scholarly journals A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Iwao Sato
Keyword(s):  
Random Walk ◽  
Zeta Function ◽  
Characteristic Polynomial ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Regular Graphs ◽  
Correlated Random Walk ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph. 

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (1) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

Deriving the transition probability matrix using computational mechanics

2018 ◽  
Vol 35 (2) ◽  
pp. 692-709 ◽  
Author(s):  
Guillermo A. Riveros ◽  
Manuel E. Rosario-Pérez
Keyword(s):  
Computational Mechanics ◽  
Transition Probability ◽  
Steel Structures ◽  
Transition Probability Matrix ◽  
Element Analysis ◽  
Content Type ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Inspection Data ◽  
Condition State

Purpose The combined effects of several complex phenomena cause the deterioration of elements in steel hydraulic structures (SHSs) within the US lock system: corrosion, cracking and fatigue, impact and overloads. Predicting the future condition state of these structures by the use of current condition state inspection data can be achieved through the probabilistic chain deterioration model. The purpose of this study is to derive the transition probability matrix using final elements modeling of a miter gate. Design/methodology/approach If predicted accurately, this information would yield benefits in determining the need for rehabilitation or replacement of SHS. However, because of the complexity and difficulties on obtaining sufficient inspection data, there is a lack of available condition states needed to formulate proper transition probability matrices for each deterioration case. Findings This study focuses on using a three-dimensional explicit finite element analysis (FEM) of a miter gate that has been fully validated with experimental data to derive the transition probability matrix when the loss of flexural capacity in a corroded member is simulated. Practical implications New methodology using computational mechanics to derive the transition probability matrices of navigation steel structures has been presented. Originality/value The difficulty of deriving the transition probability matrix to perform a Markovian analysis increases when limited amount of inspection data is available. The used state of practice FEM to derive the transition probability matrix is not just necessary but also essential when the need for proper maintenance is required but limited amount of the condition of the structural system is unknown.

scholarly journals Transition probability matrix methodology for incremental risk charge

2014 ◽  
Vol 01 (01) ◽  
pp. 1450010 ◽  
Author(s):  
Tzahi Yavin ◽  
Eugene Wang ◽  
Hu Zhang ◽  
Michael A. Clayton
Keyword(s):  
Historical Data ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Generator Matrix ◽  
Construction Methods ◽  
Discretionary Choices ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
One Year

As part of Basel II's incremental risk charge (IRC) methodology, this paper summarizes our extensive investigations of constructing transition probability matrices (TPMs) for unsecuritized credit products in the trading book. The objective is to create monthly or quarterly TPMs with predefined sectors and ratings that are consistent with the bank's Basel PDs. Constructing a TPM is not a unique process. We highlight various aspects of three types of uncertainties embedded in different construction methods: (1) the available historical data and the bank's rating philosophy; (2) the merger of one-year Basel PD and the chosen Moody's TPMs; and (3) deriving a monthly or quarterly TPM when the generator matrix does not exist. Given the fact that TPMs and specifically their PDs are the most important parameters in IRC, it is our view that banks may need to make discretionary choices regarding their methodology, with uncertainties well understood and managed.

On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

2012 ◽  
Vol 2 (4) ◽  
pp. 353-372 ◽  
Author(s):  
Hao Jiang ◽  
Xi Chen ◽  
Yushan Qiu ◽  
Wai-Ki Ching
Keyword(s):  
Maximum Entropy ◽  
Stationary Distribution ◽  
Sparse Matrix ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Efficiency And Effectiveness ◽  
Probabilistic Boolean Networks ◽  
Transition Probability Matrices ◽  
Probability Matrices

Abstract.To understand a genetic regulatory network, two popular mathematical models, Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) have been proposed. Here we address the problem of constructing a sparse Probabilistic Boolean Network (PBN) from a prescribed positive stationary distribution. A sparse matrix is more preferable, as it is easier to study and identify the major components and extract the crucial information hidden in a biological network. The captured network construction problem is both ill-posed and computationally challenging. We present a novel method to construct a sparse transition probability matrix from a given stationary distribution. A series of sparse transition probability matrices can be determined once the stationary distribution is given. By controlling the number of nonzero entries in each column of the transition probability matrix, a desirable sparse transition probability matrix in the sense of maximum entropy can be uniquely constructed as a linear combination of the selected sparse transition probability matrices (a set of sparse irreducible matrices). Numerical examples are given to demonstrate both the efficiency and effectiveness of the proposed method.

