18,176 research outputs found
Small sets and Markov transition densities
Get PDFThe theory of general state-space Markov chains can be strongly related to the case of discrete state-space by use of the notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on state-spaces with countably generated sigma-algebras, though the minorization provided by the theory concerns small sets of order n and n-step transition kernels for some unspecified n. Partly motivated by the growing importance of small sets for Markov chain Monte Carlo and Coupling from the Past, we show that in general there need be no small sets of order n = 1 even if the kernel is assumed to have a density function (though of course one can take n = 1 if the kernel density is continuous). However, n = 2 will suffice for kernels with densities (integral kernels), and in fact small sets of order 2 abound in the technical sense that the 2-step kernel density can be expressed as a countable sum of non-negative separable summands based on small sets, This can be exploited to produce a representation using a latent discrete Markov chain; indeed one might say, inside every Markov chain with measurable transition density there is a discrete state-space Markov chain struggling to escape. We conclude by discussing complements to these results, including their relevance to Harris-recurrent Markov chains and we relate the counterexample to Turan problems for bipartite graphs
Beveridge-Nelson Decomposition with Markov Switching
Get PDFThis paper considers Beveridge-Nelson decomposition in a context where the permanent and transitory components both follow a Markov switching process. Our approach incorporates Markov switching into a single source of error state-space framework, allowing business cycle asymmetries and regime switches in the long run multiplier.Beveridge-Nelson decomposition, Markov switching, Single source of error state space models
Nonlinear Markov semigroups and interacting Lévy type processes
Get PDFSemigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R-d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models
A GOODNESS OF FIT TEST FOR ERGODIC MARKOV PROCESSES
Get PDFWe introduce a goodness of fit test for ergodic Markov processes. Our test compares the data against the set of stationary densities implied by the class of models specified in the null hypothesis, and rejects if no model in the class yields a stationary density that matches with the data. No alternative needs to be specified in order to implement the test. Although our test compares densities it involves no smoothing parameters, and is powerful against 1√n local alternatives.Specification test, goodness of fit, Markov processes.
A statistical multiresolution approach for face recognition using structural hidden Markov models
Get PDFThis paper introduces a novel methodology that combines the multiresolution feature of the discrete wavelet transform (DWT) with the local interactions of the facial structures expressed through the structural hidden Markov model (SHMM). A range of wavelet filters such as Haar, biorthogonal 9/7, and Coiflet, as well as Gabor, have been implemented in order to search for the best performance. SHMMs perform a thorough probabilistic analysis of any sequential pattern by revealing both its inner and outer structures simultaneously. Unlike traditional HMMs, the SHMMs do not perform the state conditional independence of the visible observation sequence assumption. This is achieved via the concept of local structures introduced by the SHMMs. Therefore, the long-range dependency problem inherent to traditional HMMs has been drastically reduced. SHMMs have not previously been applied to the problem of face identification. The results reported in this application have shown that SHMM outperforms the traditional hidden Markov model with a 73% increase in accuracy
Speculative moves : multithreading Markov Chain Monte Carlo programs
Get PDFThe increasing availability of multi-core and multi-processor architectures provides new opportunities for improving the performance of many computer simulations. Markov Chain Monte Carlo (MCMC) simulations are widely used for approximate counting problems, Bayesian inference and as a means for estimating very high-dimensional integrals. As such MCMC has found a wide variety of applications in biology and biomedical analysis.
This paper presents a new method for reducing the runtime of Markov Chain Monte Carlo simulations by using SMP machines to speculatively perform iterations in parallel, reducing the runtime of MCMC programs whilst producing statistically identical results to conventional sequential implementations. We calculate the theoretical reduction in runtime that may be achieved using our technique under perfect conditions, and test and compare the method on a selection of multi-core and multi-processor architectures. Experiments are presented that show reductions in runtime of 35% using two cores and 55% using four cores
The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
Get PDFIto's construction of Markovian solutions to stochastic equations driven by a
Lévy noise is extended to nonlinear distribution dependent integrands aiming at
the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchine type) with
variable coeffcients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or
nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov
processes of a long known fact for ordinary diffusions
Rank-driven Markov processes
Get PDFWe study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κ N on {1,2,…,N} K . Our main results are that, under appropriate conditions on κ N , the system exhibits threshold behavior at s ∗∈[0,1], where s ∗ is a function of κ N , and the marginal distribution of a randomly selected element converges to U[s ∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest
Estimation of the stationary distribution of a semi-Markov chain
No full textThis article is concerned with the estimation of the stationary distribution of a discretetime semi-Markov process. After briefly presenting the discrete-time semi-Markov setting, wepropose an estimator of the associated stationary distribution. The main results concern theasymptotic properties of this estimator, as the sample size becomes large. A numerical exampleillustrates the asymptotic properties of the estimators
Markov-Switching Model Selection Using Kullback-Leibler Divergence
Get PDFIn Markov-switching regression models, we use Kullback-Leibler (KL) divergence between the true and candidate models to select the number of states and variables simultaneously. In applying Akaike information criterion (AIC), which is an estimate of KL divergence, we find that AIC retains too many states and variables in the model. Hence, we derive a new information criterion, Markov switching criterion (MSC), which yields a marked improvement in state determination and variable selection because it imposes an appropriate penalty to mitigate the over-retention of states in the Markov chain. MSC performs well in Monte Carlo studies with single and multiple states, small and large samples, and low and high noise. Furthermore, it not only applies to Markov-switching regression models, but also performs well in Markov- switching autoregression models. Finally, the usefulness of MSC is illustrated via applications to the U.S. business cycle and the effectiveness of media advertising.Research Methods/ Statistical Methods,
- …
