This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
SPECTRAL ESTIMATE FOR STABLE SIGNALS WITH P-ADIC TIME AND OPTIMAL SELECTION O...sipij
The spectral density of stable signals with p-adic times is already estimated under various conditions. The
estimate is made by constructing a periodogram that is subsequently smoothed by a spectral window. It is
clear that the convergence rate of this estimator depends on the bandwidth of the spectral window (called
the smoothing parameter). This work gives a method to select the smoothing parameter in an optimal way,
i.e. the estimator converges to the spectral density with the bestrate.
The method is inspired by the cross-validation method, which consists in minimizing the estimate of the
integrated square error.
ALEXANDER FRACTIONAL INTEGRAL FILTERING OF WAVELET COEFFICIENTS FOR IMAGE DEN...sipij
The present paper, proposes an efficient denoising algorithm which works well for images corrupted with
Gaussian and speckle noise. The denoising algorithm utilizes the alexander fractional integral filter which
works by the construction of fractional masks window computed using alexander polynomial. Prior to the
application of the designed filter, the corrupted image is decomposed using symlet wavelet from which only
the horizontal, vertical and diagonal components are denoised using the alexander integral filter.
Significant increase in the reconstruction quality was noticed when the approach was applied on the
wavelet decomposed image rather than applying it directly on the noisy image. Quantitatively the results
are evaluated using the peak signal to noise ratio (PSNR) which was 30.8059 on an average for images
corrupted with Gaussian noise and 36.52 for images corrupted with speckle noise, which clearly
outperforms the existing methods.
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
This document summarizes a novel algorithm for fast sparse image reconstruction from compressed sensing measurements. The algorithm uses adaptive nonlinear filtering strategies in an iterative framework. It formulates the image reconstruction problem using total variation minimization and solves it using a two-step iterative scheme. Numerical experiments show that the algorithm is efficient, stable, and fast compared to state-of-the-art methods, as it can reconstruct images from highly incomplete samples in just a few seconds with competitive performance.
This document summarizes research on the consistency and stability of linear multistep methods for solving initial value differential problems. It discusses the local truncation error and consistency conditions for convergence. The consistency condition requires that the truncation error approaches zero as the step size decreases. Stability conditions like relative and weak stability are also analyzed. It is shown that linear multistep methods satisfy the conditions of the Banach fixed point theorem, ensuring a unique solution. Specifically, a two-step predictor-corrector method is presented where the predictor provides an initial estimate that is corrected.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
SPECTRAL ESTIMATE FOR STABLE SIGNALS WITH P-ADIC TIME AND OPTIMAL SELECTION O...sipij
The spectral density of stable signals with p-adic times is already estimated under various conditions. The
estimate is made by constructing a periodogram that is subsequently smoothed by a spectral window. It is
clear that the convergence rate of this estimator depends on the bandwidth of the spectral window (called
the smoothing parameter). This work gives a method to select the smoothing parameter in an optimal way,
i.e. the estimator converges to the spectral density with the bestrate.
The method is inspired by the cross-validation method, which consists in minimizing the estimate of the
integrated square error.
ALEXANDER FRACTIONAL INTEGRAL FILTERING OF WAVELET COEFFICIENTS FOR IMAGE DEN...sipij
The present paper, proposes an efficient denoising algorithm which works well for images corrupted with
Gaussian and speckle noise. The denoising algorithm utilizes the alexander fractional integral filter which
works by the construction of fractional masks window computed using alexander polynomial. Prior to the
application of the designed filter, the corrupted image is decomposed using symlet wavelet from which only
the horizontal, vertical and diagonal components are denoised using the alexander integral filter.
Significant increase in the reconstruction quality was noticed when the approach was applied on the
wavelet decomposed image rather than applying it directly on the noisy image. Quantitatively the results
are evaluated using the peak signal to noise ratio (PSNR) which was 30.8059 on an average for images
corrupted with Gaussian noise and 36.52 for images corrupted with speckle noise, which clearly
outperforms the existing methods.
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
This document summarizes a novel algorithm for fast sparse image reconstruction from compressed sensing measurements. The algorithm uses adaptive nonlinear filtering strategies in an iterative framework. It formulates the image reconstruction problem using total variation minimization and solves it using a two-step iterative scheme. Numerical experiments show that the algorithm is efficient, stable, and fast compared to state-of-the-art methods, as it can reconstruct images from highly incomplete samples in just a few seconds with competitive performance.
This document summarizes research on the consistency and stability of linear multistep methods for solving initial value differential problems. It discusses the local truncation error and consistency conditions for convergence. The consistency condition requires that the truncation error approaches zero as the step size decreases. Stability conditions like relative and weak stability are also analyzed. It is shown that linear multistep methods satisfy the conditions of the Banach fixed point theorem, ensuring a unique solution. Specifically, a two-step predictor-corrector method is presented where the predictor provides an initial estimate that is corrected.
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, risk theory, queuing, and supply chain/social/multi-agents networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and propose two alternative importance sampling (IS) approaches [1,2] to improve the Monte Carlo (MC) estimator efficiency. The key challenge in the IS framework is to choose an appropriate change of probability measure to achieve substantial variance reduction, which often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control formulation. We pursue two alternative approaches to mitigate the curse of dimensionality when solving the resulting dynamic programming problem. In the first approach [1], we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. As an alternative, we present in [2] a dimension reduction method, based on mapping the problem to a significantly lower dimensional space via the Markovian projection (MP) idea. The output of this model reduction technique is a low dimensional SRN (potentially one dimension) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained via a discrete $L^2$ regression. By solving a resulting projected Hamilton-Jacobi-Bellman (HJB) equation for the reduced-dimensional SRN, we get projected IS parameters, which are then mapped back to the original full-dimensional SRN system, and result in an efficient IS-MC estimator of the full-dimensional SRN. Our analysis and numerical experiments verify that both proposed IS (learning based and MP-HJB-IS) approaches substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators. [1] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. Learning-based importance sampling via stochastic optimal control for stochastic reaction net-works. Statistics and Computing 33, no. 3 (2023): 58. [2] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. (2023). Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach. To appear soon.
