Aggregated semi-Markov processes incorporating time interval omission

1991 ◽  
Vol 23 (04) ◽  
pp. 772-797 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Renewal Process ◽  
Stochastic Modelling ◽  
Complete Information ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite State ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Finite State Space

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

Aggregated semi-Markov processes incorporating time interval omission

1991 ◽  
Vol 23 (4) ◽  
pp. 772-797 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Renewal Process ◽  
Stochastic Modelling ◽  
Complete Information ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite State ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Finite State Space

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

Aggregated Markov processes incorporating time interval omission

1988 ◽  
Vol 20 (03) ◽  
pp. 546-572 ◽  
Author(s):  
Frank Ball ◽  
Mark Sansom
Keyword(s):  
State Space ◽  
Renewal Process ◽  
Numerical Study ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Closed Sets ◽  
Finite State ◽  
Dynamic Stochastic ◽  
Stochastic Properties ◽  
Finite State Space

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.

Aggregated Markov processes incorporating time interval omission

1988 ◽  
Vol 20 (3) ◽  
pp. 546-572 ◽  
Author(s):  
Frank Ball ◽  
Mark Sansom
Keyword(s):  
State Space ◽  
Renewal Process ◽  
Numerical Study ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Closed Sets ◽  
Finite State ◽  
Dynamic Stochastic ◽  
Stochastic Properties ◽  
Finite State Space

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.

Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (04) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB (t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB (t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB (t) as t →∞.

Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (4) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂ t , the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂ t , and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

scholarly journals Simple derivations of properties of counting processes associated with Markov renewal processes

2005 ◽  
Vol 42 (4) ◽  
pp. 1031-1043 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne
Keyword(s):  
Renewal Process ◽  
Factorial Moment ◽  
Probability Generating Function ◽  
Stochastic Modelling ◽  
Markov Renewal Process ◽  
Counting Processes ◽  
Continuous Time Markov Chain ◽  
Factorial Moments ◽  
Asymptotic Expressions ◽  
Markov Renewal Processes

A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

Aggregated Markov processes with negative exponential time interval omission

1990 ◽  
Vol 22 (04) ◽  
pp. 802-830 ◽  
Author(s):  
Frank Ball
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Single Channel ◽  
The State ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Negative Exponential Distribution ◽  
Finite State ◽  
Negative Exponential ◽  
Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of f O(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

Aggregated Markov processes with negative exponential time interval omission

1990 ◽  
Vol 22 (4) ◽  
pp. 802-830 ◽  
Author(s):  
Frank Ball
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Single Channel ◽  
The State ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Negative Exponential Distribution ◽  
Finite State ◽  
Negative Exponential ◽  
Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of fO(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

1997 ◽  
Vol 29 (01) ◽  
pp. 92-113 ◽  
Author(s):  
Frank Ball ◽  
Sue Davies
Keyword(s):  
Markov Chain ◽  
Ion Channel ◽  
State Space ◽  
Continuous Time ◽  
Markov Models ◽  
Numerical Study ◽  
Channel Gating ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean[N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

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