64,837 research outputs found
Generation of short and long range temporal correlated noises
We present the implementation of an algorithm to generate Gaussian random
noises with prescribed time correlations that can be either long or short
ranged. Examples of Langevin dynamics with short and long range noises are
presented and discussed.Comment: 7 pages, 6 figs, submitted to J. Comp. Phy
Time-frequency transforms of white noises and Gaussian analytic functions
A family of Gaussian analytic functions (GAFs) has recently been linked to
the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This
answered pioneering work by Flandrin [2015], who observed that the zeros of the
Gabor transform of white noise had a very regular distribution and proposed
filtering algorithms based on the zeros of a spectrogram. The mathematical link
with GAFs provides a wealth of probabilistic results to inform the design of
such signal processing procedures. In this paper, we study in a systematic way
the link between GAFs and a class of time-frequency transforms of Gaussian
white noises on Hilbert spaces of signals. Our main observation is a conceptual
correspondence between pairs (transform, GAF) and generating functions for
classical orthogonal polynomials. This correspondence covers some classical
time-frequency transforms, such as the Gabor transform and the Daubechies-Paul
analytic wavelet transform. It also unveils new windowed discrete Fourier
transforms, which map white noises to fundamental GAFs. All these transforms
may thus be of interest to the research program `filtering with zeros'. We also
identify the GAF whose zeros are the extrema of the Gabor transform of the
white noise and derive their first intensity. Moreover, we discuss important
subtleties in defining a white noise and its transform on infinite dimensional
Hilbert spaces. Finally, we provide quantitative estimates concerning the
finite-dimensional approximations of these white noises, which is of practical
interest when it comes to implementing signal processing algorithms based on
GAFs.Comment: to appear in Applied and Computational Harmonic Analysi
Wavelet Shrinkage and Thresholding based Robust Classification for Brain Computer Interface
A macaque monkey is trained to perform two different kinds of tasks, memory
aided and visually aided. In each task, the monkey saccades to eight possible
target locations. A classifier is proposed for direction decoding and task
decoding based on local field potentials (LFP) collected from the prefrontal
cortex. The LFP time-series data is modeled in a nonparametric regression
framework, as a function corrupted by Gaussian noise. It is shown that if the
function belongs to Besov bodies, then using the proposed wavelet shrinkage and
thresholding based classifier is robust and consistent. The classifier is then
applied to the LFP data to achieve high decoding performance. The proposed
classifier is also quite general and can be applied for the classification of
other types of time-series data as well, not necessarily brain data
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