The point process of state transitions in a regular Markov chain

1998 ◽  
Vol 14 (4) ◽  
pp. 374-380
Author(s):  
Shi Dinghua ◽  
Guo Jinli
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
State Transitions ◽  
Regular Markov Chain

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (02) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

scholarly journals Sequential tracking of a hidden Markov chain using point process observations

2009 ◽  
Vol 119 (6) ◽  
pp. 1792-1822 ◽  
Author(s):  
Erhan Bayraktar ◽  
Michael Ludkovski
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
Hidden Markov ◽  
Hidden Markov Chain

Expected frequencies of DNA patterns using whittle's formula

1991 ◽  
Vol 28 (04) ◽  
pp. 886-892 ◽  
Author(s):  
Richard Cowan
Keyword(s):  
Markov Chain ◽  
Molecular Biology ◽  
Dna Sequences ◽  
Exact Method ◽  
State Transitions ◽  
Initial State ◽  
Expected Frequency

Given a realisation of a Markov chain, one can count the numbers of state transitions of each type. One can ask how many realisations are there with these transition counts and the same initial state. Whittle (1955) has answered this question, by finding an explicit though complicated formula, and has also shown that each realisation is equally likely. In the analysis of DNA sequences which comprise letters from the set {A, C, G, T}, it is often useful to count the frequency of a pattern, say ACGCT, in a long sequence and compare this with the expected frequency for all sequences having the same start letter and the same transition counts (or ‘dinucleotide counts' as they are called in the molecular biology literature). To date, no exact method exists; this paper rectifies that deficiency.

Texture pattern image generation by regular Markov chain

1979 ◽  
Vol 11 (4) ◽  
pp. 225-233 ◽  
Author(s):  
Ryuzo Yokoyama ◽  
Robert M. Haralick
Keyword(s):  
Markov Chain ◽  
Image Generation ◽  
Texture Pattern ◽  
Regular Markov Chain

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

On the variance to mean ratio for random variables from Markov chains and point processes

1998 ◽  
Vol 35 (2) ◽  
pp. 303-312 ◽  
Author(s):  
Timothy C. Brown ◽  
Kais Hamza ◽  
Aihua Xia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Point Process ◽  
Continuous Time ◽  
Infinitesimal Generator ◽  
Point Processes ◽  
Random Variables ◽  
Simple Point ◽  
Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

Point processes generated by transitions of Markov chains

1973 ◽  
Vol 5 (2) ◽  
pp. 262-286 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
Point Processes ◽  
Queueing System ◽  
Initial Distribution ◽  
Recurrence Time ◽  
Asynchronous Sampling ◽  
Doubly Stochastic ◽  
Special Cases ◽  
Forward Recurrence Time

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

scholarly journals Markov Chain Monte Carlo Methods for State-Space Models with Point Process Observations

2012 ◽  
Vol 24 (6) ◽  
pp. 1462-1486 ◽  
Author(s):  
Ke Yuan ◽  
Mark Girolami ◽  
Mahesan Niranjan
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
State Space ◽  
Point Process ◽  
Large Data ◽  
State Space Models ◽  
Superior Performance ◽  
Mcmc Methods ◽  
Data Sets

This letter considers how a number of modern Markov chain Monte Carlo (MCMC) methods can be applied for parameter estimation and inference in state-space models with point process observations. We quantified the efficiencies of these MCMC methods on synthetic data, and our results suggest that the Reimannian manifold Hamiltonian Monte Carlo method offers the best performance. We further compared such a method with a previously tested variational Bayes method on two experimental data sets. Results indicate similar performance on the large data sets and superior performance on small ones. The work offers an extensive suite of MCMC algorithms evaluated on an important class of models for physiological signal analysis.

scholarly journals Extensions of Markov Modulated Poisson Processes  and Their Applications to Deep Earthquakes

2021 ◽  
Author(s):  
◽  
Shaochuan Lu
Keyword(s):  
Markov Chain ◽  
New Zealand ◽  
Mutual Information ◽  
Point Process ◽  
Point Processes ◽  
Marked Point Processes ◽  
Information Rate ◽  
The Third ◽  
Deep Earthquakes ◽  
Markov Modulated

<p>The focus of this thesis is on the Markov modulated Poisson process (MMPP) and its extensions, aiming to propose appropriate statistical models for the occurrence patterns of main New Zealand deep earthquakes. Such an attempt might be beyond the scope of the MMPP and its extensions, however we hope its main patterns can be characterized by current models proposed in three parts of the thesis. The first part of the thesis is concerned with introductions and preliminaries of discrete time hidden Markov models (HMMs) and MMPP. The  exibility in model formulation and openness in model framework of HMMs are reviewed in this part, suggesting also possible extensions of MMPP. The second part of the thesis is mainly about several extensions of MMPP. One extension of MMPP is by associating each occurrence of MMPP with a mark. Such an extension is potentially useful for spatial-temporal modelling or other point  processes with marks. A special case of this type of extension is by allowing the multiple observations of MMPP synchronized together under the same Markov chain. This extension opens the possibility of modelling multiple point process observations with weak dependence. The third extension is motivated by the attempt to describe small scale temporal clustering existing in the deep earthquakes via treating the recognized aftershocks as marks which itself forms a finite point process. The rest of the second part focuses on some information theoretical aspects of MMPPs such as the entropy rate of the underlying Markov chain and observed point process respectively and their mutual information rate. A conjecture on the possible links between mutual information rate of MMPP and the Fisher information of the estimated parameters is suggested. The second part on extensions of MMPP is featured by the derivation of the likelihood and complete likelihood, parameter estimation via EM algorithm, state smoothing estimation and model evaluation through systematic applications of rescaling theory of multivariate point processes and marked point processes. The third part of the thesis includes the applications of these methods to the deep earthquakes in New Zealand. We first evaluate the data coverage, catalogue completeness and explore its descriptive characteristics and empirical properties such as epicentral distributions, depth distributions and magnitude distributions.  Clustering behavior is studied via the second order moment analysis of point processes in the chapter 8. We also apply, the stress release models and the ETAS models which are usually used for shallow earthquakes, to the New Zealand deep earthquakes and provide tentative explanations of why they are not satisfactory for the deep earth-quakes. The chapter 9 is on the applications of MMPP and its extensions to the New Zealand deep earthquakes. Conclusions and future studies are presented in chapter 10.</p>

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