The document discusses the discrete Fourier transform (DFT) and its relationship to the discrete-time Fourier transform (DTFT) and discrete Fourier series (DFS). It begins by introducing the DTFT as a theoretical tool to evaluate frequency responses, but notes it cannot be directly computed. The DFT solves this issue by sampling the frequency spectrum. It then discusses how the DFS represents periodic discrete-time signals using complex exponentials, unlike the continuous-time Fourier series. The key properties of the DFS such as linearity, time-shifting, duality, and periodic convolution are also covered. Finally, it discusses how the continuous-time Fourier transform (CTFT) of periodic signals relates to the DFS through Poisson's sum
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document discusses a study analyzing the coronas (gas discharge visualizations) of apple tree leaves and fruits using the GDV Assistant system. The researchers recorded coronas under different conditions to analyze plant vitality and stress levels. They used various machine learning algorithms to analyze the parameterized corona images. The results showed coronas provide useful information about plant stress and variety. However, they could not differentiate between organically and conventionally grown fruit that were similar in standard quality measures. The document describes the GDV Assistant system parameters, recording methodology, classification problems analyzed, machine learning methods used, and results.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
This document provides a course calendar and lecture plans for topics related to Bayesian estimation methods. The course calendar lists 12 class dates from September to December covering topics like Bayes estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms. One lecture plan provides details on the hidden Markov model, including the introduction, definition of HMMs, and problems of evaluation, decoding, and learning. Another lecture plan covers particle filters, including the sequential importance sampling algorithm, choice of proposal density, and the particle filter algorithm of sampling, weight update, resampling, and state estimation.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les cordeliers
Jere Koskela's slides
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
Simple Comparison of Convergence of GeneralIterations and Effect of Variation...Komal Goyal
The document compares the convergence of Jungck-Ishikawa and Jungck-Noor iterative procedures for solving nonlinear equations. It presents 4 examples showing that the two procedures compute solutions in the same number of iterations when the parameters α, β, and γ are equal. Graphs demonstrate the effect of varying these parameters and the initial value x0 on the number of iterations needed for convergence. Both procedures consistently converge for the examples presented.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, risk theory, queuing, and supply chain/social/multi-agents networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and propose two alternative importance sampling (IS) approaches [1,2] to improve the Monte Carlo (MC) estimator efficiency. The key challenge in the IS framework is to choose an appropriate change of probability measure to achieve substantial variance reduction, which often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control formulation. We pursue two alternative approaches to mitigate the curse of dimensionality when solving the resulting dynamic programming problem. In the first approach [1], we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. As an alternative, we present in [2] a dimension reduction method, based on mapping the problem to a significantly lower dimensional space via the Markovian projection (MP) idea. The output of this model reduction technique is a low dimensional SRN (potentially one dimension) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained via a discrete $L^2$ regression. By solving a resulting projected Hamilton-Jacobi-Bellman (HJB) equation for the reduced-dimensional SRN, we get projected IS parameters, which are then mapped back to the original full-dimensional SRN system, and result in an efficient IS-MC estimator of the full-dimensional SRN. Our analysis and numerical experiments verify that both proposed IS (learning based and MP-HJB-IS) approaches substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators. [1] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. Learning-based importance sampling via stochastic optimal control for stochastic reaction net-works. Statistics and Computing 33, no. 3 (2023): 58. [2] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. (2023). Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach. To appear soon.
The document discusses the discrete Fourier transform (DFT) and its relationship to the discrete-time Fourier transform (DTFT) and discrete Fourier series (DFS). It begins by introducing the DTFT as a theoretical tool to evaluate frequency responses, but notes it cannot be directly computed. The DFT solves this issue by sampling the frequency spectrum. It then discusses how the DFS represents periodic discrete-time signals using complex exponentials, unlike the continuous-time Fourier series. The key properties of the DFS such as linearity, time-shifting, duality, and periodic convolution are also covered. Finally, it discusses how the continuous-time Fourier transform (CTFT) of periodic signals relates to the DFS through Poisson's sum
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document discusses a study analyzing the coronas (gas discharge visualizations) of apple tree leaves and fruits using the GDV Assistant system. The researchers recorded coronas under different conditions to analyze plant vitality and stress levels. They used various machine learning algorithms to analyze the parameterized corona images. The results showed coronas provide useful information about plant stress and variety. However, they could not differentiate between organically and conventionally grown fruit that were similar in standard quality measures. The document describes the GDV Assistant system parameters, recording methodology, classification problems analyzed, machine learning methods used, and results.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
This document provides a course calendar and lecture plans for topics related to Bayesian estimation methods. The course calendar lists 12 class dates from September to December covering topics like Bayes estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms. One lecture plan provides details on the hidden Markov model, including the introduction, definition of HMMs, and problems of evaluation, decoding, and learning. Another lecture plan covers particle filters, including the sequential importance sampling algorithm, choice of proposal density, and the particle filter algorithm of sampling, weight update, resampling, and state estimation.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les cordeliers
Jere Koskela's slides
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
Simple Comparison of Convergence of GeneralIterations and Effect of Variation...Komal Goyal
The document compares the convergence of Jungck-Ishikawa and Jungck-Noor iterative procedures for solving nonlinear equations. It presents 4 examples showing that the two procedures compute solutions in the same number of iterations when the parameters α, β, and γ are equal. Graphs demonstrate the effect of varying these parameters and the initial value x0 on the number of iterations needed for convergence. Both procedures consistently converge for the examples presented.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
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Mixed Spectra for Stable Signals from Discrete Observations
1. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
DOI: 10.5121/sipij.2021.12502 21
MIXED SPECTRA FOR STABLE SIGNALS
FROM DISCRETE OBSERVATIONS
Rachid Sabre
Biogeosciences (UMR CNRS/uB 6282),
University of Burgundy, 26, Bd Docteur Petitjean, Dijon, France
ABSTRACT
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
KEYWORDS
Spectral density, stable processes, periodogram, smoothing estimate, aliasing.
1. INTRODUCTION
The multiple applications where the random signals whose variance increases indefinitely impose
the interest of using stable alpha processes by several research authors in various fields. In
particular, Stable symmetric harmonizable processes and their properties have been widely
studied by many authors such as [1] - [10] to name a few.
Concrete applications of stable symmetric processes cover a wide spectrum of fields such as:
physics, biology, electronics and electricity, hydrology, economics, communications and radar
applications., ...ect. See: [11]-[22]. In this paper we consider a symmetric alpha stable
harmonizable process = { ( ): }
X X t t R
. Alternatively X has the integral representation:
( ) = exp ( ) ( )
X t i t d
(1)
where 1< < 2
and is a complex valued symmetric -stable random measure on R with
independent and isotropic increments. The control measure is defined by ( ) =| ( ) |
m A A
(see
[4]) is called spectral measure. The estimation of the spectral density function was already
studied in different cases: by E.Masry and S.Combanis [4] when the time of the process is
continuous, by Sabre [23] when the time of the process is discrete and by R.Sabre [24]-[25]when
the time of the process is p-adic.
