arXiv subjects

Gi-Ren Liu

Publications and source records attributed to Gi-Ren Liu.

13 recordsLinked to original sources

arXiv2025

On Random Fields Associated with Analytic Wavelet Transform

Despite the broad application of the analytic wavelet transform (AWT), a systematic statistical characterization of its magnitude and phase as inhomogeneous random fields on the time-frequency domain when the input is a random process remains underexplored. In this work, we study the magnitude and phase of the AWT as random fields on the time-frequency domain when the observed signal is a deterministic function plus additive stationary Gaussian noise. We derive their marginal and joint distributions, establish concentration inequalities that depend on the signal-to-noise ratio (SNR), and analyze their covariance structures. Based on these results, we derive an upper bound on the probability of incorrectly identifying the time-scale ridge of the clean signal, explore the regularity of scalogram contours, and study the relationship between AWT magnitude and phase. Our findings lay the groundwork for developing rigorous AWT-based algorithms in noisy environments.

math.ST↗

arXiv2024

Probabilistic Analysis of Scalogram Ridges in Signal Processing

While ridges in the scalogram, determined by the squared modulus of analytic wavelet transform (AWT), is a widely accepted concept and utilized in nonstationary time series analysis, their behavior in noisy environments remains underexplored. Our object is to provide a theoretical foundation for scalogram ridges by defining ridges as a potentially set-valued random process connecting local maxima of the scalogram along the scale axis and analyzing their properties when the signal fulfills the adaptive harmonic model and is contaminated by stationary Gaussian noise. In addition to establishing several key properties of the AWT for random processes, we investigate the probabilistic characteristics of the resulting random ridge points in the scalogram. Specifically, we establish the uniqueness property of the ridge point at individual time instances and prove the upper hemicontinuity of the ridge random process. Furthermore, we derive bounds on the probability that the deviation between the ridges of noisy and clean signals exceeds a specified threshold, and these bounds depend on the signal-to-noise ratio. To achieve these ridge deviation results, we derive maximal inequalities for the complex modulus of nonstationary Gaussian processes, leveraging classical tools such as the Borell-TIS inequality and Dudley's theorem, which might be of independent interest.

math.ST↗

arXiv2023

Convergence Rate Analysis in Limit Theorems for Nonlinear Functionals of the Second Wiener Chaos

The purpose of this paper is to analyze the distribution distance between random vectors derived from the magnitude of the analytic wavelet transform of the squared envelopes of Gaussian processes and their large-scale limits. When the Hurst index of the underlying Gaussian process falls below 1/2, the large-scale limit takes the form of a combination of the second and fourth Wiener chaos. In this case, we employ the non-Stein approach to determine the convergence rate in the Kolmogorov metric. In scenarios where the Hurst index of the underlying Gaussian process exceeds 1/2, the large-scale limit is a correlated chi-distributed random process with two degrees of freedom. The convergence rate in the Wasserstein metric is derived through the application of Malliavin calculus and multidimensional Stein's method. Although the large-scale limits are the same for Hurst indices within the interval (1/2,1), notable differences in the upper bounds of convergence rates are observed among Hurst indices falling within the subintervals (1/2,3/4) and (3/4,1).

math.PR↗

arXiv2023

Ridge detection for nonstationary multicomponent signals with time-varying wave-shape functions and its applications

We introduce a novel ridge detection algorithm for time-frequency (TF) analysis, particularly tailored for intricate nonstationary time series encompassing multiple non-sinusoidal oscillatory components. The algorithm is rooted in the distinctive geometric patterns that emerge in the TF domain due to such non-sinusoidal oscillations. We term this method \textit{shape-adaptive mode decomposition-based multiple harmonic ridge detection} (\textsf{SAMD-MHRD}). A swift implementation is available when supplementary information is at hand. We demonstrate the practical utility of \textsf{SAMD-MHRD} through its application to a real-world challenge. We employ it to devise a cutting-edge walking activity detection algorithm, leveraging accelerometer signals from an inertial measurement unit across diverse body locations of a moving subject.

