s and a higher redundancy yield to a better SST for noisy data. It is interesting that a band-limited SST can also provide better statistical stability if the range of instantaneous uencies/local wave vectors is nown a priori. Let us justify this idea with the follow example. We apply the band-limited SSWPT to the D noisy image in Figure 6 and present the results in the last row of Figure 6. Comparing to the results in the second row of Figure 6, the band-limited SSWPT clearly outperforms the original SSWPT. Component test Here we present an example to validate the last observation in Section 3.. Suppose we loo at a region in the -uency or phase space domain and we now there might be only one IMT in this region. This assumption is reasonable because after the SST people might be interested in the synchrosqueezed energy in a particular region: is this corresponding to a component or just heavy noise? A straightforward solution is that, at 9

20 Figure 3: Synchrosqueezed energy distributions with s =.65 left column, s =.75 middle column and s =.875 right column. In the first row, we apply the SSWPT to clean data. In the second row, the SSWPT with a smaller redundancy is applied to the noisy data with.75n, noise in Figure. In the last row, a highly redundant SSWPT is applied to the same noisy data. each or space grid point, we only reassign those coefficients with the largest magnitude. By Theorem.8, if there is an IMT, we can obtain a setch of its instantaneous uency or local wave vector with a high probability. If there was only noise, we would obtain random reassigned energy with a high probability. Using this idea, we apply the bandlimited SSWPT with s =.65 and red = to a noisy version of the image in Figure 5 left. From left to right, Figure 7 shows the results of a noisy image with 5N, noise, a noisy image with N, noise, and an image with only noise, respectively. A reliable setch of the local wave vector is still visible even if the input image is highly noisy.

21 Figure 4: Synchrosqueezed energy distributions with s =.65 left, s =.75 middle and s =.875 right using highly redundant SSWPTs. The synchrosqueezed energy with a smaller s is smoother and the influence of impulsive noise is weaer x x x x Figure 5: Left: A D noiseless wave-lie component. Right: A noisy wave-lie component generated with white Gaussian noise 5N,. 3.. Quantitative performance analysis We quantitatively analyze the performance of the highly redundant SSWPT and compare it with other methods in this subsection in terms of statistical stability and computational efficiency. Let us revisit the wave-lie component in Figure, where 6πix+.5 cosπx fx = e is sampled in [, ] with a sampling rate 4 Hz. Its instantaneous uency is qx = 3.π sinπx. Let g = f + e be the noisy data, where e is white Gaussian noise with a distribution σ N,. To measure noise, we introduce the following signal-to-noise ratio SNR of the input data g = f + e: Varf SNR[dB]g = log. Vare An ideal -uency distribution of fx is a function Dv, x such that Dv, x = δv qx, where δv is a Dirac delta function of v. To quantify the numerical performance of various methods, we introduce the Earth mover s distance EMD [9, 3, 3] to measure the distance between a resultant -uency distribution T v, x and the ideal distribution

22 v v v x x x v v v x x x v v v x x x Figure 6: Staced synchrosqueezed energy distribution R T f v, v, x, dv of the noisy D signal in Figure 5. From left to right, s =.65,.75 and.875. From top to bottom: red =, red =, and red = with a restricted uency band from to Hz. Dv, x. At each x, after the discretization and normalization of T v, x, we obtain a D discrete distribution T v, x. Similarly, we compute the discrete version Dv, x of Dv, x. At each x, we compute the D EMD between T v, x and Dv, x. The average EMD at all x is defined as the distance also denoted as EMD between T v, x and Dv, x in this paper. A smaller EMD means a better -uency concentration to the ground true instantaneous uency and fewer noise fluctuations. First, we compare the performance of the SSWPT with different redundancy parameters red for noisy data with different SNRs. We compare the EMD between their resultant uency distributions and the ideal one. Figure 9 top-left shows the EMD as functions in the variable of red. The EMD functions decrease fast when the red increases to. All the SSWPTs with red have almost the same efficiency. Second, we compare the performance of the SSWPT with the same redundancy parameters red = but different geometric scaling parameter s for noisy data with different noise level, e.g. σ ranging from to 4, i.e., SNR from to 5. We compare the EMD between their resultant -uency distributions and the ideal one. Figure 9 top-right shows the EMD as functions in the variable of σ for different s. It shows that when the