In this paper we consider a general case where the spectral measure is the sum of an absolutely
continuous measure with respect to Lebesgue measure and a discrete measure:
2. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
22
=1
( ) = ( )
q
i wi
i
d x dx c
+
where is a Dirac measure, the specral density is nonnegative integrable and bounded
function. i
c is an unknown positive real number and i
w is an unknown real number. Assume
that 0
i
w . The discrete measurement comes from the repeated random energy jumps during the
experimental measurements. Spectral density represents the distribution of the energy carried by
the signal.
Our objective is to propose a nonparametric estimator of the spectral density after discrete
sampling of the process ( )
X t . This work is motivated by the fact that in practice it is impossible
to observe the process over a continuous time interval. However, we sampled the process at
equidistant times, i.e., =
n
t n , > 0
. It is known that aliasing of occurs. For more details
about aliasing phenomenon, see [26]. To avoid this difficulty, we suppose that the spectral
density is vanishing for | |>
where is a nonnegative real number. We create an
estimate of the spectral density based on smoothing methods. We show that it is asymptotically
unbiased and consistent.
Briefly, the organization of this paper will be as follows: in the second section two technical
lemmas will be presented as well as a preiodogram and we will show that this periodogram is an
asymptotically unbiased but inconsistent estimator. In the third section, the periodogram will be
smoothed by two well chosen spectral windows to estimate the spectral density at the jump
points. We show that the smoothing periodogram is a consistent estimator. The fourth section
gives conclusions and working perspectives.
2. THE PERIODOGRAM AND ITS PROPRIETIES
In this section we give some basic notations and properties of the Jackson's polynomial kernel.
Let N is the size of sample of X . Let k and n are the numbers satisfying:
1
1= 2 ( 1)
2
N k n with n N k N
− −
if
1
=
2
k then 1 1
= 2 1,
n n n N
− .
The Jackson's polynomial kernel is defined by: ( )
( ) = ( )
N
N N
H A H
where
2 2
( )
,
,
sin sin
1 1
2 2
( ) = = .
2
sin sin
2 2
k k
N
k n
k n
n n
H with q d
q
−
where
1
,
= ( )
N N
A B
−
with ( )
, = ( ) .
N
N
B H d
−
We cite two lemmas which are used in this paper. Their proof are given in [23].
Lemma 2.1 There is a non negative function k
h such as:
3. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
23
( 1)
( )
= ( 1)
( ) = cos( )
k n
N
k
m k n
m
H h m
n
−
− −
Lemma 2.2 Let
2
, ,
sin
2
= = | | | ( ) | ,
sin
2
k
N N N
n
B d and J u H u du
− −
where
]0,2],
then
2
2 1
,
2 1
2
2 0 < < 2
4 1
< < 2
2 1 2
k
k
N
k
n if
B
k
n if
k k
−
−
−
and
2
2 2 1
, 2
2
1 1 1
< <
2 ( 2 1) 2 2
2 1 1
< < 2
2 ( 1)(2 1) 2
k
k k
N k
k
if
k n k k
J
k
if
k n k
+
−
+
+
− +
+
+ − −
This paragraph gives a periodogram and develops its proprieties. Consider the process ( ),
X t
defined in (1), observed at instants =
j
t j , =1,2,...
j N and
2
=
, where is a real
number strictly greater than 2 . Define the periodogram ˆ
N
I on ] , [
− as follows:
( )
,
ˆ ( ) = | | , 0 < <
2
p
N p N
I C I p
Where
1 = ( 1)
= ( 1)
( ) = [ ] e[ exp{ ( )} ( ( 1) )],
' '
n k n
' '
N N k
'
n k n
n
I A R h i n X n k n
n
−
− −
− + −
The normalisation constant ,
p
C is given by , /
,
=
[ ]
p
p p
p
D
C
F C
, with
| |
,
1 1
1 cos( ) 1
= a =
| | | |
a
u
p p
p p
u e
D du nd F du
u u
−
+ +
− −
− −
.
Lemma 2.3
4. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
24
The characteristic function of ( )
N
I ,
exp ( )
N
E irI , converges to
,1 ,2
exp[ | | ( ( ) ( )].
N N
C r
− + where
( )
,1( ) = | |
N N
y
H y dy
−
−
and ,2
=1
( ) = ( )
q
N i N i
i
c H w
−
Proof
From (1) and the expression of N
I , we have:
1 = ( 1)
= ( 1)
( ) = [ ] e exp [ ( )] exp [ ( 1)] ( ).
' '
n k n
'
N N k
R
'
n k n
n
I A R h i n u i uk n d u
n
−
− −
− −
Using [1] and the definition of the Jackson polynomial kernel we obtain that the characteristic
function is the form:
exp ( ) = exp | | ( ) .
a
N a N
E irI C r
−
(2)
where ,1 ,2
( ) = ( ) ( )
N N N
+ with
( )
,1( ) = | |a
N N
R
v
H v dv
−
and ,2
=1
( ) = ( )
q
N i N i
i
c H w
−
( )
,1( ) = | |a
N N
R
v
H v dv
−
( )
(2 1)
(2 1)
= | | .
j
a
N
j
j Z
v
H v dv
+
−
−
Putting = 2
v y j
− and using the fact that N
H is 2 -periodic, we get
( )
( ) = | | ( ) ,
a
N N j
j Z
H y y dy
−
−
where
2
( ) = .
j
y
y j
−
Let j be an integer
such that
2
< <
y j
−
− . Since <
and | |<
y , we get
1
| |< <1
2 2
j
+ and then
= 0
j . Therefore
( )
,1( ) = | | .
N N
y
H y dy
−
−
(3)
Theorem 2.4 Let < <
− then ( )
ˆ ( ) = ,
p
N N
E I
Proof
From the following equality used in [4], for all real x and 0 < < 2
p ,
1 1
1 1
1 cos( ) 1
| | = = e ,
| | | |
ixu
p
p p
p p
xu e
x D du D R du
u u
− −
+ +
− −
− −
(4)
5. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
25
and replacing x by N
I , we obtain
/ 1
,
1 exp{ ( )}
1
ˆ ( ) = e ,
[ ] | |
N
N p p
p a
iuI
I R du
F C u
+
−
−
(5)
The equation (5) and the definition of the , ,
p
F give
/ 1
,
/
1 exp | | ( )
1
ˆ ( ) = .
