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Sparse sequences of neural spikes are posited to underlie aspects of working memory, motor production, and learning. Discovering these sequences in an unsupervised manner is a longstanding problem in statistical neuroscience. Promising…
Time series analysis is crucial in fields like finance, economics, environmental science, and biomedical engineering, aiding in forecasting, pattern identification, and understanding underlying mechanisms. While traditional time-domain…
We investigate the benefits and challenges of utilizing the frequency information in differential equation identification. Solving differential equations and Fourier analysis are closely related, yet there is limited work in exploring this…
The analysis of chaotic signals with time-frequency methods is considered. For this purpose, two new transformations are presented which consist in the decomposition of a signal onto an orthogonal set of respectively linear and hyperbolic…
Discrete-time fractional-order dynamical systems (DT-FODS) have found innumerable applications in the context of modeling spatiotemporal behaviors associated with long-term memory. Applications include neurophysiological signals such as…
Time-frequency distributions have been used to provide high resolution representation in a large number of signal processing applications. However, high resolution and accurate instantaneous frequency (IF) estimation usually depend on the…
Detrend fluctuation analysis (DFA) has become a choice method for effective analysis of a broad variety of nonstationary signals. We show in the present article that, provided the nonstationary fluctuations occur at a large enough time…
The extraction of oscillatory components and their properties from different time-frequency representations, such as windowed Fourier transform and wavelet transform, is an important topic in signal processing. The first step in this…
We consider the problem of reconstructing a discrete-time signal (sequence) with continuous-valued components corrupted by a known memoryless channel. When performance is measured using a per-symbol loss function satisfying mild regularity…
We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…
Nonlinear stochastic differential equations provide one of the mathematical models yielding 1/f noise. However, the drawback of a single equation as a source of 1/f noise is the necessity of power-law steady-state probability density of the…
We study the classical problem of recovering a multidimensional source signal from observations of nonlinear mixtures of this signal. We show that this recovery is possible (up to a permutation and monotone scaling of the source's original…
Functional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise.…
There is a class of physical filtration processes where the input is adequately modeled by a continuous periodic function f (x) of bounded variation over its period, and the output depends only on certain harmonics of the Fourier expansion…
A new technique for calculating the time-evolution, correlations and steady state spectra for nonlinear stochastic differential equations is presented. To illustrate the method, we consider examples involving cubic nonlinearities in an…
Stochastic averaging allows for the reduction of the dimension and complexity of stochastic dynamical systems with multiple time scales, replacing fast variables with statistically equivalent stochastic processes in order to analyze…
Context: Several approaches to estimate frequency, phase and amplitude errors in time series analyses were reported in the literature, but they are either time consuming to compute, grossly overestimating the error, or are based on…
Normal human heart rate shows complex fluctuations in time, which is natural, since heart rate is controlled by a large number of different feedback control loops. These unpredictable fluctuations have been shown to display fractal…
Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate processes. Parameters like the spectral density matrix and its inverse, the coherence…
Fluorescence spectroscopy is an image correlation technique to analyze and characterize the molecular dynamics from a sequence of fluorescence images. Many image correlation techniques have been developed for different applications [1]. But…