scholarly journals High-order Vector Markov Chain with Partial Connections in Data Analysis

2017 ◽  
Vol 46 (3-4) ◽  
pp. 37-45 ◽  
Author(s):  
Yuriy Kharin ◽  
Michail Maltsew
Keyword(s):  
Mathematical Model ◽  
Time Series ◽  
Markov Chain ◽  
Data Analysis ◽  
Probability Distribution ◽  
Discrete Time ◽  
Model Parameters ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Conditional Probability Distribution

A new mathematical model for discrete time series is proposed: homogenous vector Markov chain of the order s with partial connections. Conditional probability distribution for this model is determined only by a few components of previous vector states. Probabilistic properties of the model are given: ergodicity conditions and conditions under which the stationary probability distribution is uniform. Consistent statistical estimators for model parameters are constructed.

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (03) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (3) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

scholarly journals Recurrence time distribution and temporal clustering properties of a cellular automaton modelling landslide events

2013 ◽  
Vol 20 (6) ◽  
pp. 1071-1078 ◽  
Author(s):  
E. Piegari ◽  
R. Di Maio ◽  
A. Avella
Keyword(s):  
Time Series ◽  
Probability Distribution ◽  
Cellular Automaton ◽  
Time Window ◽  
Fano Factor ◽  
Recent Analysis ◽  
Recurrence Time ◽  
Model Parameters ◽  
Reasonable Prediction ◽  
First Time

Abstract. Reasonable prediction of landslide occurrences in a given area requires the choice of an appropriate probability distribution of recurrence time intervals. Although landslides are widespread and frequent in many parts of the world, complete databases of landslide occurrences over large periods are missing and often such natural disasters are treated as processes uncorrelated in time and, therefore, Poisson distributed. In this paper, we examine the recurrence time statistics of landslide events simulated by a cellular automaton model that reproduces well the actual frequency-size statistics of landslide catalogues. The complex time series are analysed by varying both the threshold above which the time between events is recorded and the values of the key model parameters. The synthetic recurrence time probability distribution is shown to be strongly dependent on the rate at which instability is approached, providing a smooth crossover from a power-law regime to a Weibull regime. Moreover, a Fano factor analysis shows a clear indication of different degrees of correlation in landslide time series. Such a finding supports, at least in part, a recent analysis performed for the first time of an historical landslide time series over a time window of fifty years.

scholarly journals Two-Sex Branching Processes with Offspring and Mating in a Random Environment

2009 ◽  
Vol 46 (04) ◽  
pp. 993-1004
Author(s):  
S. Ma ◽  
M. Molina
Keyword(s):  
Mathematical Model ◽  
Probability Distribution ◽  
Discrete Time ◽  
Random Environment ◽  
Generating Functions ◽  
Branching Processes ◽  
Random Vectors ◽  
Environmental Process ◽  
Mating Function ◽  
The Mathematical Model

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

STATISTICAL PROPERTIES OF INTENSITY FLUCTUATION OF SATURATION LASER MODEL DRIVEN BY CROSS-CORRELATED ADDITIVE AND MULTIPLICATIVE NOISES

2010 ◽  
Vol 24 (14) ◽  
pp. 2175-2188 ◽  
Author(s):  
PING ZHU ◽  
YI JIE ZHU
Keyword(s):  
Probability Distribution ◽  
Cross Correlation ◽  
Laser Intensity ◽  
Laser System ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Model Driven ◽  
Multiplicative Noises ◽  
Laser Model ◽  
The Stability

Statistical properties of the intensity fluctuation of a saturation laser model driven by cross-correlation additive and multiplicative noises are investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution Pst(I), the relaxation time Tc, and the normalized variance λ2(0) of the system. By numerical computation, we discussed the effects of the cross-correlation strength λ, the cross-correlation time τ, the quantum noise intensity D, and the pump noise intensity Q for the fluctuation of the laser intensity. Above the threshold, λ weakens the stationary probability distribution, speeds up the startup velocity of the laser system from start status to steady work, and attenuates the stability of laser intensity output; however, τ strengthens the stationary probability distribution and strengths the stability of laser intensity output; when λ < 0, τ speeds up the startup; on the contrast, when λ > 0, τ slows down the startup. D and Q make the relaxation time exhibit extremum structure, that is, the startup time possesses the least values. At the threshold, τ cannot generate the effects for the saturation laser system, λ expedites the startup velocity and weakens the stability of laser intensity output. Below threshold, the effects of λ and τ not only relate to λ and τ, but also relate to other parameters of the system.

scholarly journals Statistical Mechanics of Unconfined Systems: Challenges and Lessons

2021 ◽  
Vol 3 (1) ◽  
pp. 8
Author(s):  
Bruno Arderucio Costa ◽  
Pedro Pessoa
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
A Priori ◽  
Ideal Gas ◽  
Stationary Probability ◽  
De Sitter ◽  
Stationary Probability Distribution ◽  
Local Measurements ◽  
Priori Information

Motivated by applications of statistical mechanics in which the system of interest is spatially unconfined, we present an exact solution to the maximum entropy problem for assigning a stationary probability distribution on the phase space of an unconfined ideal gas in an anti-de Sitter background. Notwithstanding the gas’ freedom to move in an infinite volume, we establish necessary conditions for the stationary probability distribution solving a general maximum entropy problem to be normalizable and obtain the resulting probability for a particular choice of constraints. As a part of our analysis, we develop a novel method for identifying dynamical constraints based on local measurements. With no appeal to a priori information about globally defined conserved quantities, it is therefore applicable to a much wider range of problems.

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