Sparse Markov Chains for Sequence Data

With a significant development of big data analysis and cloud-fog-edge computing, human-centered computing (HCC) has been a hot research topic worldwide. Essentially, HCC is a cross-disciplinary research domain, in which the core idea is to build an efficient interaction among persons, cyber space, and real world. Inspired by the improvement of HCC on big data analysis, we intend to involve related core and technologies to help solve one of the most important issues in the real world, i.e., flood prediction. To minimize the negative impacts brought by floods, researchers pay special attention to improve the accuracy of flood forecasting with quantity of technologies including HCC. However, historical flood data is essentially imbalanced. Imbalanced data causes machine learning classifiers to be more biased towards patterns with majority samples, resulting in poor classification of pattern with minority samples. In this paper, we propose a novel Synthetic Minority Over-sampling Technique (SMOTE)-Boost-based sparse Bayesian model to perform flood prediction with both high accuracy and robustness. The proposed model consists of three modules, namely, SMOTE-based data enhancement, AdaBoost training strategy, and sparse Bayes model construction. In SMOTE-based data enhancement, we adopt a SMOTE algorithm to effectively cover diverse data modes and generate more samples for prediction pattern with minority samples, which greatly alleviates the problem of imbalanced data by involving experts’ analysis and users’ intentions. During AdaBoost training strategy, we propose a specifically designed AdaBoost training strategy for sparse Bayesian model, which not only adaptively and inclemently increases prediction ability of Bayesian model, but also prevents its over-fitting performance. Essentially, the design of AdaBoost strategy helps keep balance between prediction ability and model complexity, which offers different but effective models over diverse rivers and users. Finally, we construct a sparse Bayesian model based on AdaBoost training strategy, which could offer flood prediction results with high rationality and robustness. We demonstrate the accuracy and effectiveness of the proposed model for flood prediction by conducting experiments on a collected dataset with several comparative methods.

In this paper, a new classification method based on relevance vector machine (RVM) is used in the MPSK signals classification. Compared with the support vector machine (SVM), RVM is sparse model in the Bayesian framework, not only the solution is highly sparse, but also it does not need to adjust model parameter and its kernel functions don't need to satisfy Mercer's condition. The fourth order cumulants of the received signals are used as the classification vector firstly, and then multi-class classifier of RVM is designed. We first introduce the sparse Bayesian classification model, then transform the RVM learning to the maximization of marginal likelihood, and select the fast sequential sparse Bayesian learning algorithm. Through the experiment compared with SVM classifier proves the advantage of RVM.

Feature fusion and distillation embedded sparse Bayesian learning model for in-situ foreknowledge of robotic machining errors

In this chapter we consider a Markov chain {ξ k , k ∈ Z +} having a finite state space but random transition probabilities. Of interest here is the asymptotic behavior of the chain and its transition probabilities as n → ∞ under the assumption that the transition probabilities of the Markov chain are close to those of a homogeneous Markov chain having nonrandom transition probabilities.

Posterior impropriety of some sparse Bayesian learning models

A novel polynomial dimension decomposition method based on sparse Bayesian learning and Bayesian model averaging

The newly emerging sampling methodology of compressed sensing opens a door to obtain compressed data directly. How to efficiently reconstruct the original signal from the compressed data becomes a new challenge problem. Many reconstruction works have been proposed on mono-view images by exploring the sparsity of the original image, how ever it is a challenge to efficiently explore the correlations between different views in compressed multiview imaging systems. With the aid of inter-view disparity information at receiver end, a joint reconstruction approach is presented for independently captured view point images via compressed imaging.

Mathematically, stochastic hybrid systems can be described by Markov processes with discrete and continuous behaviours. This chapter provides in an incremental way the necessary mathematical background for understanding such complex systems. The chapter starts with discrete and continuous-time Markov chains. In this book, a Markov chain is understood as a stochastic process with the Markov property (or memoryless property, i.e. its future evolution depends only on the current state) defined on a discrete (finite or countable) state space. The Markov property is illustrated in the Chapman-Kolmogorov equation satisfied by the transition probabilities. Usually, in the continuous-time Markov case, the evolution of these transition probabilities is described by forward/backward Kolmogorov equations. These equations are expressed in terms of the stochastic matrix (called also infinitesimal generator) associated to the Markov chain. This is the matrix of transition rates and the practical use of Markov chains resides in our ability of handling it.The second part of this chapter is dedicated to Markov processes defined on continuous state spaces that can be equipped with additional structures as sigma-algebras, metrics, topologies, norms or others like specific algebraic structures (lattice, vector space). Characterisations of such processes are usually given using functional analysis operators like: the operator semigroup/resolvent, the infinitesimal generator or other operators that can be associated to a Markov process. At an abstract level, these operators describe the evolution the evolution of transition probabilities for a Markov process. Chapman-Kolmogorov equation, Kolmogorov equations and other properties of these operators (e.g. Dynkin formula, martingale problem) represent the natural tools for studying such Markov processes.KeywordsMarkov ChainTopological SpaceMarkov ProcessMarkov PropertyPolish SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.