[ ] | |
= ( ) .
a N
N p p
R
p a
p
N
C u
EI du
F C u
+
− −
3. SMOOTHING PERIODOGRAM
In order to obtain a consistent estimate of [ ( )]
p
, we smooth the periodogram via spectral
windows depending on whether is a jump point or not( )
i
w
.
(1)
1 2
(2) (1)
( ) { , , , }
( ) = ( ) ( )
1
N q
N N N
f if w w w
f f cf
else
c
−
−
where (1) (1) ˆ
( ) = ( ) ( )
N N N
f W u I u du
−
−
and (2) (2) ˆ
( ) = ( ) ( ) .
N N N
f W u I u du
−
−
The spectral windows (1)
N
W and (2)
N
W are defined by: (1) (1) (1)
( ) = ( )
N N N
W x M W M x and
(2) (2)) (2)
( ) = ( )
N N N
W x M W M x with W is an even nonnegative, continuous function , vanishing for
| |>1
such that
1
1
( ) =1
W u du
−
. The bandwidths (1)
)
N
M and (2)
)
N
M satisfying:
(2)
(1)
N
N
M
c
M
=
( )
lim
i
N N
M
→ = +,
( )
= 0
lim
i
N
N
M
N
→ for 1,2
i = ,
(2)
(1)
= 0
lim
N
N
N
M
M
→+ and such that
(2) (1)
(1) (1)
1 1
( ) = ( ) ,
N N
N N
W M W M
M M
−
.
Th following theorem shows that ( )
N
f is an asymptotically unbiased estimator of [ ( )]
p
for
< <
− and 1 2
{ , , , }
q
w w w
.
Theorem 3.1
Let < <
− , such that 1 2
{ , , , }
q
w w w
. Then, ( )
( ) ( ) = o 1 .
p
N
E f
−
6. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
26
If satisfies the hypothesis ( ) ( ) ( ) ,
x y cste x y
−
− − with < 2 1
k
− , then,
(2 1) (1)
(1) (2 1) (1) 2 1
1 1
= 0
( ) [ ( )] = .
1 1 1
= 0
k
p
N
N
k k
N N
O if
n M
E f
O if
M n M n
−
− −
+
−
+ +
Proof
It is easy to see that:
(1) (1) ˆ
( ) = ( ) ( ) .
N N N N
R
E f M W M u E I u du
−
Let (1)
( ) =
N
M u v
− , we obtain:
1
(1)
1
[ ( )] = ( ) .
p
N N
N
v
E f W v dv
M
−
−
(7)
Since
1
1
( ) =1
W u du
−
and the inequality (3), we get:
1
(1)
1
( ) ( ) ( ) ( ) .
p
p
N N
N
v
E f W v dv
M
−
− − −
As < 1
p
, we obtain
,1 ,2
(1) (1) (1)
( ) ( ) |
p p p
N N N
N N N
v v v
M M M
− − − − + −
We now examine the limit of ,1 (1)
N
N
v
M
−
. From (3) we get:
,1 (1) (1)
= .
N N
N N
v v u
H u du
M M
−
− − −
Let (1)
=
N
v
u y
M
− −
, we obtain:
7. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
27
,1 (1) (1)
(2 1)
(1)
(2 1)
= ( )
= ( ) .
N N
R
N N
j
N
j
j Z N
v v y
H y dy
M M
v y
H y dy
M
+
−
− − +
− +
(8)
Let 2 =
y j s
− . Since | (.) |
N
H
is 2 −periodic function, we get
,1 (1) (1)
2
= ( ) .
N N
j Z
N N
v v s
H s j ds
M M
+
−
− − + +
The function is uniformly continuous on [ , ]
− and since | |
N
H
is a kernel, the right hand
side of the last equality converges to
2
.
j Z
j
+
Let j be an integer such that
2
< <
j
+
− . The definition of implies that | |<| |<
. It is easy to see that
| |< 1
j and then = 0
j . Since (1)
N
H is a kernel, we obtain that ,1 (1)
N
N
v
M
−
converges to
( )
. On the other hand, (1)
,2 (1) (1)
=1
=
q
N N i
i
N N
v v
H w
M M
− − −
Since i
w is different from and from the lemma 2.2, we get
1
2 (1)
2 1
,2 (1)
=1
2 1
2 , where cte =inf sin .
2
k i
q
N
k
N i
i
N
v
w
M
v
n c
M cte
−
−
− −
−
Therefore, ,2 (1) 2 1
1
=
N k
N
v
O
M n
−
−
. Thus, we have
( ) ( ) = (1).
p
N
E f o
−
The rate of convergence:
Assume that the spectral density satisfies the hypothesis H . Let
/
=| ( ( )) |=| [ ( )] [ ( )] |
p
N N
F Bias f E f
− . It follows that
1
1
1
(1) (1)
1
( ) [ ( )] ( ) .
2
p
p
N N
N N
p v v
F W v dv
M M
−
−
−
− + − −
8. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
28
Since N
N
v
M
−
converges to ( )
, in order to study the rate of the convergence for F
we examine the rate of convergence of
1
(1)
1
( ) ( )
N
N
v
W v dv
M
−
− −
. Indeed, from (3), we
obtain (1) (1)
= | | .
N N
N N
v v y
H y dy
M M
−
− − +
Let ( ) (1)
, = ( )
N N
N
v
M
− −
. Putting (1)
=
N
v
t y
M
− − +
and using the condition
H , we get ( ) ( )
(1)
1 (1)
(1)
, .
v
MN
v
N N
N
MN
v t
C H t dt
M
− +
− −
+
We can maximize as follows:
( )
1 1
1 (1)
1 1
1
(1)
1
1
1
1 (1)
1
( ) ( , ) 2 ( ) | |
2 ( )
N
N
v
MN
v N
MN
W v dv C W v v dv
M
C
W v H t t dtdv
− −
− +
− − −
+
The second integral of the right hand side is bounded as follows:
( ) ( )
( )
( )
(1)
| | | |
(1)
(1)
| | | |
(1)
.
v
MN
v
v N N
M
M N
N
N
v
MN
N
H t t dt H t t dt
H t t dt
H t t dt
− +
−
− − −
− −
−
+ +
+
+
(9)
Since | (.) |
N
H is even, the first and the last integrals in the right hand side of (9) are equal. As
N
v
M
converges to zero and < <
, for a large N we have:
9. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
29
( ) ( )
| | | | | | | |
(1) (1)
(1)
2
,
(1)
(2 )
| |
(2 )
sin | |
v v
M M
N N
N N
N
k
'
N
N
H t t dt H t dt
M
B
M
+ + + +
+
+ +
The lemma 2.1 gives:
( )
2 1
| | | |
(1)
(1) 2 1
1
O 0,
=
1
O = 0,
k
v
MN
N
k
N
if
n
H t t dt
if
M n
−
+ +
−
Thus, we obtain the result of the theorem.
