Random walks on a dodecahedron

1980 ◽  
Vol 17 (02) ◽  
pp. 373-384 ◽  
Author(s):  
G. Letac ◽  
L. Takács
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Regular Dodecahedron

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn +1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X 0 = A] and P[In = A | I 0 = A]. The methods used are applicable to other solids.

Random walks on a dodecahedron

1980 ◽  
Vol 17 (2) ◽  
pp. 373-384 ◽  
Author(s):  
G. Letac ◽  
L. Takács
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Regular Dodecahedron

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn+1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X0 = A] and P[In = A | I0 = A]. The methods used are applicable to other solids.

scholarly journals Random shuffles on trees using extended promotion

2019 ◽  
Vol 29 (03) ◽  
pp. 561-580
Author(s):  
Svetlana Poznanović ◽  
Kara Stasikelis
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Matrix ◽  
Transition Probabilities ◽  
Minimal Ideal ◽  
The Self ◽  
Linear Extensions ◽  
Rooted Forest ◽  
Over Time ◽  
Self Organizing

The Tsetlin library is a very well-studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. In this paper, we consider a generalization which can be interpreted as a self-organizing library in which the arrangements of books on each shelf are restricted to be linear extensions of a fixed poset. The moves on the books are given by the extended promotion operators of Ayyer, Klee and Schilling while the shelves, bookcases, etc. evolve according to the move-to-back moves as in the the self-organizing library of Björner. We show that the eigenvalues of the transition matrix of this Markov chain are [Formula: see text] integer combinations of the transition probabilities if the posets that prescribe the restrictions on the book arrangements are rooted forests or more generally, if they consist of ordinal sums of a rooted forest and so called ladders. For some of the results, we show that the monoids generated by the moves are either [Formula: see text]-trivial or, more generally, in [Formula: see text] and then we use the theory of left random walks on the minimal ideal of such monoids to find the eigenvalues. Moreover, in order to give a combinatorial description of the eigenvalues in the more general case, we relate the eigenvalues when the restrictions on the book arrangements change only by allowing for one additional transposition of two fixed books.

Lattice-valued random walks with Markov chain dependent steps

1979 ◽  
Vol 86 (1) ◽  
pp. 115-126 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Three Dimensional ◽  
Mean Square Displacement ◽  
Building Model ◽  
Finite State ◽  
The Mean ◽  
Regenerative Phenomenon ◽  
Finite State Space

AbstractThe probability of ever returning to the origin and the mean square displacement after n steps are studied for some lattice-valued random walks, whose successive steps constitute a Markov chain on a finite state space with transition probabilities of a simple kind, and such that the returns to the origin form a regenerative phenomenon. The case of walks on a diamond lattice with no immediate reversals is included: this example is relevant as a polymer chain building model. The numerical evaluation of the return probabilities of some three-dimensional walks is discussed and examples given.

Data Mining Technique Applied To DNA Sequencing

2015 ◽  
Vol 2 (7) ◽  
pp. 18
Author(s):  
R. Jamuna
Keyword(s):  
Data Mining ◽  
Dna Methylation ◽  
Markov Chain ◽  
Human Genome ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Effective Means ◽  
Cpg Islands ◽  
Chain Model ◽  
The Given

CpG islands (CGIs) play a vital role in genome analysis as genomic markers.  Identification of the CpG pair has contributed not only to the prediction of promoters but also to the understanding of the epigenetic causes of cancer. In the human genome [1] wherever the dinucleotides CG occurs the C nucleotide (cytosine) undergoes chemical modifications. There is a relatively high probability of this modification that mutates C into a T. For biologically important reasons the mutation modification process is suppressed in short stretches of the genome, such as ‘start’ regions. In these regions [2] predominant CpG dinucleotides are found than elsewhere. Such regions are called CpG islands. DNA methylation is an effective means by which gene expression is silenced. In normal cells, DNA methylation functions to prevent the expression of imprinted and inactive X chromosome genes. In cancerous cells, DNA methylation inactivates tumor-suppressor genes, as well as DNA repair genes, can disrupt cell-cycle regulation. The most current methods for identifying CGIs suffered from various limitations and involved a lot of human interventions. This paper gives an easy searching technique with data mining of Markov Chain in genes. Markov chain model has been applied to study the probability of occurrence of C-G pair in the given   gene sequence. Maximum Likelihood estimators for the transition probabilities for each model and analgously for the  model has been developed and log odds ratio that is calculated estimates the presence or absence of CpG is lands in the given gene which brings in many  facts for the cancer detection in human genome.

scholarly journals Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 729
Author(s):  
Miquel Montero
Keyword(s):  
Random Walks ◽  
Stereographic Projection ◽  
Transition Probabilities ◽  
Simple Random Walk ◽  
General Introduction ◽  
Elliptic Case ◽  
One Step ◽  
Underlying Geometry ◽  
Step Transition ◽  
Wide Family

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

scholarly journals Calibration of Transition Intensities for a Multistate Model: Application to Long-Term Care

Risks ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 37
Author(s):  
Manuel L. Esquível ◽  
Gracinda R. Guerreiro ◽  
Matilde C. Oliveira ◽  
Pedro Corte Real
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Long Term Care ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Chain Model ◽  
Chain Process ◽  
Term Care ◽  
National Network

We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.

scholarly journals A Bayesian model for binary Markov chains

2004 ◽  
Vol 2004 (8) ◽  
pp. 421-429 ◽  
Author(s):  
Souad Assoudou ◽  
Belkheir Essebbar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Bayesian Estimation ◽  
Bayesian Model ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Jeffreys Prior ◽  
Data Set

This note is concerned with Bayesian estimation of the transition probabilities of a binary Markov chain observed from heterogeneous individuals. The model is founded on the Jeffreys' prior which allows for transition probabilities to be correlated. The Bayesian estimator is approximated by means of Monte Carlo Markov chain (MCMC) techniques. The performance of the Bayesian estimates is illustrated by analyzing a small simulated data set.

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.

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