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.

Optimal stopping rules for correlated random walks with a discount

2004 ◽  
Vol 41 (2) ◽  
pp. 483-496 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Optimal Stopping ◽  
Constant Factor ◽  
Stopping Rules ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
Optimal Rule ◽  
Correlated Random Walks ◽  
Optimal Stopping Rules

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

Order and Homogeneity of Family Specific Brand-switching Processes

1966 ◽  
Vol 3 (1) ◽  
pp. 48-54 ◽  
Author(s):  
William F. Massy
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Transition Probability ◽  
Empirical Work ◽  
Brand Choice ◽  
First Order ◽  
Brand Switching ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Crucial Assumption

Most empirical work on Markov processes for brand choice has been based on aggregative data. This article explores the validity of the crucial assumption that underlies such analyses, i.e., that all the families in the sample follow a Markov process with the same or similar transition probability matrices. The results show that there is a great deal of diversity among families’ switching processes, and that many of them are of zero rather than first order.

Return probabilities for correlated random walks

1986 ◽  
Vol 23 (1) ◽  
pp. 201-207
Author(s):  
Gillian Iossif
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Integer Lattice ◽  
Asymptotic Estimates ◽  
Correlated Random Walk ◽  
Correlated Random Walks ◽  
Previous Step

A correlated random walk on a d-dimensional integer lattice is studied in which, at any stage, the probabilities of the next step being in the various possible directions depend on the direction of the previous step. Using a renewal argument, asymptotic estimates are obtained for the probability of return to the origin after n steps.

scholarly journals 4525 Estimates of Dose Response using the Dixon Up-and-Down Method and BOIN study designs

2020 ◽  
Vol 4 (s1) ◽  
pp. 48-48
Author(s):  
Charles Gene Minard
Keyword(s):  
Decision Making ◽  
Standard Deviation ◽  
Effective Dose ◽  
Transition Probability ◽  
Experimental Unit ◽  
The Mean ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Study Designs ◽  
Dose Levels

OBJECTIVES/GOALS: The Dixon up-and-down method (U/D), original developed for testing explosives, is especially common in anesthesia research studies. The objective of this research is to compare the performance of the U/D method for obtaining and analyzing sensitivity data with that of the Bayesian Optimal Interval (BOIN) method. METHODS/STUDY POPULATION: A simulation study will compare the performance of the U/D method with the BOIN design. The two study designs offer alternative decision-making algorithms with respect to the dose at which the next experimental unit is treated. These alternative decisions may impact the precision of point estimates of the mean and standard deviation of the effective dose to elicit a response. Transition probability matrices are developed, and maximum likelihood estimates of the unknown parameters assessed for accuracy. For simulation, the effective dose is assumed to be randomly distributed with a known mean and standard deviation. Fixed dose levels are defined, and decisions for what level the next experimental unit should be treated at are defined by the Dixon up-and-down method and the BOIN design. For the U/D method, the stimulus is increased by one level in the absence of a response or decreased if a response occurs from an initial stimulus. A target toxicity probability of 0.50 is used to define the dose escalation or de-escalation rules for the application of the BOIN design. RESULTS/ANTICIPATED RESULTS: A feature of both methods is that the consecutive observations are concentrated about the mean value of the effective dose. However, the BOIN design tends to be more concentrated between these two dose levels. In the presence of severe adverse events, the BOIN design can choose to eliminate doses that are too toxic whereas the U/D design cannot eliminate any dose levels. Transition probability matrices are defined and parameters for the distribution of the effective dose are estimated using maximum likelihood estimation. Mean squared errors for the estimated mean and standard deviations compare the two study designs. DISCUSSION/SIGNIFICANCE OF IMPACT: The BOIN design offers an alternative method for decision-making compared with the U/D method. The BOIN design tends to concentrate dose levels about the mean more than the U/D. This may provide better estimates of the mean and standard deviation of the effective dose for eliciting a response in some circumstances.

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