Non-parametric system identification with Gaussian processes for underwater vehicles is explored in this research with the purpose of modelling autonomous underwater vehicle(AUV) dynamics with a low amount of data. Mu...Non-parametric system identification with Gaussian processes for underwater vehicles is explored in this research with the purpose of modelling autonomous underwater vehicle(AUV) dynamics with a low amount of data. Multi-output Gaussian processes and their aptitude for modelling the dynamic system of an underactuated AUV without losing the relationships between tied outputs are used. The simulation of a first-principle model of a Remus 100 AUV is employed to capture data for the training and validation of the multi-output Gaussian processes. The metric and required procedure to carry out multi-output Gaussian processes for AUV with 6 degrees of freedom(DoF) is also shown in this paper. Multi-output Gaussian processes compared with the popular technique of recurrent neural network show that multi-output Gaussian processes manage to surpass RNN for non-parametric dynamic system identification in underwater vehicles with highly coupled DoF with the added benefit of providing the measurement of confidence.展开更多
A novel model named Multi-scale Gaussian Processes (MGP) is proposed. Motivated by the ideas of multi-scale representations in the wavelet theory, in the new model, a Gaussian process is represented at a scale by a li...A novel model named Multi-scale Gaussian Processes (MGP) is proposed. Motivated by the ideas of multi-scale representations in the wavelet theory, in the new model, a Gaussian process is represented at a scale by a linear basis that is composed of a scale function and its different translations. Finally the distribution of the targets of the given samples can be obtained at different scales. Compared with the standard Gaussian Processes (GP) model, the MGP model can control its complexity conveniently just by adjusting the scale pa-rameter. So it can trade-off the generalization ability and the empirical risk rapidly. Experiments verify the fea-sibility of the MGP model, and exhibit that its performance is superior to the GP model if appropriate scales are chosen.展开更多
In order to meet the demand of online optimal running, a novel soft sensor modeling approach based on Gaussian processes was proposed. The approach is moderately simple to implement and use without loss of performance...In order to meet the demand of online optimal running, a novel soft sensor modeling approach based on Gaussian processes was proposed. The approach is moderately simple to implement and use without loss of performance. It is trained by optimizing the hyperparameters using the scaled conjugate gradient algorithm with the squared exponential covariance function employed. Experimental simulations show that the soft sensor modeling approach has the advantage via a real-world example in a refinery. Meanwhile, the method opens new possibilities for application of kernel methods to potential fields.展开更多
Data-driven paradigms are well-known and salient demands of future wireless communication. Empowered by big data and machine learning techniques,next-generation data-driven communication systems will be intelligent wi...Data-driven paradigms are well-known and salient demands of future wireless communication. Empowered by big data and machine learning techniques,next-generation data-driven communication systems will be intelligent with unique characteristics of expressiveness, scalability, interpretability, and uncertainty awareness, which can confidently involve diversified latent demands and personalized services in the foreseeable future. In this paper, we review a promising family of nonparametric Bayesian machine learning models,i.e., Gaussian processes(GPs), and their applications in wireless communication. Since GP models demonstrate outstanding expressive and interpretable learning ability with uncertainty, they are particularly suitable for wireless communication. Moreover, they provide a natural framework for collaborating data and empirical models(DEM). Specifically, we first envision three-level motivations of data-driven wireless communication using GP models. Then, we present the background of the GPs in terms of covariance structure and model inference. The expressiveness of the GP model using various interpretable kernels, including stationary, non-stationary, deep and multi-task kernels,is showcased. Furthermore, we review the distributed GP models with promising scalability, which is suitable for applications in wireless networks with a large number of distributed edge devices. Finally, we list representative solutions and promising techniques that adopt GP models in various wireless communication applications.展开更多
Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the...Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) > x) is considered, as x → ∞.展开更多
