scholarly journals On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

2005 ◽  
Vol 37 (4) ◽  
pp. 938-962 ◽  
Author(s):  
Sergio Albeverio ◽  
Shuji Kawasaki
Keyword(s):  
Limit Theorem ◽  
Moment Method ◽  
Wavelet Coefficient ◽  
General Setting ◽  
Maximum Likelihood Estimators ◽  
Positive Definite ◽  
Wavelet Coefficients ◽  
Localization Property ◽  
Stationary Increments ◽  
Coefficient Decay

We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

scholarly journals On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

2005 ◽  
Vol 37 (04) ◽  
pp. 938-962
Author(s):  
Sergio Albeverio ◽  
Shuji Kawasaki
Keyword(s):  
Limit Theorem ◽  
Moment Method ◽  
Wavelet Coefficient ◽  
General Setting ◽  
Maximum Likelihood Estimators ◽  
Positive Definite ◽  
Wavelet Coefficients ◽  
Localization Property ◽  
Stationary Increments ◽  
Coefficient Decay

We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

2020 ◽  
pp. 1-25
Author(s):  
Kerry Fendick
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Markov Process ◽  
Maximum Likelihood Estimators ◽  
Single Server ◽  
Single Server Queue ◽  
Stationary Increments ◽  
Transient Distribution ◽  
Independent Increments ◽  
Server Queue

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.

scholarly journals Limit theorems for multivariate Brownian semistationary processes and feasible results

2019 ◽  
Vol 51 (03) ◽  
pp. 667-716
Author(s):  
Riccardo Passeggeri ◽  
Almut E. D. Veraart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Laws Of Large Numbers ◽  
Stationary Increments ◽  
Large Numbers ◽  
Weak Laws

AbstractIn this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

Characterization of local regions for wavelet-based image denoising using a statistical approach

2020 ◽  
Vol 18 (03) ◽  
pp. 2050011
Author(s):  
Rajiv Verma ◽  
Rajoo Pandey
Keyword(s):  
Image Denoising ◽  
Statistical Approach ◽  
Wavelet Coefficient ◽  
Threshold Value ◽  
Vital Role ◽  
Wiener Filtering ◽  
Original Signal ◽  
Wavelet Coefficients ◽  
Local Window

The shape of local window plays a vital role in the estimation of original signal variance, which is used to shrink the noisy wavelet coefficients in wavelet-based image denoising algorithms. This paper presents an anisotropic-shaped region-based Wiener filtering (ASRWF) and BayesShrink (ASRBS) algorithms, which exploit the region characteristics to estimate the original signal variance using a statistical approach. The proposed approach divides the region centered on a noisy wavelet coefficient into various non-overlapping subregions. The Euclidean distance-based measure is considered to obtain the similarities between reference subregion and adjacent subregions. An appropriate threshold value is estimated by applying a statistical approach on these distances and the sets of similar and dissimilar subregions are obtained from a defined region. Thus, an anisotropic-shaped region is obtained by neglecting the dissimilar subregions in a defined region. The variance of every similar subregion is calculated and then averaged to estimate the original signal variance to shrink noisy wavelet coefficients effectively. Finally, the estimated signal variance is utilized in Wiener filtering and BayesShrink algorithms to improve the denoising performance. The performance of the proposed algorithms is analyzed qualitatively and quantitatively on standard images for different noise levels.

Application of a New Kind of Wavelet Threshold De-Noising Method in Partial Discharge Signals

2014 ◽  
Vol 889-890 ◽  
pp. 780-785
Author(s):  
Cheng Long Xu ◽  
Hong Yu ◽  
Bi Qiang Du ◽  
Jun Li ◽  
Ze Kun Liu ◽  
...  
Keyword(s):  
Wavelet Transform ◽  
Simulation Experiment ◽  
Wavelet Coefficient ◽  
Partial Discharge ◽  
Noise Signal ◽  
Selection Method ◽  
Wavelet Coefficients ◽  
Threshold Selection ◽  
Whole Process ◽  
Maximum Value

In order to more effectively remove noise in partial discharge signals, it is proposed a new threshold selection method in this paper. This method firstly takes the signals before the partial discharge starting to happen as only contain noise signal, and then applies a wavelet transform to the only contain noise signal. Secondly record every detail part and the maximum value of wavelet coefficients of last layer approximation part, and take this value as its layer threshold. And then applies a wavelet transform to the partial discharge signals which contains noises. Next is to process wavelet coefficient of each layer using the selected threshold. Finally, the already handled wavelet coefficients is used to reconstruction the signals. The whole process of threshold choosing is automatic without human intervention. Simulation experiment show that compared with the traditional threshold selection method, this method can be better to remove the noise of the partial discharge signals, and it has a strong practical value.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (01) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

MATHEMATICAL IMPLEMENTATION OF HYBRID FAST FOURIER TRANSFORM AND DISCRETE WAVELET TRANSFORM FOR DEVELOPING GRAPHICAL USER INTERFACE USING VISUAL BASIC FOR SIGNAL PROCESSING APPLICATIONS

2012 ◽  
Vol 12 (05) ◽  
pp. 1240031 ◽  
Author(s):  
MOUSA K. WALI ◽  
M. MURUGAPPAN ◽  
R. BADLISHAH AHMMAD
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Input Signal ◽  
Wavelet Decomposition ◽  
Wavelet Coefficient ◽  
Visual Basic ◽  
Discrete Wavelet ◽  
Wavelet Coefficients ◽  
End User ◽  
Decomposition Level

In recent years, the application of discrete wavelet transform (DWT) on biosignal processing has made a significant impact on developing several applications. However, the existing user-friendly software based on graphical user interfaces (GUI) does not allow the freedom of saving the wavelet coefficients in .txt or .xls format and to analyze the frequency spectrum of wavelet coefficients at any desired wavelet decomposition level. This work describes the development of mathematical models for the implementation of DWT in a GUI environment. This proposed software based on GUI is developed under the visual basic (VB) platform. As a preliminary tool, the end user can perform "j" level of decomposition on a given input signal using the three most popular wavelet functions — Daubechies, Symlet, and Coiflet over "n" order. The end user can save the output of wavelet coefficients either in .txt or .xls file format for any further investigations. In addition, the users can gain insight into the most dominating frequency component of any given wavelet decomposition level through fast Fourier transform (FFT). This feature is highly essential in signal processing applications for the in-depth analysis on input signal components. Hence, this GUI has the hybrid features of FFT with DWT to derive the frequency spectrum of any level of wavelet coefficient. The novel feature of this software becomes more evident for any signal processing application. The proposed software is tested with three physiological signal — electroencephalogram (EEG), electrocardiogram (ECG), and electromyogram (EMG) — samples. Two statistical features such as mean and energy of wavelet coefficient are used as a performance measure for validating the proposed software over conventional software. The results of proposed software is compared and analyzed with MATLAB wavelet toolbox for performance verification. As a result, the proposed software gives the same results as the conventional toolbox and allows more freedom to the end user to investigate the input signal.

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