A different construction of Gaussian fields from Markov chains: Dirichlet covariances

We study a class of Gaussian random fields with negative correlations. These fields are easy to simulate. They are defined in a natural way from a Markov chain that has the index space of the Gaussian field as its state space. In parallel with Dynkin's investigation of Gaussian fields having covariance given by the Green's function of a Markov process, we develop connections between the occupation times of the Markov chain and the prediction properties of the Gaussian field. Our interest in such fields was initiated by their appearance in random matrix theory.     

Energy discriminant analysis, quantum logic, and fuzzy sets

In this paper, we show that quantum logic of linear subspaces can be used for recognition of random signals by a Bayesian energy discriminant classifier. The energy distribution on linear subspaces is described by the correlation matrix of the probability distribution. We show that the correlation matrix corresponds to von Neumann density matrix in quantum theory. We suggest the interpretation of quantum logic as a fuzzy logic of fuzzy sets. The use of quantum logic for recognition is based on the fact that the probability distribution of each class lies approximately in a lower-dimensional subspace of feature space. We offer the interpretation of discriminant functions as membership functions of fuzzy sets. Also, we offer the quality functional for optimal choice of discriminant functions for recognition from some class of discriminant functions.     

Localic subspaces and colimits of localic spaces

A spectral space is localic if it corresponds to a frame under Stone Duality. This class of spaces was introduced by the author (under the name ’locales’) as the topological version of the classical frame theoretic notion of locales, see Johnstone and also Picado and Pultr). The appropriate class of subspaces of a localic space are the localic subspaces. These are, in particular, spectral subspaces. The following main questions are studied (and answered): Given a spectral subspace of a localic space, how can one recognize whether the subspace is even localic? How can one construct all localic subspaces from particularly simple ones? The set of localic subspaces and the set of spectral subspaces are both inverse frames. The set of localic subspaces is known to be the image of an inverse nucleus on the inverse frame of spectral subspaces. How can the inverse nucleus be described explicitly? Are there any special properties distinguishing this particular inverse nucleus from all others? Colimits of spectral spaces and localic spaces are needed as a tool for the comparison of spectral subspaces and localic subspaces.     

THE DECOMPOSITION OF STATE SPACE FOR MARKOV CHAIN IN RANDOM ENVIRONMENT

This paper is a continuation of [8] and [9]. The author obtains the decomposition of state space X of an Markov chain in random environment by making use of the results in [8] and [9], gives three examples, random walk in random environment, renewal process in random environment and queue process in random environment, and obtains the decompositions of the state spaces of these three special examples.     

A quickly convergent continuous ant colony optimization algorithm with Scout Ants

Many studies on ants behavior have demonstrated that their food searching process starts with Scout Ants’ scouting all around for food. In this paper, we propose a novel Scout Ant Continuous Optimization (SACO) algorithm which can simulate the food searching process of the Scout Ants. In this algorithm, the solution space of an optimization problem is divided into m subspaces. One Scout Ant is assigned to each subspace, and a total number of m Scout Ants in m subspaces will cooperate in the whole food searching process. The location of a Scout Ant in a subspace corresponds to a solution in that subspace. When an ant moves, the change of its position is driven by two factors. The first factor is the independent, random ergodic search with a small moving step in the ant’s assigned subspace, and the second is the collaborative, global search inspired by the global heuristic information accumulated among m ants. Each of these two factors is weighed by an appropriate weight to balance its contribution to the moving step size. This balanced computation helps adapt optimization problems with different features. Our numerical experiments have demonstrated that, in addition to the high accuracy and success rate in seeking the optimized solutions, our algorithm also has very fast convergence speed and impressive performance in optimization applications in high-dimensional spaces.     

多圆盘的加权Bergman空间上的不变子空间和约化子空间

本文主要研究多圆盘的加权Bergman 空间上的不变子空间和约化子空间, 给出了某些解析Toeplitz 算子的极小约化子空间的完全刻画, 以及一类解析Toeplitz 算子Tzi (1≤i≤n) 的不变子空间的Beurling 型定理.     

Inner-approximating domains of attraction for discrete-time switched systems via Multi-step multiple Lyapunov-like functions

In this paper, we propose an iterative approach for estimating the domains of attraction for a class of discrete-time switched systems, where the state space is divided into several disjoint regions and each region is described by polynomial inequalities. At first, we introduce the basic concepts of Multi-step state subsequence, Multi-step state subspace, Multi-step basin of attraction and Multi-step multiple Lyapunov-like function. Secondly, beginning with an initial inner estimation, a theoretical framework is proposed for estimating the domain of attraction by iteratively calculating the Multi-step multiple Lyapunov-like functions. Thirdly, notice that the Multi-step state subspaces may be empty sets such that the corresponding constraints in the theoretical framework are redundant, we propose a numerical approach based on the homotopy continuation method to pre-check the non-emptiness of the Multi-step state subspaces, and then under-approximatively realize the framework by using S-procedure and sum of squares programming. At last, we implement our iterative approach and apply it to three discrete-time switched system examples with comparisons to existing methods in the literatures. These computation and comparison results show the advantages of our method.     

Robust dimension reduction,fusion frames,and Grassmannian packings

We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first consider the linear minimum mean-squared error (LMMSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the mean-squared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equi-dimensional tight fusion frames. We call such fusion frames equi-distance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equi-distance tight fusion frames.     

Gaussian and non-Gaussian random fields associated with Markov processes

To every Markov process with a symmetric transition density, there correspond two random fields over the state space: a Gaussian field (the free field) φ and the occupation field T which describes amount of time the particle spends at each state. A relation between these two random fields is established which is useful both for the field theory and theory of Markov processes.     