Let X={X(t),t∈R+} be a centered space anisotropic Gaussian process values in R^(d) with non-stationary increments,whose components are independent but may not be identically distributed.Under certain conditions,then ...Let X={X(t),t∈R+} be a centered space anisotropic Gaussian process values in R^(d) with non-stationary increments,whose components are independent but may not be identically distributed.Under certain conditions,then almost surely c_(1)≤ϕ-m(X([0,1]))≤c_(2),where ϕ denotes the exact Hausdorff measure associated with function ϕ(s)=s1/α_(k)+Σ_(i=1)k(1-αi/αk)log log 1/s for some 1≤k≤d,(α_(1),…,α_(d))∈(0,1]^(d).We also obtain the exact Hausdorff measure of the graph of X on[0,1].展开更多
Gaussian process(GP)is a stochastic process that has been successfully applied in finance,black-box modeling of biosystems,machine learning,geostatistics,multitask learning or robotics and reinforcement learning.Effec...Gaussian process(GP)is a stochastic process that has been successfully applied in finance,black-box modeling of biosystems,machine learning,geostatistics,multitask learning or robotics and reinforcement learning.Effectively estimating the spectral density function(SDF)and degree of memory(DOM)of a long-memory stationary GP(LMSGP)is needed to get accurate information about the process.The practice demonstrated that the periodogram estimator(PE)and lag window estimator(LWE)that are the extremely used estimators of the SDF and DOM have some drawbacks,especially for LMSGPs.The behaviors of the PEs and LWEs are soundly investigated numerically;however,the theoretical justifications are limited and thus the challenge to improve them is still daunting.This paper gives a closer look at the theoretical justifications of the efficiency of the LWEs that provides new sufficient conditions(NSCs)for improving the LWEs of the SDF and DOM for LMSGPs.The precision,the convergence rate of the bias and variance,and the asymptotic distributions of the LWEs under the NSCs are studied.A comparison study among the LWEs under the NSCs,the LWEs without the NSCs and the PEs is given to investigate the significance of the NSCs.The main theoretical and simulation results show that:the LWEs under theNSCs are asymptotically unbiased and consistent and better than the LWEswithout the NSCs and the PEs,and the asymptotic distributions of the LWEs under the NSCs are chi-square for SDF and normal for DOM.展开更多
Reduced-order models offer a cost-effective and accurate approach to analyzing high-dimensional combustion problems.These surrogate models are built in a data-driven manner by combining computational fluid dynamics si...Reduced-order models offer a cost-effective and accurate approach to analyzing high-dimensional combustion problems.These surrogate models are built in a data-driven manner by combining computational fluid dynamics simulations with Proper Orthogonal Decomposition(POD)for dimensionality reduction and Gaussian Process Regression(GPR)for nonlinear regression.However,these models can yield physically inconsistent results,such as negative mass fractions.As a linear decomposition method,POD complicates the enforcement of constraints in the reduced space,while GPR lacks inherent provisions to ensure physical consistency.To address these challenges,this study proposes a novel constrained reduced-order model framework that enforces physical consistency in predictions.Dimensionality reduction is achieved by downsampling the dataset through low-cost Singular Value Decomposition(lcSVD)using optimal sensor placement,ensuring that the retained data points preserve physical information in the reduced space.We integrate finite-support parametric distribution functions,such as truncated Gaussian and beta distribution scaled to the interval[a,b],into the GPR framework.These bounded likelihood functions explicitly model the observational noise in the bounded space and use variational inference to approximate analytically intractable posterior distributions,producing GP estimations that satisfy physical constraints by construction.We validate the proposed methods using a synthetic dataset and a benchmark case of one-dimensional laminar NH3/H2 flames.The results show that the thermo-chemical state predictions comply with physical constraints while maintaining the high accuracy of unconstrained reduced-order models.展开更多
In this paper,we are concerned with the asymptotic behavior,as u→∞,of P{sup_t∈|0,T|X_u(t)>u},where X_u(t),t∈|0,T|,u>0 is a family of centered Gaussian processes with continuous trajectories.A key application...In this paper,we are concerned with the asymptotic behavior,as u→∞,of P{sup_t∈|0,T|X_u(t)>u},where X_u(t),t∈|0,T|,u>0 is a family of centered Gaussian processes with continuous trajectories.A key application of our findings concerns P{sup_t∈|0,T|(X(t)+g(t))>u},as u→∞,for X a centered Gaussian process and g some measurable trend function.Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.展开更多
In this paper,we establish some limit theorems on the increments of an l^p-valued multi- parameter Gaussian process under weaker conditions than those of Cs(?)rg(?)-Shao theorems published in Ann.Probab.(1993).
