scholarly journals Markov chain models of cancer metastasis

2018 ◽  
Author(s):  
Jeremy Mason ◽  
Paul K. Newton
Keyword(s):  
Markov Chain ◽  
Longitudinal Data ◽  
Cancer Metastasis ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Metastatic Cancer ◽  
Markov Network ◽  
Markov Chain Models ◽  
Main Challenge ◽  
Markov Matrix

Abstract.We describe the use of Markov chain models for the purpose of quantitative forecasting of metastatic cancer progression. Each site (node) in the Markov network (directed graph) is an organ site where a secondary tumor could develop with some probability. The Markov matrix is an N x N matrix where each entry represents a transition probability of the disease progressing from one site to another during the course of the disease. The initial state-vector has a 1 at the position corresponding to the primary tumor, and 0s elsewhere (no initial metastases). The spread of the disease to other sites (metastases) is modeled as a directed random walk on the Markov network, moving from site to site with the estimated transition probabilities obtained from longitudinal data. The stochastic model produces probabilistic predictions of the likelihood of each metastatic pathway and corresponding time sequences obtained from computer Monte Carlo simulations. The main challenge is to empirically estimate the N^2 transition probabilities in the Markov matrix using appropriate longitudinal data.

scholarly journals Infinitely divisible random transition probabilities with application to dependent Markov chains

1984 ◽  
Vol 25 (4) ◽  
pp. 463-472
Author(s):  
Peter L. Chesson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
White Noise ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Rates ◽  
Infinitely Divisible ◽  
Random Transition ◽  
Transition Probability Matrices ◽  
Probability Matrices

AbstractRandom transition probability matrices with stationary independent factors define “white noise” environment processes for Markov chains. Two examples are considered in detail. Such environment processes can be used to construct several Markov chains which are dependent, have the same transition probabilities and are jointly a Markov chain. Transition rates for such processes are evaluated. These results have application to the study of animal movements.

The robustness of positive recurrence and recurrence of Markov chains under perturbations of the transition probabilities

1975 ◽  
Vol 12 (04) ◽  
pp. 744-752 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Finite Number ◽  
Transition Probabilities ◽  
General State ◽  
Transition Matrices ◽  
Positive Recurrence ◽  
Markov Chain Models ◽  
Countable Space ◽  
General State Space

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.

scholarly journals On the Convergence Rate of Bonus-Malus Systems

1992 ◽  
Vol 22 (2) ◽  
pp. 217-223 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Convergence Rate ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Limit Probability ◽  
Step Transition

AbstractUnder certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

Some explicit results for correlated random walks

1989 ◽  
Vol 26 (4) ◽  
pp. 757-766 ◽  
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Equilibrium Distribution ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Correlated Random Walks ◽  
Finite State ◽  
Direction Of Movement ◽  
Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

The square root law in the embedding detection problem for Markov chains with unknown matrix of transition probabilities

2019 ◽  
Vol 29 (1) ◽  
pp. 59-68
Author(s):  
Artem V. Volgin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Classical Model ◽  
Transition Probability Matrix ◽  
Square Root ◽  
Detection Problem ◽  
Asymptotic Growth

Abstract We consider the classical model of embeddings in a simple binary Markov chain with unknown transition probability matrix. We obtain conditions on the asymptotic growth of lengths of the original and embedded sequences sufficient for the consistency of the proposed statistical embedding detection test.

MARKOV CHAIN MODELING AND ANALYSIS OF COMPLICATED PHENOMENA IN COUPLED CHAOTIC OSCILLATORS

2010 ◽  
Vol 19 (04) ◽  
pp. 801-818 ◽  
Author(s):  
YOSHIFUMI NISHIO ◽  
YUTA KOMATSU ◽  
YOKO UWATE ◽  
MARTIN HASLER
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Chain Model ◽  
Chaotic Oscillators ◽  
Modeling And Analysis ◽  
Markov Chain Models ◽  
Simulation Results ◽  
Markov Chain Modeling

In this paper, we propose a Markov chain modeling of complicated phenomena observed from coupled chaotic oscillators. Once we obtain the transition probability matrix from computer simulation results, various statistical quantities can be easily calculated from the model. It is shown that various statistical quantities are easily calculated by using the Markov chain model. Various features derived from the Markov chain models of chaotic wandering of synchronization states and switching of clustering states are compared with those obtained from computer simulations of original circuit equations.

Degradation modelling and life expectancy using Markov chain model for carriageway

2018 ◽  
Vol 35 (6) ◽  
pp. 1268-1288 ◽  
Author(s):  
Kong Fah Tee ◽  
Ejiroghene Ekpiwhre ◽  
Zhang Yi
Keyword(s):  
Markov Chain ◽  
Life Expectancy ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Transition Probability ◽  
Condition Assessment ◽  
Cumulative Distribution ◽  
Chain Model ◽  
Content Type ◽  
Pairwise Correlation

PurposeAutomated condition surveys have been recently introduced for condition assessment of highway infrastructures worldwide. Accurate predictions of the current state, median life (ML) and future state of highway infrastructures are crucial for developing appropriate inspection and maintenance strategies for newly created as well as existing aging highway infrastructures. The paper aims to discuss these issues.Design/methodology/approachThis paper proposes Markov Chain based deterioration modelling using a linear transition probability (LTP) matrix method and a median life expectancy (MLE) algorithm. The proposed method is applied and evaluated using condition improvement between the two successive inspections from the Surface Condition Assessment of National Network of Roads survey of the UK Pavement Management System.FindingsThe proposed LTP matrix model utilises better insight than the generic or decoupling linear approach used in estimating transition probabilities formulated in the past. The simulated LTP predicted conditions are portrayed in a deterioration profile and a pairwise correlation. The MLs are computed statistically with a cumulative distribution function plot.Originality/valueThe paper concludes that MLE is ideal for projecting half asset life, and the LTP matrix approach presents a feasible approach for new maintenance regime when more certain deterioration data become available.

Some explicit results for correlated random walks

1989 ◽  
Vol 26 (04) ◽  
pp. 757-766 ◽  
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Equilibrium Distribution ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Correlated Random Walks ◽  
Finite State ◽  
Direction Of Movement ◽  
Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

scholarly journals Comparison between the predicted performance curve and the Markov Chain models for structural performance of infrastructure components

2019 ◽  
Vol 289 ◽  
pp. 08006
Author(s):  
Nabil Semaan ◽  
Youssef Dib
Keyword(s):  
Markov Chain ◽  
Service Life ◽  
Transition Probability ◽  
Structural Performance ◽  
Deterioration Model ◽  
Deterioration Rate ◽  
Markov Chain Models ◽  
Probability Matrices ◽  
Inspection Data ◽  
Very High

This paper compares the PPC model to a Markov Chain (MC) stochastic deterioration model. First, inspection data from the Société de Transport de Montréal (STM) is gathered and analyzed. Then Transition Probability Matrices (TPM) are developed, and, using Matlab, MC deterioration curves are developed. Comparison between MC and the PPC deterioration curves is performed for subway station walls and slabs. The comparison has shown that the useful service life can be as low as 2 years for components having many inspection history records, and very high as 30 years for components having very few inspection history records. The PPC model has always a higher useful service life estimate. Also, the MC has a ten times higher deterioration rate (0.2 per year) compared to the PPC model (0.02 per year). It can be concluded that the MC deterioration model requires a high amount of inspection data, and it is mathematically difficult to generate since most practicing managers and engineers have no background in Markov Chain modeling.

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.

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