Theorem 3.2. Let a real nuber belonging to ] , [
− , and = i
w
. Choose k such that
( )
2
(1)
2 1
= 0
lim
k
N
N k
M
n
→ −
. Then,
i) [ ( )] [ ( )] = (1)
p
N
E f o
−
ii) If satisfiies the hypothesis ( ) ( ) ( ) ,
x y cste x y
−
− − with
1
< < 2
2k
+
, then
( )
( )
( )
2
(1)
2 1
(2)
2
(1)
(2) 2 1
1
0 < 1
[ ( )] [ ( )] =
1
1< 2
k
N
k
p
N
N k
N
k
N
M
O if
n
M
E f
M
O if
M n
−
−
+
−
+
Proof :
The form of estimator gives:
10. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
30
(2)
(2) (1)
(1)
(2)
(1)
( ) ( )
[ ( )] = [ ( )]
1
N
N N p
N
N N
N
N
M
W u W u
M
E f u du
M
M
−
− − −
−
1 1
(1) (1)
1
1 1 2 3
(1)
(1)
[ ( )] =
M M
N N
N
M
M N
N
E f E E E
− +
− +
−
+ + = + +
Put
1
(2) (2) (1)
(1)
1
2 (2)
(1)
(1)
[ ] [ ]
= , = [ ( )] .
1
p
N n N
MN
N
N
MN
N
M W M v W M v
u v E v dv
M
M
−
−
− −
−
Therefore 2 0
E =
for a large N.
1
(2) (1)
(2)
1 (2)
(1)
= [ ( )][ ( )]
1
p
M
N N
N N
N
N
M
E W M u u du
M
M
−
−
− −
−
1
(2) (1)
(1)
(2)
(1)
[ ( )][ ( )]
1
p
M
N N
N N
N
N
M
W M u u du
M
M
−
−
−
−
Put (2)
( ) =
N
M u v
− in the first integral and put (1)
( ) = ,
N
M u w
− in the second integral and for
a large N, we have (1) (2)
( ) >1 ( ) >1.
N N
M et M
+ + As W is null outside of [ 1,1]
− , for
large N, the second integral of 1
E is zero.Therefore,
1
(2)
1 (2) (2)
(1)
(1)
1
= ( ) .
1
p
N
MN
N N
MN
N
v
E W v dv
M M
M
−
−
(10)
(2) (2)
(2) (1)
1 1
3 (2) (2)
(1) (1)
(1) (1)
= [ ( )][ ( )] [ ( )][ ( )]
1 1
p p
N N
N N N N
N N
M M
N N
N N
M M
E W M u u du W M u u du
M M
M M
+ +
− − −
− −
Putting (2)
( ) =
N
M u v
− in the first integral and (1)
( ) =
N
M u w
− in the second integral, we
obtain
(2)
(2)
(1)
(1) 1
(2) (1)
3 (2) (2)
(2) (1)
( ) ( )
(1) (1)
1
= ( ) ( ) .
1 1
p p
N
MN
M N
N
N N
M M
N N
N N
N N
N N
M
M
v w
E W v dv W w dw
M M
M M
M M
−
−
− −
− − −
− −
11. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
31
(2) (1)
For large we have, ( ) < 1 ( ) < 1.
N N
N M and M
− − − −
1
(2)
3 (2) (2)
(1)
(1)
1
= ( ) .
1
p
N
MN
N N
MN
N
v
E W v dv
M M
M
+
−
(11)
It is easy to show that for a large N
1
(2)
(2)
(1)
(1)
1 1
( ) = .
2
1
MN
N
MN
N
W v dv
M
M
−
(12)
1 3
[ ( )] [ ( )] = [ ( )]
p p
N
E f E E
− + − 1 3
1 1
[ ( )] [ ( )]
2 2
p p
E E
− + −
From (10) and (12), for a large N, we have
1
(2)
1 ,1
(2) (2) (2)
2
(1)
(1)
1 1
[ ( )] ( ) ( )
2
1
p
p
N N
MN
N N N
MN
N
v v
E W v dv
M M M
M
− − + − −
−
As 1
p
, we obtain
1
(2)
1 ,1
(2) (2)
(1)
(1)
1 1
[ ( )] ( ) ( )
2
1
p
p
N
MN
N N
MN
N
v
E W v dv
M M
M
− − −
−
+
1
(2) ,2
(2) (2)
(1)
(1)
1
( ) .
1
p
N
MN
N N
MN
N
v
W v dv
M M
M
−
−
On the other hand,
1 1
(2) ,1 ,1
(2) (2)
0
(1)
( ) ( ) ( ) ( )
p p
N N
MN N N
MN
v v
W v dv W v dv
M M
− −
For all belonging to ] , [
− ,1 (2)
N
N
v
M
converges to ( )
, uniformly in
[ 1,1].
v − Therefore,
1
(2) ,1
(2) (2)
(1)
(1)
1
( ) ( )
1
p
N
MN
N N
MN
N
v
W v dv
M M
M
−
−
converge to zero.
12. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
32
Since = ,
i
w ,2 2
(2)
=1 ,
=
(2)
1
1
sin
2
m
N i k
m
N N
m i i m
N
q
a
v
M B v
w w
M
−
2
,
(2)
1
.
1
sin
2
i
k
N
N
a
B v
M
+
For all =
m i ,
( )
2 2
[ 1,1]
(2)
1 1
= .
sup
lim
1
1 sin
sin
2
2
k k
N v
i m
i m
N
v w w
w w
M
→+ −
−
−
Thus, for large N, we get
( )
2 2
=1 =1
, ,
= =
(2) {1,2, , } { }
1 1
.