The dynamic soft sensor based on a single Gaussian process regression(GPR) model has been developed in fermentation processes.However,limitations of single regression models,for multiphase/multimode fermentation proce...The dynamic soft sensor based on a single Gaussian process regression(GPR) model has been developed in fermentation processes.However,limitations of single regression models,for multiphase/multimode fermentation processes,may result in large prediction errors and complexity of the soft sensor.Therefore,a dynamic soft sensor based on Gaussian mixture regression(GMR) was proposed to overcome the problems.Two structure parameters,the number of Gaussian components and the order of the model,are crucial to the soft sensor model.To achieve a simple and effective soft sensor,an iterative strategy was proposed to optimize the two structure parameters synchronously.For the aim of comparisons,the proposed dynamic GMR soft sensor and the existing dynamic GPR soft sensor were both investigated to estimate biomass concentration in a Penicillin simulation process and an industrial Erythromycin fermentation process.Results show that the proposed dynamic GMR soft sensor has higher prediction accuracy and is more suitable for dynamic multiphase/multimode fermentation processes.展开更多
Let {X（t）, t ≥ 0} be a standard（zero-mean, unit-variance） stationary Gaussian process with correlation function r（·） and continuous sample paths. In this paper, we consider the maxima M（T） = max{X（t）, ...Let {X（t）, t ≥ 0} be a standard（zero-mean, unit-variance） stationary Gaussian process with correlation function r（·） and continuous sample paths. In this paper, we consider the maxima M（T） = max{X（t）, t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M（TT） exists under some additional conditions related to the correlation function r（·）.展开更多
Low pressure chemical vapor deposition(LPCVD) is one of the most important processes during semiconductor manufacturing.However,the spatial distribution of internal temperature and extremely few samples makes it hard ...Low pressure chemical vapor deposition(LPCVD) is one of the most important processes during semiconductor manufacturing.However,the spatial distribution of internal temperature and extremely few samples makes it hard to build a good-quality model of this batch process.Besides,due to the properties of this process,the reliability of the model must be taken into consideration when optimizing the MVs.In this work,an optimal design strategy based on the self-learning Gaussian process model(GPM) is proposed to control this kind of spatial batch process.The GPM is utilized as the internal model to predict the thicknesses of thin films on all spatial-distributed wafers using the limited data.Unlike the conventional model based design,the uncertainties of predictions provided by GPM are taken into consideration to guide the optimal design of manipulated variables so that the designing can be more prudent Besides,the GPM is also actively enhanced using as little data as possible based on the predictive uncertainties.The effectiveness of the proposed strategy is successfully demonstrated in an LPCVD process.展开更多
We study the moduli of continuity of a class of N-parameter Gaussian processes and get some results on＇the packing dimension of the set of their fast points.
We study the asymptotic relation among the maximum of continuous weakly and strongly dependent stationary Gaussian process, the maximum of this process sampled at discrete time points, and the partial sum of this proc...We study the asymptotic relation among the maximum of continuous weakly and strongly dependent stationary Gaussian process, the maximum of this process sampled at discrete time points, and the partial sum of this process. It is shown that these two extreme values and the sum are asymptotically independent if the grid of the discrete time points is sufficiently sparse and the Gaussian process is weakly dependent, and asymptotically dependent if the grid points are Pickands grids or dense grids.展开更多
By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.
Let{X k(t),t≥0},k=1,2,…,be a sequence of independent Gaussian processes withσk 2(h)=E(X k(t+h)-X k(t))2.Putσ(p,h)=(∑∞k=1σk p(h))1/p,p≥1.The author establishes the large increment results for boundedσ(p,h).
In this paper, we get three local continuity moduli theorems for the almost surely continuous, stationary increments Gaussian process {Y(t), t0}, the partial sum processes X(t,N)= (t) of infinite dimensional Ornstein-...In this paper, we get three local continuity moduli theorems for the almost surely continuous, stationary increments Gaussian process {Y(t), t0}, the partial sum processes X(t,N)= (t) of infinite dimensional Ornstein-Uhlenbeck processes {Xk(t), t0}, and lp-valued Gaussian processes {Y(t), t0}={Xk(t), t0}, separately. The first theorem implies the local continuity modulus theorem for the series X(t)=, Xk(t) of infinite dimensional OrnsteinUhlenbeck processes which has been obtained in [3].展开更多
文摘Non-parametric system identification with Gaussian processes for underwater vehicles is explored in this research with the purpose of modelling autonomous underwater vehicle(AUV) dynamics with a low amount of data. Multi-output Gaussian processes and their aptitude for modelling the dynamic system of an underactuated AUV without losing the relationships between tied outputs are used. The simulation of a first-principle model of a Remus 100 AUV is employed to capture data for the training and validation of the multi-output Gaussian processes. The metric and required procedure to carry out multi-output Gaussian processes for AUV with 6 degrees of freedom(DoF) is also shown in this paper. Multi-output Gaussian processes compared with the popular technique of recurrent neural network show that multi-output Gaussian processes manage to surpass RNN for non-parametric dynamic system identification in underwater vehicles with highly coupled DoF with the added benefit of providing the measurement of confidence.