23 v v v x x x v v v x x x Figure 7: Top row: The synchrosqueezed energy distribution of the highly redundant bandlimited SSWPT with a uency band to Hz. Bottom row: reassigned wave pacet coefficients with the largest magnitude at a space location. Left column: 5N, noise. Middle column: N, noise. Right column: noise only Figure 8: Left and middle: The multitaper -uency reassignment using an arithmetic mean MRSA and a geometric mean MRSG with tapers. Right: The highly redundant SSWPT with s =.75 and red =. data is clean, the SSWPT with s =, which is essentially the SSWT with high redundancy, has the best performance. However, once the data is noisy, the SSWPT with s =.75 has the best overall performance. This observation agrees with our theorems in Section. If s is larger, the instantaneous uency estimation is more accurate if there is no noise. But the probability for a good estimation is smaller if there is noise. If s is smaller, the estimation accuracy is worse but the probability for a good estimation is higher. In theory, it is interesting to find an optimal s that maes a good balance. In practice, a heuristic choice for the optimal s is.75. Third, we compare the performance of the standard SSWT, SSSTFT in [] and the 3

24 highly redundant SSWPT. Again, we choose the parameters in the example of Figure because they result in good visualization of -uency distribution. These transforms are applied to noisy data with different noise level. We compare the EMD between their resultant -uency distributions and the ideal one. The comparison is shown in Figure 9 bottom-left. The highly redundant SSWPT has better -uency concentration than the standard SSWT and SSSTFT in all examples, even if in the noiseless case. Finally, we compare the multitaper -uency reassignment using an arithmetic mean MRSA and a geometric mean MRSG in[46] with the highly redundant SSWPT. A MATLAB pacage of the MRSA and the MRSG is available on the authors homepage. The -uency distribution of these three methods are visualized in Figure 8. We visualize their results using the same discrete grid in the -uency domain. To mae a fair comparison, the number of tapers are chosen to be for the MRSA and the MRSG, and the redundancy parameter red is for the SSWPT with s =.75. For one realization of this experiment, the running for the MRSA or the MRSG is 55.3 seconds, while the running for the SSWPT is only.43 seconds. A visual inspection of Figure 8 shows that the SSWPT also outperforms the MRSA and the MRSG: clear spectral energy at the boundary and better -uency concentration. To quantify this comparison, we apply the EMD again and the results are shown in Figure 9 bottom-right. In most cases, the SSWPT with s =.75 gives a smaller EMD. When σ is near 4, i.e., the SNR is near 5, the MRSG with tapers and the SSWPT have comparable performance. 4 Conclusion In theory, the statistical analysis in this paper has analyzed the statistical properties of a wide range of compactly supported synchrosqueezed transforms in multidimensional spaces, considering zero mean stationary Gaussian random process and small perturbation. Guided by these properties, this paper has presented several approaches to improve the performance of these synchrosqueezed transforms under heavy noise. A MATLAB pacage SynLab for these algorithms is available at Acnowledgments. This wor was partially supported by the National Science Foundation under award DMS-8465 and the U.S. Department of Energys Advanced Scientific Computing Research program under award DE-FC-3ER634/DE-SC949 for Lexing Ying. H. Yang is also supported by a AMS Simons travel grant. He thans Jean-Baptiste Tary and Miro van der Baan for the discussion of benchmar signals, thans Lexing Ying, Charles K. Chui, Jianfeng Lu and Segey Fomel for the discussion of the implementation and the application of synchrosqueezed transforms. References [] [] [3] F. Auger and P. Flandrin. Improving the readability of -uency and -scale representations by the reassignment method. Signal Processing, IEEE Transactions on, 435:68 89,

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