Sparse Bayesian factor models are routinely implemented for parsimonious dependence modeling and dimensionality reduction in high-dimensional applications. We provide theoretical understanding of such Bayesian procedures in terms of posterior convergence rates in inferring high-dimensional covariance matrices where the dimension can be larger than the sample size. Under relevant sparsity assumptions on the true covariance matrix, we show that commonly-used point mass mixture priors on the factor loadings lead to consistent estimation in the operator norm even when $p\gg n$. One of our major contributions is to develop a new class of continuous shrinkage priors and provide insights into their concentration around sparse vectors. Using such priors for the factor loadings, we obtain similar rate of convergence as obtained with point mass mixture priors. To obtain the convergence rates, we construct test functions to separate points in the space of high-dimensional covariance matrices using insights from random matrix theory; the tools developed may be of independent interest. We also derive minimax rates and show that the Bayesian posterior rates of convergence coincide with the minimax rates upto a $\sqrt{\log n}$ term.

This review article is concerned to provide a global context to several works on the fitting of continuous time nonhomogeneous Markov chains with finite state space and also to point out some selected aspects of two techniques previously introduced—estimation and calibration—relevant for applications and used to fit a continuous time Markov chain model to data by the adequate selection of parameters. The denomination estimation suits the procedure better when statistical techniques—e.g., maximum likelihood estimators—are employed, while calibration covers the case where, for instance, some optimisation technique finds a best approximation parameter to ensure good model fitting. For completeness, we provide a short summary of well-known important notions and results formulated for nonhomogeneous Markov chains that, in general, can be transferred to the homogeneous case. Then, as an illustration for the homogeneous case, we present a selected Billingsley’s result on parameter estimation for irreducible chains with finite state space. In the nonhomogeneous case, we quote two recent results, one of the calibration type and the other with more of a statistical flavour. We provide an ample set of bibliographic references so that the reader wanting to pursue her/his studies will be able to do so more easily and productively.

The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the singularity of their transition probabilities, while topological and functional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the investigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimensional or weak-* ergodic processes. To this end, we first develop local counterparts of zero-two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that admit asymptotic couplings or that are locally mixing in the sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary absolutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. Finally, we show that our abstract results can be applied to infinite-dimensional Markov processes that arise in several settings, including dissipative stochastic partial differential equations, stochastic spin systems and stochastic differential delay equations.

Hyperspectral unmixing is a technique that can be used to find the materials and their corresponding proportions present in observed mixed pixels. It has been widely used for hyperspectral data exploitation. Traditionally, this problem is solved using the constrained least squares optimization, which would lead to a dense solution. In recent years, researchers have observed that the number of materials in a mixed pixel is generally much smaller than the whole number of materials present in a given scene. They proposed to solve the problem through sparse unmixing approaches, which generally lead to constrained optimization. These sparse unmixing approaches have received considerable attention over the past few years. From the perspective of Bayesian inference, these methods assume fixed priors. In this article, we propose to solve the spectral unmixing problem by using sparse Bayesian learning (SBL) framework. In contrast to the approaches that use fixed priors, SBL incorporates a hierarchical parameterized prior that encourages sparsity in representation in its Bayesian inference to obtain a sparse solution. Experimental results on simulated data sets and a real hyperspectral scene demonstrate that the proposed method can achieve good performance compared with some state-of-the-art sparse unmixing algorithms. Our results also show that the proposed method can achieve good performance with a set of default parameter settings.

Identifying agents in a linear mixture is a fundamental problem in spectral sensing applications including chemical and biological agent identification. In general, the size of the spectral signature library is usually much larger than the number of agents really present. Based on this fact, the sparsity of the mixing coefficient vector can be utilized to help improve the identification performance. In this paper, we propose a new agent identification method by using a sparse Bayesian model. The proposed iterative algorithm takes into account the nonnegativity of the abundance fractions and is proved to be convergent. Numerical studies with a set of ultraviolet (UV) to infrared (IR) spectra are carried out for demonstration. The effect of the signature mismatch is also studied using a group of terahertz (THz) spectra.

In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.