文摘A novel model named Multi-scale Gaussian Processes (MGP) is proposed. Motivated by the ideas of multi-scale representations in the wavelet theory, in the new model, a Gaussian process is represented at a scale by a linear basis that is composed of a scale function and its different translations. Finally the distribution of the targets of the given samples can be obtained at different scales. Compared with the standard Gaussian Processes (GP) model, the MGP model can control its complexity conveniently just by adjusting the scale pa-rameter. So it can trade-off the generalization ability and the empirical risk rapidly. Experiments verify the fea-sibility of the MGP model, and exhibit that its performance is superior to the GP model if appropriate scales are chosen.
文摘In order to meet the demand of online optimal running, a novel soft sensor modeling approach based on Gaussian processes was proposed. The approach is moderately simple to implement and use without loss of performance. It is trained by optimizing the hyperparameters using the scaled conjugate gradient algorithm with the squared exponential covariance function employed. Experimental simulations show that the soft sensor modeling approach has the advantage via a real-world example in a refinery. Meanwhile, the method opens new possibilities for application of kernel methods to potential fields.
基金supported in part by the National Key R&D Program of China with grant No. 2018YFB1800800by the Basic Research Project No. HZQB-KCZYZ-2021067 of Hetao Shenzhen-HK S&T Cooperation Zone+3 种基金by Natural Science Foundation of China (NSFC) with grants No. 92067202 and No. 62106212by Shenzhen Outstanding Talents Training Fund 202002by Guangdong Research Projects No. 2017ZT07X152 and No. 2019CX01X104by China Postdoctoral Science Foundation with grant No. 2020M671899。
文摘Data-driven paradigms are well-known and salient demands of future wireless communication. Empowered by big data and machine learning techniques,next-generation data-driven communication systems will be intelligent with unique characteristics of expressiveness, scalability, interpretability, and uncertainty awareness, which can confidently involve diversified latent demands and personalized services in the foreseeable future. In this paper, we review a promising family of nonparametric Bayesian machine learning models,i.e., Gaussian processes(GPs), and their applications in wireless communication. Since GP models demonstrate outstanding expressive and interpretable learning ability with uncertainty, they are particularly suitable for wireless communication. Moreover, they provide a natural framework for collaborating data and empirical models(DEM). Specifically, we first envision three-level motivations of data-driven wireless communication using GP models. Then, we present the background of the GPs in terms of covariance structure and model inference. The expressiveness of the GP model using various interpretable kernels, including stationary, non-stationary, deep and multi-task kernels,is showcased. Furthermore, we review the distributed GP models with promising scalability, which is suitable for applications in wireless networks with a large number of distributed edge devices. Finally, we list representative solutions and promising techniques that adopt GP models in various wireless communication applications.
基金Supported by the Scientific Research Fund of Sichuan Provincial Education Department(12ZB082)the Scientific research cultivation project of Sichuan University of Science&Engineering(2013PY07)+1 种基金the Scientific Research Fund of Shanghai University of Finance and Economics(2017110080)the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing(2018QZJ01)
文摘Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) > x) is considered, as x → ∞.
基金partially supported by National Natural Science Foundation of China(12371150,12071003)Natural Science Foundation of Anhui Polytechnic University(2022AH050955,2022YQQ025,Xjky2022163)Zhejiang Province Philosophy and Social Science Planning Routine Subject(24NDJC131YB).
文摘Let X={X(t),t∈R+} be a centered space anisotropic Gaussian process values in R^(d) with non-stationary increments,whose components are independent but may not be identically distributed.Under certain conditions,then almost surely c_(1)≤ϕ-m(X([0,1]))≤c_(2),where ϕ denotes the exact Hausdorff measure associated with function ϕ(s)=s1/α_(k)+Σ_(i=1)k(1-αi/αk)log log 1/s for some 1≤k≤d,(α_(1),…,α_(d))∈(0,1]^(d).We also obtain the exact Hausdorff measure of the graph of X on[0,1].