math.NA↗

arXiv2023

Gaussian Approximation for the Moving Averaged Modulus Wavelet Transform and its Variants

The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases, and we provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.

math.PR↗

arXiv2021

When Scattering Transform Meets Non-Gaussian Random Processes, a Double Scaling Limit Result

We explore the finite dimensional distributions of the second-order scattering transform of a class of non-Gaussian processes when all the scaling parameters go to infinity simultaneously. For frequently used wavelets, we find a coupling rule for the scale parameters of the wavelet transform within the first and second layers such that the limit exists. The coupling rule is explicitly expressed in terms of the Hurst index of the long range dependent inputs. More importantly, the spectral density function of the limiting process can be explicitly expressed in terms of the Hurst index of the long-range dependent input process and the Fourier transform of the mother wavelet. To obtain these results, we first show that the scattering transform of a class of non-Gaussian processes can be approximated by the second-order scattering transform of Gaussian processes when the scale parameters are sufficiently large. Secondly, for each approximation process, which is still a non-Gaussian random process, we show that its doubling scaling limit exists by the moment method. The long-range dependent non-Gaussian model considered in this work is realized by real data, and assumptions are supported by numerical results.

math.PR↗

arXiv2021

Asymptotic Analysis of Higher-order Scattering Transform of Gaussian Processes

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein's method to achieve the goal.

math.PR↗

arXiv2020

Central and Non-central Limit Theorems arising from the Scattering Transform and its Neural Activation Generalization

Motivated by analyzing complicated and non-stationary time series, we study a generalization of the scattering transform (ST) that includes broad neural activation functions, which is called neural activation ST (NAST). On the whole, NAST is a transform that comprises a sequence of ``neural processing units'', each of which applies a high pass filter to the input from the previous layer followed by a composition with a nonlinear function as the output to the next neuron. Here, the nonlinear function models how a neuron gets excited by the input signal. In addition to showing properties like non-expansion, horizontal translational invariability and insensitivity to local deformation, the statistical properties of the second order NAST of a Gaussian process with various dependence and (non-)stationarity structure and its interaction with the chosen high pass filters and activation functions are explored and central limit theorem (CLT) and non-CLT results are provided. Numerical simulations are also provided. The results explain how NAST processes complicated and non-stationary time series, and pave a way towards statistical inference based on NAST under the non-null case.

stat.ML↗

arXiv2019

Explore intrinsic geometry of sleep dynamics and predict sleep stage by unsupervised learning techniques

We propose a novel unsupervised approach for sleep dynamics exploration and automatic annotation by combining modern harmonic analysis tools. Specifically, we apply diffusion-based algorithms, diffusion map (DM) and alternating diffusion (AD) algorithms, to reconstruct the intrinsic geometry of sleep dynamics by reorganizing the spectral information of an electroencephalogram (EEG) extracted from a nonlinear-type time frequency analysis tool, the synchrosqueezing transform (SST). The visualization is achieved by the nonlinear dimension reduction properties of DM and AD. Moreover, the reconstructed nonlinear geometric structure of the sleep dynamics allows us to achieve the automatic annotation purpose. The hidden Markov model is trained to predict the sleep stage. The prediction performance is validated on a publicly available benchmark database, Physionet Sleep-EDF [extended] SC* and ST*, with the leave-one-subject-out cross validation. The overall accuracy and macro F1 achieve 82:57% and 76% in Sleep-EDF SC* and 77.01% and 71:53% in Sleep-EDF ST*, which is compatible with the state-of-the-art results by supervised learning-based algorithms. The results suggest the potential of the proposed algorithm for clinical applications.

eess.SP↗

arXiv2019

Diffuse to fuse EEG spectra -- intrinsic geometry of sleep dynamics for classification

