scholarly journals The Predictive Power of Transition Matrices

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2096
Author(s):  
André Berchtold
Keyword(s):  
Markov Chain ◽  
Transition Matrix ◽  
Predictive Power ◽  
Distribution Model ◽  
Transition Matrices ◽  
Association Measure ◽  
New Class ◽  
Transition Distribution ◽  
Respective Contribution ◽  
Available Information

When working with Markov chains, especially if they are of order greater than one, it is often necessary to evaluate the respective contribution of each lag of the variable under study on the present. This is particularly true when using the Mixture Transition Distribution model to approximate the true fully parameterized Markov chain. Even if it is possible to evaluate each transition matrix using a standard association measure, these measures do not allow taking into account all the available information. Therefore, in this paper, we introduce a new class of so-called "predictive power" measures for transition matrices. These measures address the shortcomings of traditional association measures, so as to allow better estimation of high-order models.

ACCELERATING MONTE CARLO MARKOV PROCESSES

COSMOS ◽  
2005 ◽  
Vol 01 (01) ◽  
pp. 87-94 ◽  
Author(s):  
CHII-RUEY HWANG
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Process ◽  
Markov Processes ◽  
Transition Matrix ◽  
Asymptotic Variance ◽  
Suitable Condition ◽  
Transition Matrices ◽  
Finite Set ◽  
The Continuum

Let π be a probability density proportional to exp - U(x) in S. A convergent Markov process to π(x) may be regarded as a "conceptual" algorithm. Assume that S is a finite set. Let X0,X1,…,Xn,… be a Markov chain with transition matrix P and invariant probability π. Under suitable condition on P, it is known that [Formula: see text] converges to π(f) and the corresponding asymptotic variance v(f, P) depends only on f and P. It is natural to consider criteria vw(P) and va(P), defined respectively by maximizing and averaging v(f, P) over f. Two families of transition matrices are considered. There are four problems to be investigated. Some results and conjectures are given. As for the continuum case, to accelerate the convergence a family of diffusions with drift ∇U(x) + C(x) with div(C(x)exp - U(x)) = 0 is considered.

Credit Transition and Structural Shocks

2021 ◽  
Author(s):  
Camilla Ferretti ◽  
Giampaolo Gabbi ◽  
Piero Ganugi ◽  
Pietro Vozzella
Keyword(s):  
Markov Chain ◽  
Credit Risk ◽  
Transaction Costs ◽  
Adverse Selection ◽  
Transition Matrix ◽  
Transition Matrices ◽  
Structural Shocks ◽  
Long Run ◽  
Economic Downturns ◽  
Lending Behavior

Credit risk involves not only the complexity of screening but also monitoring and estimating rating transition. The adoption of inadequate transition matrices causes a misevaluation of credit risk, a consequent misallocation of capital, with the prospect that the lending process will be affected by increasing transaction costs and limited rationality, especially after a shock. Comparing the mover–stayer and the Markov chain approaches to estimate the SME rating transition matrix, we find that the risk of a structural credit shock imposes flexible estimates not constrained by the long-run trajectory of borrowers. Improved migration estimation mitigates adverse selection in banks’ lending behavior. This conclusion is particularly true during economic downturns with the consequence of reducing the cyclicality and empowering the resilience of banks.

scholarly journals Markov chains, ${\mathscr R}$-trivial monoids and representation theory

2015 ◽  
Vol 25 (01n02) ◽  
pp. 169-231 ◽  
Author(s):  
Arvind Ayyer ◽  
Anne Schilling ◽  
Benjamin Steinberg ◽  
Nicolas M. Thiéry
Keyword(s):  
Markov Chain ◽  
General Theory ◽  
Markov Chains ◽  
Random Walks ◽  
Transition Matrix ◽  
Coxeter Groups ◽  
Mixing Time ◽  
New Class ◽  
Free Tree ◽  
Sandpile Models

We develop a general theory of Markov chains realizable as random walks on [Formula: see text]-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom–Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

A New Method for Modeling the Behavior of Finite Population Evolutionary Algorithms