基金supported by the UIC Research Grants with No.of(R201810,R201912 and R202010)the Curriculum Development and Teaching Enhancement with No.of(UICR0400046-21CTL)+1 种基金the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science,BNU-HKBU United International College with No.of(2022B1212010006)in part by Guangdong Higher Education Upgrading Plan(2021-2025)with No.of(UIC R0400001-22).
文摘Gaussian process(GP)is a stochastic process that has been successfully applied in finance,black-box modeling of biosystems,machine learning,geostatistics,multitask learning or robotics and reinforcement learning.Effectively estimating the spectral density function(SDF)and degree of memory(DOM)of a long-memory stationary GP(LMSGP)is needed to get accurate information about the process.The practice demonstrated that the periodogram estimator(PE)and lag window estimator(LWE)that are the extremely used estimators of the SDF and DOM have some drawbacks,especially for LMSGPs.The behaviors of the PEs and LWEs are soundly investigated numerically;however,the theoretical justifications are limited and thus the challenge to improve them is still daunting.This paper gives a closer look at the theoretical justifications of the efficiency of the LWEs that provides new sufficient conditions(NSCs)for improving the LWEs of the SDF and DOM for LMSGPs.The precision,the convergence rate of the bias and variance,and the asymptotic distributions of the LWEs under the NSCs are studied.A comparison study among the LWEs under the NSCs,the LWEs without the NSCs and the PEs is given to investigate the significance of the NSCs.The main theoretical and simulation results show that:the LWEs under theNSCs are asymptotically unbiased and consistent and better than the LWEswithout the NSCs and the PEs,and the asymptotic distributions of the LWEs under the NSCs are chi-square for SDF and normal for DOM.
基金support of the HECO2-Axe-1 project,funded by the Walloon Region under the REPowerEU initiative,Bel-giumfinancial support from the ERC proof of concept grant INVENT(grant agreement No:101123406).
文摘Reduced-order models offer a cost-effective and accurate approach to analyzing high-dimensional combustion problems.These surrogate models are built in a data-driven manner by combining computational fluid dynamics simulations with Proper Orthogonal Decomposition(POD)for dimensionality reduction and Gaussian Process Regression(GPR)for nonlinear regression.However,these models can yield physically inconsistent results,such as negative mass fractions.As a linear decomposition method,POD complicates the enforcement of constraints in the reduced space,while GPR lacks inherent provisions to ensure physical consistency.To address these challenges,this study proposes a novel constrained reduced-order model framework that enforces physical consistency in predictions.Dimensionality reduction is achieved by downsampling the dataset through low-cost Singular Value Decomposition(lcSVD)using optimal sensor placement,ensuring that the retained data points preserve physical information in the reduced space.We integrate finite-support parametric distribution functions,such as truncated Gaussian and beta distribution scaled to the interval[a,b],into the GPR framework.These bounded likelihood functions explicitly model the observational noise in the bounded space and use variational inference to approximate analytically intractable posterior distributions,producing GP estimations that satisfy physical constraints by construction.We validate the proposed methods using a synthetic dataset and a benchmark case of one-dimensional laminar NH3/H2 flames.The results show that the thermo-chemical state predictions comply with physical constraints while maintaining the high accuracy of unconstrained reduced-order models.
基金supported by Swiss National Science Foundation (Grant No. 200021166274)the National Science Centre (Poland) (Grant No. 2015/17/B/ST1/01102) (2016–2019)
文摘In this paper,we are concerned with the asymptotic behavior,as u→∞,of P{sup_t∈|0,T|X_u(t)>u},where X_u(t),t∈|0,T|,u>0 is a family of centered Gaussian processes with continuous trajectories.A key application of our findings concerns P{sup_t∈|0,T|(X(t)+g(t))>u},as u→∞,for X a centered Gaussian process and g some measurable trend function.Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.
基金supported by NSFC(10131040)supported by SRFDP(2002335090)+1 种基金supported by KRF(2001-042-D00008)supported by KRF(2001-042-D00008)
文摘In this paper,we establish some limit theorems on the increments of an l^p-valued multi- parameter Gaussian process under weaker conditions than those of Cs(?)rg(?)-Shao theorems published in Ann.Probab.(1993).
