scholarly journals A Model of Indel Evolution by Finite-State, Continuous-Time Machines

Genetics ◽  
2020 ◽  
Vol 216 (4) ◽  
pp. 1187-1204
Author(s):  
Ian Holmes
Keyword(s):  
Differential Equations ◽  
Continuous Time ◽  
Markov Models ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Coarse Graining ◽  
Automata Theory ◽  
State Spaces ◽  
Finite State ◽  
Related Approach

We introduce a systematic method of approximating finite-time transition probabilities for continuous-time insertion-deletion models on sequences. The method uses automata theory to describe the action of an infinitesimal evolutionary generator on a probability distribution over alignments, where both the generator and the alignment distribution can be represented by pair hidden Markov models (HMMs). In general, combining HMMs in this way induces a multiplication of their state spaces; to control this, we introduce a coarse-graining operation to keep the state space at a constant size. This leads naturally to ordinary differential equations for the evolution of the transition probabilities of the approximating pair HMM. The TKF91 model emerges as an exact solution to these equations for the special case of single-residue indels. For the more general case of multiple-residue indels, the equations can be solved by numerical integration. Using simulated data, we show that the resulting distribution over alignments, when compared to previous approximations, is a better fit over a broader range of parameters. We also propose a related approach to develop differential equations for sufficient statistics to estimate the underlying instantaneous indel rates by expectation maximization. Our code and data are available at https://github.com/ihh/trajectory-likelihood.

scholarly journals Approximation of Indel Evolution by Differential Calculus of Finite State Automata

2020 ◽  
Author(s):  
Ian Holmes
Keyword(s):  
Differential Equations ◽  
Markov Models ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Coarse Graining ◽  
Automata Theory ◽  
Sufficient Statistics ◽  
State Spaces ◽  
Finite State ◽  
Related Approach

AbstractWe introduce a systematic method of approximating finite-time transition probabilities for continuous-time insertion-deletion models on sequences. The method uses automata theory to describe the action of an infinitesimal evolutionary generator on a probability distribution over alignments, where both the generator and the alignment distribution can be represented by Pair Hidden Markov Models (Pair HMMs). In general, combining HMMs in this way induces a multiplication of their state spaces; to control this, we introduce a coarse-graining operation to keep the state space at a constant size. This leads naturally to ordinary differential equations for the evolution of the transition probabilities of the approximating Pair HMM. The TKF model emerges as an exact solution to these equations for the special case of single-residue indels. For the general case, the equations can be solved by numerical integration. Using simulated data we show that the resulting distribution over alignments, when compared to previous approximations, is a better fit over a broader range of parameters. We also propose a related approach to develop differential equations for sufficient statistics to estimate the underlying instantaneous indel rates by Expectation-Maximization. Our code and data are available at https://github.com/ihh/trajectory-likelihood.

scholarly journals On the total reward variance for continuous-time Markov reward chains

2006 ◽  
Vol 43 (04) ◽  
pp. 1044-1052 ◽  
Author(s):  
Nico M. Van Dijk ◽  
Karel Sladký
Keyword(s):  
Growth Rate ◽  
Discrete Time ◽  
Continuous Time ◽  
Asymptotically Linear ◽  
State Spaces ◽  
Total Reward ◽  
Finite State ◽  
Markov Reward ◽  
Discrete Time Case

As an extension of the discrete-time case, this note investigates the variance of the total cumulative reward for continuous-time Markov reward chains with finite state spaces. The results correspond to discrete-time results. In particular, the variance growth rate is shown to be asymptotically linear in time. Expressions are provided to compute this growth rate.

scholarly journals On Determining the Order of Markov Dependence of an Observed Process Governed by a Hidden Markov Model

2002 ◽  
Vol 10 (3) ◽  
pp. 241-251 ◽  
Author(s):  
R.J. Boys ◽  
D.A. Henderson
Keyword(s):  
Markov Chain ◽  
Markov Models ◽  
Transition Probabilities ◽  
Hidden Markov ◽  
Real Data ◽  
Information Criteria ◽  
Monte Carlo Algorithm ◽  
Model Parameters ◽  
Markov Dependence ◽  
Finite State