1
1 sin
inf
sin
2
2
m m
k k
m m
N N
m i m i
i m
i m m q i
N
q q
a a
B B
v w w
w w
M
−
+
−
−
The lemma 2.1 gives 2 2 1
=1 ,
=
(2)
1 1
= .
1
sin
2
m
k k
m N
m i i m
N
q
a
O
B n
v
w w
M
−
−
(13)
For large N, we have ]0,1] (2) (2)
= <
supv
N N
v
M M
. Consequently
2
2 2
, ,
(2)
(2)
1
.
1
sin
2
k
i i
k k
N N
N
N
a a
B B v
v
M
M
As (1) (2) (2)
N N N
v
M M M
, we obtain 2
,
(2)
1
1
sin
2
i
k
N
N
a
B v
M
2
2
,
(1)
.
k
i
k
N
N
a
B
M
Frome the lemma 2.2 , we obtain
13. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
33
( )
2
(1)
2 2 1
,
(2)
1
= .
1
sin
2
k
N
i
k k
N
N
M
a
O
B n
v
M
−
+ −
(14)
Therefore, we get
( )
2
(1)
1
(2) ,2
(2) (2) 2 1
(2 1)
(1)
(1)
2 1
( ) =
1
p
p
p k
N
N p k
M k
N
N N
MN
N
M
v
W v dv O
M M n
n
M
−
−
+
−
(15)
Choosing (1)
N
M such that
( )
2
(1)
2 1
k
N
k
M
n
−
converges to 0 . For example (1)
= b
N
M n with
1
0 < <1
2
b
k
− . Thus, [ ( )] [ ( )] = 0.
lim
p
N
N
E f
→+
−
Theorem 4.2 Let < <
− such that ( ) > 0
. Then,
( ))
N
var f converges to zero.
If (1)
= c
N
M n with 2 2
1 1
< <
2 2
c
k
, then (1 2 )
1
( )) = O .
N c
var f
n
−
Proof
Fisrt suppose that 1 2
{ , , , }
q
w w w
. It is clear that the variance of N
f can be written as
follows:
(1) (1)
1
2
ˆ ˆ
[ ( )] = ( ) ( )c ( ), ( ) .
' ' '
N N N N N
R
var f W u W u ov I u I u dudu
− −
Since W is zero for | |>1
, for large N , we have
1
1 1
1 1 1 1
1
ˆ ˆ
[ ( )] = , ( ) ( ) .
'
' '
N N N
N N
x x
var f cov I I W x W x dx dx
M M
−
− −
Define two subsets of the 2
[ 1,1]
− by:
•
2
1 1 1 1 1
= ( , ) [ 1,1] ; | |> ,
' '
N
L x x x x
− −
•
2
2 1 1 1 1
= ( , ) [ 1,1] ; | | ,
' '
N
L x x x x
− −
14. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
34
where N
is a nonnegative real, converging to 0 . We split the integral into an integral over the
subregion 2
L and an integral over 1
L : 1 2
2 1
var[ ( )] = = .
N L L
f J J
+ +
Cauchy-Schwartz inequality and theorem 3.1, give
1 1 1 1 1
| |
1 1
( ) ( ) .
' '
'
x x N
J C W x W x dx dx
−
where C is constant. Thus, we obtain
1 = ( )
N
J O (16)
It remains to show that 2
J converges to zero. For simplicity, we define
1 1
1 2
(1) (1)
= ; = ,
'
N N
x x
M M
− − and 1 1
(1) (1)
ˆ ˆ
( ) = c , .
'
N N
N N
x x
C ov I I
M M
− −
We first show that ( )
C converges to zero uniformly in 1 1
, [ 1,1]
'
x x − . Indeed, from lemma
2.3, we have
( )
( ) | | ( )
1 /
, 1
e
ˆ ˆ
( ) ( ) = [ ] .
| |
iuI v C u v
N N
p
N N p p
R e e
EI v I v F C du
u
−
− −
+
−
−
−
Thus, the expression of the covariance becomes
( )
2 2
2
, 2
=1
2 1 2
1
=1
1 2
( ) = cos ( )
exp | | ( )
| |
p
p k N k
R
k
k N k p
k
C F C E u I
du du
C u
u u
−
−
+
− −
Using the following equality: 2cos cos = cos( ) cos( )
x y x y x y
+ + − , we have
( )
1
2 2
=1
=1
1
2 1
=1
1
cos ( ) = exp ( ) ( ) ( )
2
1
exp ( ) ( 1) ( ) ( ) .
2
k N k k N k
k
k
k
k N k
k
E u I C u H v d v
C u H v d v
+
− −
+ − − −
By substituting the expression for ( )
C and changing the variable 2
u to 2
( )
u
− in the second
term, we obtain
( )
2
2 1 2
, 2 1
1 2
( ) = ,
| |
p
'
K K
p p
R
du du
C F C e e
u u
−
− − −
+
−
(17)
15. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
35
where
1
2
=1
= ( ) ( ) ( )
k N k
k
R
K C u H v d v
−
and
2
=1
= | | ( ) ( )
'
k N k
k R
K C u H v v dv
−
Since , > 0
'
K K , exp{ }
'
K K ' ' '
e e K K K K K
− −
− − − − , we obtain:
2
1 2
2 | | ( ; ),
'
N
K K C u Q
− where
2 2
1 2 1 2
( ; ) = | ( ) | | ( ) | ( )
N N N
Q H u H u d u
−
− −
Now, let us show that 1 2
( ; )
N
Q converges to zero. Indeed, since is bounded on [ , ]
− ,
we have
2
1 2 1 2
2
1 2
=1
( ; ) sup( ) | ( ) ( ) |
( ) ( )
N N N
q
i N i N i
i
Q H u H u du
c H w H w
−
− −
+ − −
(18)
From the definition of N
H , we can write
( ) ( )
( )
( )
( )
( )
1 2
2
1 2
,
1 2
sin sin
1 2 2
| | = .
1 1
sin sin
2 2
k k
N N '
N
n n
v v
H v H v dv dv
B
v v
− −
− −
− −
− −
a) First step:
We show that the denominators of the first and second terms under the last integral do not vanish
for the same v, so we suppose thatv exists, belonging to [ , ]
− and , '
z z Z
such as:
1 2
= 2 = 2 '
v z and v z
− − . Since 1 2
=
, then z and z are different.
Therefore, ( )
1 2
=
2
'
z z
− − . Hence, 1 2
1
| |= .