Markov Chains in a Field of Traps

We consider a Markov chain on a countable state space, on which is placed a random field of traps, and ask whether the chain gets trapped almost surely. We show that the quenched problem (when the traps are fixed) is equivalent to the annealed problem (when the traps are updated each unit of time) and give a criterion for almost sure trapping versus positive probability of nontrapping. The hypotheses on the Markov chain are minimal, and in particular, our results encompass the results of den Hollander, Menshikov and Volkov (1995).     

How to count and guess well: Discrete adaptive filters

A discrete state and time Markov chain is observed through a finite state function which is subject to random perturbations. Such a situation is often called a Hidden Markov Model. A general filter is obtained which provides recursive updates of estimates of processes related to the Markov chain given the observations. In the unnormalized, or Zakai, form this provides particularly simple equations. Specializing this result provides recursive estimates and smoothers for the state of the process, for the number of jumps from one state to another, for the occupation time in any state and for a process related to the observations. These results allow a re-estimation of the parameters of the model, so that our procedures are adaptive or self tuning to the data. The main contributions of this paper are the introduction of an equivalent measure under which the observation values are independent and identically distributed, and the use of the idempotent property when the state space of the Markov chain is identified with canonical unit vectors in a Euclidean space.This research was supported by NSERC Grant A7964.     

The lumpability property for a family of Markov chains on poset block structures

We construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We essentially use combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associated with a Gelfand pair, the spectral analysis of the corresponding lumped Markov chain is completely determined by the decomposition of the group action into irreducible submodules.     

Tree-Indexed Markov Chains in Random Environment and Some of Their Strong Limit Properties

In this paper, the authors first introduce the tree-indexed Markov chains in random environment, which takes values on a general state space. Then, they prove the existence of this stochastic process, and develop a class of its equivalent forms. Based on this property, some strong limit theorems including conditional entropy density are studied for the tree-indexed Markov chains in random environment.     

基于Kernel Discriminant Canonical Correlation(KDCC)的多观测样本分类算法

针对多观测样本分类问题,提出一种基于Kernel Discriminant CanonicalCorrelation(KDCC)来实现多观测样本分类的模型.该算法首先把原空间样本非线性的投影到高维特征空间,通过KPCA得到核子空间,然后在高维特征空间定义一个使类内核子空间的相关性最大,同时使类间核子空间的相关性最小的KDCC矩阵,通过迭代法训练出最优的KDCC矩阵,把每个核子空间投影到KDCC矩阵上得到转换核子空间,采用典型相关性作为转换核子空间之间的相似性度量,并采用最近邻准则作为多观测样本的分类决策,从而实现多观测样本的分类.在三个数据库上进行了一系列实验,实验结果表明提出的方法对于多观测样本分类具有可行性和有效性.     

Hypercyclic Subspaces on Fréchet Spaces Without Continuous Norm

Known results about hypercyclic subspaces concern either Fréchet spaces with a continuous norm or the space ω. We fill the gap between these spaces by investigating Fréchet spaces without continuous norm. To this end, we divide hypercyclic subspaces into two types: the hypercyclic subspaces M for which there exists a continuous seminorm p such that ${M \cap {\rm ker} p = \{0\}}$ and the others. For each of these types of hypercyclic subspaces, we establish some criteria. This investigation permits us to generalize several results about hypercyclic subspaces on Fréchet spaces with a continuous norm and about hypercyclic subspaces on ω. In particular, we show that each infinite-dimensional separable Fréchet space supports a mixing operator with a hypercyclic subspace.     

On some applications of Wald's identity to dams

It is known that the main difficulty in applying the Markovian analogue of Wald's Identity is the presence, in the Identity, of the last state variable before the random walk is terminated. In this paper we show that this difficulty can be overcome if the underlying Markov chain has a finite state space. The absorption probabilities thus obtained are used, by employing a duality argument, to derive time-dependent and limiting probabilities for the depletion process of a dam with Markovian inputs.The second problem that is considered here is that of a non-homogeneous but cyclic Markov chain. An analogue of Wald's Identity is obtained for this case, and is used to derive time- dependent and limiting probabilities for the depletion process with inputs forming a non- homogeneous (cyclic) Markov chain.     

次可分解算子的不变子空间格的丰富性

证明了一类次可分解算子的不变子空间格是丰富的,并举例说明存在Hilbert空间上的有界线性算子T,它有无穷多个不变子空间,但是它的不变子空间格Lat(T)不丰富．     

Nearly-Optimal Asset Allocation in Hybrid Stock Investment Models

This work develops a class of stock-investment models that are hybrid in nature and involve continuous dynamics and discrete-event interventions. In lieu of the usual geometric Brownian motion formulation, hybrid geometric Brownian motion models are proposed, in which both the expected return and the volatility depend on a finite-state Markov chain. Our objective is to find nearly-optimal asset allocation strategies so as to maximize the expected returns. The use of the Markov chain stems from the motivation of capturing the market trends as well as various economic factors. To incorporate these economic factors into the models, the underlying Markov chain inevitably has a large state space. To reduce the complexity, a hierarchical approach is suggested, which leads to singularly-perturbed switching diffusion processes. By aggregating the states of the Markov chains in each weakly irreducible class into a single state, limit switching diffusion processes are obtained. Using such asymptotic properties, nearly-optimal asset allocation policies are developed.     

p-adic Gibbs measures and Markov random fields on countable graphs

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures     

平稳遍历马氏链部分和序列最小值分布的渐近估计及其应用

本文给出了有限状态平稳遍历Markov链部分和序列最小值分布的一个渐近估计式并利用它对一类Athreya-KarlinBPRE．灭种概率的渐近行为作出估计．