This paper describes a Bayesian approach to determining the order of a finite state Markov chain whose transition probabilities are themselves governed by a homogeneous finite state Markov chain. It extends previous work on homogeneous Markov chains to more general and applicable hidden Markov models. The method we describe uses a Markov chain Monte Carlo algorithm to obtain samples from the (posterior) distribution for both the order of Markov dependence in the observed sequence and the other governing model parameters. These samples allow coherent inferences to be made straightforwardly in contrast to those which use information criteria. The methods are illustrated by their application to both simulated and real data sets.

scholarly journals On the total reward variance for continuous-time Markov reward chains

2006 ◽  
Vol 43 (4) ◽  
pp. 1044-1052 ◽  
Author(s):  
Nico M. Van Dijk ◽  
Karel Sladký
Keyword(s):  
Growth Rate ◽  
Discrete Time ◽  
Continuous Time ◽  
Asymptotically Linear ◽  
State Spaces ◽  
Total Reward ◽  
Finite State ◽  
Markov Reward ◽  
Discrete Time Case

As an extension of the discrete-time case, this note investigates the variance of the total cumulative reward for continuous-time Markov reward chains with finite state spaces. The results correspond to discrete-time results. In particular, the variance growth rate is shown to be asymptotically linear in time. Expressions are provided to compute this growth rate.

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

1990 ◽  
Vol 22 (1) ◽  
pp. 111-128 ◽  
Author(s):  
P. K. Pollett ◽  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Invariant Manifold ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Centre Manifold ◽  
Continuous Time Markov Chains ◽  
Linear Differential

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

1990 ◽  
Vol 22 (01) ◽  
pp. 111-128 ◽  
Author(s):  
P. K. Pollett ◽  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Invariant Manifold ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Centre Manifold ◽  
Continuous Time Markov Chains ◽  
Linear Differential

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables

1955 ◽  
Vol 51 (3) ◽  
pp. 433-441 ◽  
Author(s):  
D. R. Cox
Keyword(s):  
Differential Equations ◽  
Stochastic Processes ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Laplace Transforms ◽  
Supplementary Variables ◽  
Discrete States

ABSTRACTCertain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.

A canonical representation for aggregated Markov processes

1998 ◽  
Vol 35 (02) ◽  
pp. 313-324 ◽  
Author(s):  
Bret Larget
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Markov Models ◽  
Sufficient Conditions ◽  
Canonical Representation ◽  
Regularity Conditions ◽  
Deterministic Function ◽  
Finite State ◽  
Canonical Representations

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.

scholarly journals A Continuous Time Representation of smFRET for the Extraction of Rapid Kinetics

2020 ◽  
Author(s):  
Zeliha Kilic ◽  
Ioannis Sgouralis ◽  
Wooseok Heo ◽  
Kunihiko Ishii ◽  
Tahei Tahara ◽  
...  
Keyword(s):  
Single Molecule ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Hidden Markov ◽  
Photon Emission ◽  
Simulated Data ◽  
Emission Rates ◽  
Markov Jump ◽  
Conformational States ◽  
Donor And Acceptor

AbstractOur goal is to learn kinetic rates from single molecule FRET (smFRET) data even if these exceed the data acquisition rate. To achieve this, we develop a variant of our recently proposed hidden Markov jump process (HMJP) with which we learn transition kinetics from parallel measurements in donor and acceptor channels. Our HMJP generalizes the hidden Markov model (HMM) paradigm in two critical ways: (1) it deals with physical smFRET systems as they switch between conformational states in continuous time; (2) it estimates the transition rates between conformational states directly without having recourse to transition probabilities or assuming slow dynamics (as is necessary of the HMM). Our continuous time treatment learns transition kinetics and photon emission rates for dynamical regimes inaccessible to the HMM which treats system kinetics in discrete time. We validate the robustness of our framework on simulated data and demonstrate its performance on experimental data from FRET labeled Holliday junctions.

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