'
z z
w
− − As 1 2 =0
limN
→ − ,
consequently, for a large N we get:
1
<
2
'
z z
− .
Thus, we obtain a contradiction with the fact that z and '
z are different integers.
b) Second step:
Asuume there exist q points, 1 2
, , , [ , ]
q
V V V − such as:
16. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
36
for 1
=1,2, , 2
j
j q V Z
− then 1 j
V
Z
w w
− , and we assume that there exist '
q points
1 2
, , , [ , ]
' ' '
'
q
V V V − such that for =1,2, , '
i q 2
'
i
V
Z
w w
− . Showing that, | |=
j
V ,
| |=
i
V for 1 j q
and 1 '
i q
. Indeed, 1< < 0
w
−
− and 0 < <1
w
+
because
> 2
w . Hence Z
w
−
and Z
w
+
. For a large N , we get that
1
< <1 ,
E E
w w w w
− −
− +
where E
x is the integer part of x . Hence,
1
Z
w w
− . In the same manner, we show that 1
Z
w w
+ . Similarly, it can be shown that
2
Z
w
+
. Thus, | |=
j
V and | |=
i
V .
c) Third step:
We classify j
V and '
i
V by increasing order:
1 2
< < < < <
j j j '
q q
V V V
+
− , and we write
the integral in the following manner:
( )
( )
( )
( )
1 2 1
1 2, 3, 4
=1 =1
1 2 1
sin sin
2 2
=
1 1
sin sin
2 2
k k
' '
q q q q
i i
i i
n n
v v
dv I I I I
v v
+ + −
−
− −
+ + +
− −
where
17. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
37
( )
( )
( )
( )
( )
( )
( )
( )
1 2
( )
1
1
1
1 2
1 2
( )
2, ( )
1 2
3, ( )
sin sin
2 2
=
1 1
sin sin
2 2
sin sin
2 2
=
1 1
sin sin
2 2
=
k k
V N
j
k k
V N
ji
i V N
ji
Vj
i V N
ji
n n
v v
I dv
v v
n n
v v
I dv
v v
I
−
−
+
−
+
− −
− −
− −
− −
( )
( )
( )
( )
( )
( )
( )
( )
1 2
( )
1
1 2
1 2
4 ( )
1 1 2
sin sin
2 2
1 1
sin sin
2 2
sin sin
2 2
=
1 1
sin sin
2 2
k k
N
i
k k
V N
j '
q q
n n
v v
dv
v v
n n
v v
I dv
v v
−
+
+
+
− −
− −
− −
− −
where ( )
N
is a nonnegative real number converging to zero and satisfying:
1 1 2 2
< ( ) < ( ) < ( ) < ( ) < < ( ) < ( ) < ,
j j j j j j
' '
q q q q
V N V N V N V N V N V N
+ +
− − + − + − + a
nd 1 2
( ) < .
2
N
−
Showing that the first integral converges to zero. For a large N , it easy to see that 1 <
.
Without loss of generality, we assume that 1 = 2
j i
i
V k
− for all i , with i
k Z
. The fact that
there is no v between − and
1
( )
j
V N
− on which the denominators are vanishing, gives
1
1
1 2 1 2
( ) 1
.
( ) ( ( ))
( ) ( )
inf sin , sin inf sin , sin
2 2 2 2
j
k k
k k
j
V N
I
V N
N
− +
+ − + +
Substituting
1
j
V in the last inequality, we get
2 1 2 1
( ( )) | ( ) |
sin = sin
2 2
k k
j
V N N
− + − +
. For a large N ,we obtain
18. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
38
2 1
| ( ) | ( ) ( )
<
2 2 2
N N N
− +
+ − .
On the other hand, two cases are possible:
1. 2 1 > 0
− , then we have 2 1 2 1
| ( ) |= ( ) > ( )
N N N
− + − +
2. 2 1 < 0
− , since 2 1
| |> 2 ( )
N
− , we have
2 1 1 2
| ( ) |= ( ) > ( )
N N N
− + − − .
Therefore, 2 1
| ( ) |
( ) ( )
< <
2 2 2
N
N N
− +
− . For a large N , we have
1
( ) < 2 ( )
N
− + and 2
( ) < 2 ( )
N
− + . Then,
1
( )
( ) ( )
< <
2 2 2
N N
+
− and 2
( )
( ) ( )
< < .
2 2 2
N N
+
−
Therefore,
1
1 2
( )
.
( )
sin
2
j
k
V N
I
N
− +
For the integral 2,i
I , we bound the first fraction under integral by k
n
:
( )
( )
2, ( )
2
1
1
sin
2
V N
j
k i
i k
V N
ji
I n dv
v
+
−
−
. By substituting for ji
V in the last
inequality and putting
2
=
k
v u
− , we get
( )
( )
1
2, ( )
1
2
1
1
sin
2
N
k
i k
N
I n du
u
+
−
−
. Since 1
| |< ( )
u N
− , it is easy to see that
:
2 1
2 2 1 1 2 1
| |
| | | | | | | | ( ) >
2
u u N
−
− − − − − −
Since ( )
N
converges to zero, for a large N we have 2 1
2
( ) < | |
2
N
− −
, therefore
2 1 2 2 1
| | | | | | ( )
0 < < < < .
4 2 2
u N
− − − +
Consequently:
19. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
39
2,
1 2 1 2
2 ( )
,
| | | | ( )
inf sin ; sin
4 2
k
i
N n
I
N
− − +
where 2 1
2 1
| |=
N
x x
M
−
− . Then, for a large N , we have 2 1
3
< 2 .
2
− Therefore,
2 1 2 1 2 1
| | | | ( ) | |
< < .
4 2 4
N
− − + −
− Thus, we bound the integral as follows:
2,
2 1
2 ( )
| |)
sin
4
k
i k
N n
I
−
Since there is no v between ( )
ji
V N
+ and
1
( )
ji
V N
+
− on which the denominators are
vanishing, we get:
1
3,
2 ( )
,
j j
i i
i
V V N
I
A B
+
− −
where
1 1 1
2 2 1
( ) ( )
= inf sin , sin
2 2
( ) ( )
= inf sin , sin
2 2
k k
j j
i i
k k
j j
i i
V N V N
A
V N V N
B
+
+
− − − +
− − − +
It follows from the hypothesis on ( )
N
that
1 2 1 2 1 2
| | | ( ) | | |
( ) | ( ) | | ( ) | ( )
< < < < .