We propose a novel algorithm for sleep dynamics visualization and automatic annotation by applying diffusion geometry based sensor fusion algorithm to fuse spectral information from two electroencephalograms (EEG). The diffusion geometry approach helps organize the nonlinear dynamical structure hidden in the EEG signal. The visualization is achieved by the nonlinear dimension reduction capability of the chosen diffusion geometry algorithms. For the automatic annotation purpose, the {support vector machine} is trained to predict the sleep stage. The prediction performance is validated on a publicly available benchmark database, Physionet Sleep-EDF [extended] SC$^*$ {(SC = Sleep Cassette)} and ST$^*$ {(ST = Sleep Telemetry)}, with the leave-one-subject-out cross validation. When we have a single EEG channel (Fpz-Cz), the overall accuracy, macro F1 and Cohen's kappa achieve $82.72\%$,$75.91\%$ and $76.1\%$ respectively in Sleep-EDF SC$^*$ and $78.63\%$, $73.58\%$ and $69.48\%$ in Sleep-EDF ST$^*$. This performance is compatible {with} the state-of-the-art results. When we have two EEG channels (Fpz-Cz and Pz-Oz), the overall accuracy, macro F1 and Cohen's kappa achieve $84.44\%$,$78.25\%$ and $78.36\%$ respectively in Sleep-EDF SC$^*$ and $79.05\%$, $74.73\%$ and $70.31\%$ in Sleep-EDF ST$^*$. The results suggest the potential of the proposed algorithm in practical applications.

eess.SP↗

arXiv2015

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projection with Null Initialization

Alternating projection (AP) of various forms, including the Parallel AP (PAP), Real-constrained AP (RAP) and the Serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed for initial guess and proved to produce asymptotically accurate initialization for the case of Gaussian random measurement. Numerical experiments show that the null initialization produces more accurate initial guess than the spectral initialization and that AP converges faster to the true object than other iterative schemes for non-convex optimization such as the Wirtinger Flow. In numerical experiments, AP with the null initialization converges globally to the true object.

physics.data-an↗

arXiv2014

Multi-scaling Limits for Relativistic Diffusion Equations with Random Initial Data

Let $u(t,\mathbf{x}),\ t>0,\ \mathbf{x}\in \mathbb{R}^{n},$ be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatial-fractional parameter $α\in (0,2)$ and the mass parameter $\mathfrak{m}> 0$, subject to a random initial condition $u(0,\mathbf{x})$ which is characterized as a subordinated Gaussian field. In this article, we study the large-scale and the small-scale limits for the suitable space-time re-scalings of the solution field $u(t,\mathbf{x})$. Both the Gaussian and the non-Gaussian limit theorems are discussed. The small-scale scaling involves not only to scale on $u(t,\mathbf{x})$ but also to re-scale the initial data; this is a new-type result for the literature. Moreover, in the two scalings the parameter $α\in (0,2)$ and the parameter $\mathfrak{m}> 0$ paly distinct roles for the scaling and the limiting procedures.

math.PR↗

arXiv2010

Homogenization of Fractional Kinetic Systems with Random Initial Data

Let ${\bf{w}}(t,x):=(u,v)(t,x),\ t>0,\ x\in \mathbb{R}^{n},$ be the $\mathbb{R}^2$-valued spatial-temporal random field ${\bf{w}}=(u, v)$ arising from a certain two-equation system of fractional kinetic equations of reaction-diffusion type, with given random initial data $u(0,x)$ and $v(0,x).$ The space-fractional derivative is characterized by the composition of the inverses of the Riesz potential and the Bessel potential. We discuss two scaling limits, the macro and the micro, for the homogenization of ${\bf{w}}(t,x)$, and prove that the rescaled limit is a singular field of multiple Itô-Wiener integral type, subject to suitable assumptions on the random initial conditions. In the two scaling procedures, the Riesz and the Bessel parameters play distinctive roles. Moreover, since the component fields $u,v$ are dependent on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this components dependence. The time-fractional system is also considered, in which the Mittag-Leffler function is used.

math.PR↗