基金Supported by the Natural Science Foundation of Jiangsu Province of China(BK20130531)the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD[2011]6)Jiangsu Government Scholarship
文摘The dynamic soft sensor based on a single Gaussian process regression(GPR) model has been developed in fermentation processes.However,limitations of single regression models,for multiphase/multimode fermentation processes,may result in large prediction errors and complexity of the soft sensor.Therefore,a dynamic soft sensor based on Gaussian mixture regression(GMR) was proposed to overcome the problems.Two structure parameters,the number of Gaussian components and the order of the model,are crucial to the soft sensor model.To achieve a simple and effective soft sensor,an iterative strategy was proposed to optimize the two structure parameters synchronously.For the aim of comparisons,the proposed dynamic GMR soft sensor and the existing dynamic GPR soft sensor were both investigated to estimate biomass concentration in a Penicillin simulation process and an industrial Erythromycin fermentation process.Results show that the proposed dynamic GMR soft sensor has higher prediction accuracy and is more suitable for dynamic multiphase/multimode fermentation processes.
基金Supported by National Science Foundation of China(Grant No.11326175)Research Start-up Foundation of Jiaxing University(Grant No.70512021)
文摘Let {X（t）, t ≥ 0} be a standard（zero-mean, unit-variance） stationary Gaussian process with correlation function r（·） and continuous sample paths. In this paper, we consider the maxima M（T） = max{X（t）, t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M（TT） exists under some additional conditions related to the correlation function r（·）.
基金Supported by the National High Technology Research and Development Program of China(2014AA041803)the National Natural Science Foundation of China(61320106009)
文摘Low pressure chemical vapor deposition(LPCVD) is one of the most important processes during semiconductor manufacturing.However,the spatial distribution of internal temperature and extremely few samples makes it hard to build a good-quality model of this batch process.Besides,due to the properties of this process,the reliability of the model must be taken into consideration when optimizing the MVs.In this work,an optimal design strategy based on the self-learning Gaussian process model(GPM) is proposed to control this kind of spatial batch process.The GPM is utilized as the internal model to predict the thicknesses of thin films on all spatial-distributed wafers using the limited data.Unlike the conventional model based design,the uncertainties of predictions provided by GPM are taken into consideration to guide the optimal design of manipulated variables so that the designing can be more prudent Besides,the GPM is also actively enhanced using as little data as possible based on the predictive uncertainties.The effectiveness of the proposed strategy is successfully demonstrated in an LPCVD process.
文摘We study the moduli of continuity of a class of N-parameter Gaussian processes and get some results on＇the packing dimension of the set of their fast points.
基金The authors would like to thank the referees for their careful reading and helpful comments that improved the quality of the paper. This work was supported by the National Natural Science Foundation of China （Grant No. 11326175）, the Natural Science Foundation of Zhejiang Province （Nos. LQ14A010012, LY15A010019）, the Natural Science Foundation of 3iangsu Higher Education Institution of China （No. 14KJB110023）, and the Research Foundation of SUST.
文摘We study the asymptotic relation among the maximum of continuous weakly and strongly dependent stationary Gaussian process, the maximum of this process sampled at discrete time points, and the partial sum of this process. It is shown that these two extreme values and the sum are asymptotically independent if the grid of the discrete time points is sufficiently sparse and the Gaussian process is weakly dependent, and asymptotically dependent if the grid points are Pickands grids or dense grids.
基金Research supported by NSFC (10401037)supported by SRFDP (2002335090) China Postdoctoral Science Foundation
文摘By estimating small ball probabilities for l^P-valued Gaussian processes, a Chung-type law of the iterated logarithm of l^P-valued Gaussian processes is given.
文摘Let{X k(t),t≥0},k=1,2,…,be a sequence of independent Gaussian processes withσk 2(h)=E(X k(t+h)-X k(t))2.Putσ(p,h)=(∑∞k=1σk p(h))1/p,p≥1.The author establishes the large increment results for boundedσ(p,h).
文摘In this paper, we get three local continuity moduli theorems for the almost surely continuous, stationary increments Gaussian process {Y(t), t0}, the partial sum processes X(t,N)= (t) of infinite dimensional Ornstein-Uhlenbeck processes {Xk(t), t0}, and lp-valued Gaussian processes {Y(t), t0}={Xk(t), t0}, separately. The first theorem implies the local continuity modulus theorem for the series X(t)=, Xk(t) of infinite dimensional OrnsteinUhlenbeck processes which has been obtained in [3].