2010 ◽  
Vol 18 (3) ◽  
pp. 451-489 ◽  
Author(s):  
Tatsuya Motoki
Keyword(s):  
Markov Chain ◽  
Evolutionary Algorithms ◽  
Finite Population ◽  
Fitness Landscape ◽  
Markov Chain Model ◽  
Search Space ◽  
Chain Model ◽  
Transition Matrices ◽  
Memory Space ◽  
Exact Model

As practitioners we are interested in the likelihood of the population containing a copy of the optimum. The dynamic systems approach, however, does not help us to calculate that quantity. Markov chain analysis can be used in principle to calculate the quantity. However, since the associated transition matrices are enormous even for modest problems, it follows that in practice these calculations are usually computationally infeasible. Therefore, some improvements on this situation are desirable. In this paper, we present a method for modeling the behavior of finite population evolutionary algorithms (EAs), and show that if the population size is greater than 1 and much less than the cardinality of the search space, the resulting exact model requires considerably less memory space for theoretically running the stochastic search process of the original EA than the Nix and Vose-style Markov chain model. We also present some approximate models that use still less memory space than the exact model. Furthermore, based on our models, we examine the selection pressure by fitness-proportionate selection, and observe that on average over all population trajectories, there is no such strong bias toward selecting the higher fitness individuals as the fitness landscape suggests.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

scholarly journals Uncertainty-Aware Dissimilarity Measures for Interval-Valued Fuzzy Sets

2020 ◽  
Vol 28 (05) ◽  
pp. 757-768
Author(s):  
Emilio Torres-Manzanera ◽  
Pavol Král ◽  
Vladimír Janiš ◽  
Susana Montes
Keyword(s):  
Fuzzy Sets ◽  
Dissimilarity Measures ◽  
New Class ◽  
Available Information ◽  
Interval Valued

Dissimilarities are a very usual way to compare two fuzzy sets and also two interval-valued fuzzy sets. In both cases, the dissimilarity between two sets is a number. In this work, we introduce a generalization of the notion of dissimilarity for interval-valued fuzzy sets such that it assumes values on the set of subintervals instead of the set of numbers. This seems to be more realistic taking into account the available information. We also investigate its relationship with the classical notions of dissimilarity between fuzzy sets and we obtain that the new class is richer than the existing one.

Determining Equivalent Resistance and Nodal Potentials of a Resistive Circuit Using a Reduced Transition Matrix

2020 ◽  
Vol 02 (01) ◽  
pp. 2050004
Author(s):  
Je-Young Choi
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Transition Matrix ◽  
Present Approach ◽  
Voltage Source ◽  
Equivalent Resistance ◽  
Electrical Circuits ◽  
Electric Potentials ◽  
Dominant Eigenvector

Several methods have been developed in order to solve electrical circuits consisting of resistors and an ideal voltage source. A correspondence with random walks avoids difficulties caused by choosing directions of currents and signs in potential differences. Starting from the random-walk method, we introduce a reduced transition matrix of the associated Markov chain whose dominant eigenvector alone determines the electric potentials at all nodes of the circuit and the equivalent resistance between the nodes connected to the terminals of the voltage source. Various means to find the eigenvector are developed from its definition. A few example circuits are solved in order to show the usefulness of the present approach.

scholarly journals A note on repeated sequences in Markov chains

1987 ◽  
Vol 19 (03) ◽  
pp. 739-742 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Transition Matrix ◽  
Repeated Sequences ◽  
Markov Renewal Process ◽  
Conditional Mean ◽  
Sojourn Times

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

scholarly journals Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

2019 ◽  
Vol 71 (6) ◽  
pp. 1351-1366
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Representation Theory ◽  
Functional Equations ◽  
Transition Matrix ◽  
Principal Series ◽  
Surprising Result ◽  
Transition Matrices ◽  
Natural Basis ◽  
Series Representations ◽  
The Matrix ◽  
Lusztig Polynomials

AbstractA problem in representation theory of $p$-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

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