2 2 2 2 2 2 2
N
N N N N
− − − −
− + −
The definition of ji
V gives
20. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
40
2
1 1
< .
( ) ( )
sin
sin 2
2
k k
ji
V N N
− −
Similarly we bound the other terms:
1
3, 2
2 ( )
.
( )
sin
2
j j
i i
i k
V V N
I
N
+
− −
We can show by same majoration that :
1
4
2 ( )
,
j j
i i
V V N
I
E F
+
− −
where
1
1
2
2
( )
= inf sin , sin
2 2
( )
= inf sin , sin
2 2
k
k
jq q
k
k
jq q
V N
E
V N
F
+
+
− −
−
− −
−
Since ( )
N
converges to zero, for a large N , we have 1
2
( ) < | |
2
N
− −
, and
1 2
2
( ) < | |
2
N
− −
. It follows that: 4 2
( )
.
( )
sin
2
jq q
k
V N
I
N
+
− −
We recapitulate, from the
previous increases, we obtain
( )
( )
( )
( )
1 2
1
2
=1 2 1
1 2
sin sin ( ) 2 ( )
2 2
1 1 ( ) | |
sin sin sin sin
2 2 2 4
k k
k
q q
j
k k
i
n n
v v V N n N
dv
N
v v
+
−
− −
+ −
+
−
− −
1
1
2 2
=1
( )
( )
( ) ( )
sin sin
2 2
q q
j
j j q q
i i
k k
i
V N
V V N
N N
+ −
+
+
− −
− −
+ +
After simplification, we have
21. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
41
( )
( )
( )
( )
1 2
2
2 1
1 2
sin sin
2 2( 1) ( ) 2 ( )( )
2 2
.
1 1 ( ) | |
sin sin sin sin
2 2 2 4
k k
k
k k
n n
v v
q q N n N q q
dv
N
v v
−
− −
− + + +
+
−
− −
Using the following inequality
| |
sin
2
x x
, we get
( )
( )
( )
( ) ( )
2
1 2
2
2 1
1 2
sin sin
2 2 ( )( )(2 )
2 2
.
1 1 ( ) | |
sin sin
2 2
k k
k k k
k k
N
n n
v v
n N q q
dv
N x x
v v
M
−
− −
+
+
−
− −
The lemma 2.1 gives
( )
( )
( )
( )
( )
1 2
,
1 2
2 2
2
2 1
1
sin sin
1 2 2
1 1
sin sin
2 2
1 2 2 ( )( )(2 )
2 ( )
k k
N
k k k
k k
k
k N
N
n n
v v
dv
B
v v
N q q
n N
n
M
−
−
−
− −
− −
+
+
.
(19)
In order to obtain the convergence of the last expression to zero, we choose
( ) = , > 0
N n
−
, such as
2
2 1
1
1
= 0 = 0.
lim lim
k
k
k
n n
k N
N
n
and
n
n
M
−
→ →
+ −
(20)
Thus, from (18) N
Q converges to zero. On the other hand,
2
2 | | 1 2
, 1
1 2
( ) | | ,
| |
p
K K K
p p
du du
C F C K K e
u u
−
− − −
+
− −
−
where ( )
2
1 2
=1
| | | | ( ; )
k N k N
k
K K K C u Q
− − − −
.
We denote by: ( )
( , ) 1 2
= ( ; )
N k N k N
Q
− . It follows from (18) and (20) that ( , )
N k
converges to ( )
. Hence,
22. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
42
2 2
2
, 1 2 ( , )
0 1
=1 2
( ) 2 ( ; )4 exp ( ) | | .
( )
p
k
p a N k N k
p
k
k
du
C F C C Q C u
u
−
−
+ −
−
Putting ( )
1
, = ,
k N k
u v
we obtain
2
| |
2
2 1 2
, 1
1
2 2
( ,1) ( ,2)
( ; )
( ) 2 .
| |
C v
p
N
p p
p
N N
Q e
C F C C dv
v
−
−
−
−
− + −
(20)
Since ( ) > 0
, ( )
C converges uniformly in 1 1
, [ 1,1]
x x − to zero. From (20), we obtain
2 2 (1 ) 1
1
1 1
= .
k
k
k N
N
J O
n
n
M
− −
+ −
+
Thus, [ ( )]
N
var f converges to zero and then, ( )
N
f
is an asymptotically unbiased and consistent estimator.
4. CONCLUSIONS
This paper gives an estimate of the spectral density of a mixed continuous-time stable process
from observations at discrete instants. To avoid the phenomenon of aliasing we have assumed
that the spectral density is a compact support. The applications of these processes are found in
various fields. For example:
- The study of soil cracking where the observed signal is the resistance of the soil. This signal
encounters random jumps due to the encounter of certain stones in the ground. The spectral
measurement will therefore be composed of two parts, one continuous and the other discrete.
This last corresponds to the resistance jumps encountered during the measurement.
- The growth of fruits on a tree can be seen as a continuous distribution, and when there is a fall
of a fruit, the other fruits remaining on the tree absorb more energy and their growth will have a
jump in value.
As perspective of this work is to find the optimum smoothing parameters to have a better rate of
convergence. For this purpose, the cross-validation method will be the most appropriate tool.
We intend to complete this work by studying the case where the process is observed with random
errors. For this, we will use the deconvolution methods known for their efficiency in the presence
of random errors.
We think to give an estimator of the mode of the spectral density representing the frequency
where the spectral density reaches the maximum of energy.
23. Signal & Image Processing: An International Journal (SIPIJ) Vol.12, No.5, October 2021
43
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AUTHORS
Rachid Sabre received the PhD degree in statistics from the University of Rouen, France, in 1993 and
Habilitation to direct research (HdR) from the University of Burgundy, Dijon, France, in 2003. He joined
Agrosup Dijon, France, in 1995, where he is an Associate Professor. From 1998 through 2010, he served as
a member of Institut de Mathématiques de Bourgogne, France. He was a member of the Scientific Council
AgroSup Dijon from 2009 to 2013. From 2012 to 2019, he has been a member of Laboratoire Electronic,
Informatique, and Image (LE2I), France. Since 2019 he has been a member of Laboratory Biogeosciences
UMR CNRS University Burgundy. He is author/co-author of numerous papers in scientific and technical
journals and conference proceedings. His research interests lie in areas of statistical process and spectral
analysis for signal and image processing.