Application of the method of overlapping intervals to maximum likelihood estimates of the transition probabilities of a homogeneous Markov chain

1988 ◽  
Vol 41 (1) ◽  
pp. 857-861
Author(s):  
D. V. Sin'kevich
Keyword(s):  
Markov Chain ◽  
Maximum Likelihood ◽  
Transition Probabilities ◽  
Maximum Likelihood Estimates ◽  
Homogeneous Markov Chain

An adaptive Markov Chain approach for probabilistic forecasting of categorical events

2020 ◽  
pp. 1-31
Author(s):  
Edward C.D. Pope ◽  
David B. Stephenson ◽  
David R. Jackson
Keyword(s):  
Markov Chain ◽  
Space Weather ◽  
Weather Forecasting ◽  
Transition Probabilities ◽  
Parametric Estimation ◽  
Lead Times ◽  
Homogeneous Markov Chain ◽  
Probabilistic Prediction ◽  
Geomagnetic Disturbances ◽  
Memory Parameter

Abstract Categorical probabilistic prediction is widely used for terrestrial and space weather forecasting as well as for other environmental forecasts. One example is a warning system for geomagnetic disturbances caused by space weather, which are often classified on a 10-level scale. The simplest approach assumes that the transition probabilities are stationary in time – the Homogeneous Markov Chain (HMC). We extend this approach by developing a flexible Non-Homogeneous Markov Chain (NHMC) model using Bayesian non-parametric estimation to describe the time-varying transition probabilities. The transition probabilities are updated using a modified Bayes’ rule that gradually forgets transitions in the distant past, with a tunable memory parameter. The approaches were tested by making daily geomagnetic state forecasts at lead times of 1-4 days and verified over the period 2000-2019 using the Rank Probability Score (RPS). Both HMC and NHMC models were found to be skilful at all lead times when compared with climatological forecasts. The NHMC forecasts with an optimal memory parameter of ~100 days were found to be substantially more skilful than the HMC forecasts, with an RPS skill for the NHMC of 10.5% and 5.6% for lead times of 1 and 4 days ahead, respectively. The NHMC is thus a viable alternative approach for forecasting geomagnetic disturbances, and could provide a new benchmark for producing operational forecasts. The approach is generic and applicable to other forecasts including discrete weather regimes or hydrological conditions, e.g. wet and dry days.

Countable non-homogeneous Markov chains: asymptotic behaviour

1977 ◽  
Vol 9 (3) ◽  
pp. 542-552 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Asymptotic Properties ◽  
Homogeneous Markov Chain ◽  
Homogeneous Case ◽  
Main Concept ◽  
State Classification ◽  
Ratio Limit

The paper deals with asymptotic properties of the transition probabilities of a countable non-homogeneous Markov chain, the main concept used in the proofs being that of the tail σ-field of the chain. A state classification similar to that existing in the homogeneous case is given and a strong ratio limit property is shown to parallel the basic limit theorem for positive homogeneous chains. Some global asymptotic properties for null chains are also derived.

On a class of non-homogeneous Markov chains

1982 ◽  
Vol 92 (3) ◽  
pp. 527-534 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Image Position

AbstractSuppose that {Xn} is a countable non-homogeneous Markov chain andIf converges for any i, l, m, j with , thenwhenever lim , whereas if converges, thenwhere and . The behaviour of transition probabilities between various groups of states is studied and criteria for recurrence and transience are given.

Finite non-homogeneous Markov chains: Asymptotic behaviour

1976 ◽  
Vol 8 (3) ◽  
pp. 502-516 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Stochastic Matrices ◽  
Weak Ergodicity ◽  
General Picture ◽  
Markov Chain Theory ◽  
Chain Theory ◽  
Probabilistic Nature

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

Finite non-homogeneous Markov chains: Asymptotic behaviour

1976 ◽  
Vol 8 (03) ◽  
pp. 502-516 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Stochastic Matrices ◽  
Weak Ergodicity ◽  
General Picture ◽  
Markov Chain Theory ◽  
Chain Theory ◽  
Probabilistic Nature

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

mixmcm: A community-contributed command for fitting mixtures of Markov chain models using maximum likelihood and the EM algorithm

2019 ◽  
Vol 19 (2) ◽  
pp. 294-334 ◽  
Author(s):  
Legrand D. F. Saint-Cyr ◽  
Laurent Piet
Keyword(s):  
Markov Chain ◽  
Maximum Likelihood ◽  
Transition Probabilities ◽  
Finite Mixture Models ◽  
Unobserved Heterogeneity ◽  
Expectation Maximization Algorithm ◽  
Optimal Number ◽  
Mixing Distribution ◽  
Markov Chain Models ◽  
The Em Algorithm

Markov chain models and finite mixture models have been widely applied in various strands of the academic literature. Several studies analyzing dynamic processes have combined both modeling approaches to account for unobserved heterogeneity within a population. In this article, we describe mixmcm, a community-contributed command that fits the general class of mixed Markov chain models, accounting for the possibility of both entries into and exits from the population. To account for the possibility of incomplete information within the data (that is, unobserved heterogeneity), the model is fit with maximum likelihood using the expectation-maximization algorithm. mixmcm enables users to fit the mixed Markov chain models parametrically or semiparametrically, depending on the specifications chosen for the transition probabilities and the mixing distribution. mixmcm also allows for endogenous identification of the optimal number of homogeneous chains, that is, unobserved types or “components”. We illustrate mixmcm‘s usefulness through three examples analyzing farm dynamics using an unbalanced panel of commercial French farms.

On sums of random variables defined on a two-state Markov chain

1970 ◽  
Vol 7 (3) ◽  
pp. 761-765 ◽  
Author(s):  
H. J. Helgert
Keyword(s):  
Markov Chain ◽  
Transition Probabilities ◽  
Random Variables ◽  
Homogeneous Markov Chain ◽  
Sums Of Random Variables ◽  
Image Position

Assume the sequence of random variables x0, x1, x2, ··· forms a two-state, homogeneous Markov chain with transition probabilities and initial probabilities

First-passage times for the partial sums of a sequence of geometric distributions

1982 ◽  
Vol 19 (02) ◽  
pp. 430-432
Author(s):  
A. J. Woods
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
First Passage Times ◽  
Partial Sums ◽  
Homogeneous Markov Chain ◽  
First Passage ◽  
Epidemic Process ◽  
Passage Times

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

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