The censored Markov chain and the best augmentation

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.

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A Comparison of Alternative Approaches for the Synthetic Generation of a Wind Speed Time Series

Journal of Solar Energy Engineering ◽

10.1115/1.2929974 ◽

1991 ◽

Vol 113 (4) ◽

pp. 280-289 ◽

Cited By ~ 31

Author(s):

F. C. Kaminsky ◽

R. H. Kirchhoff ◽

C. Y. Syu ◽

J. F. Manwell

Keyword(s):

Time Series ◽

Markov Chain ◽

Wind Speed ◽

Probability Distribution ◽

Transition Probability ◽

Transition Probability Matrix ◽

Statistical Properties ◽

Embedded Markov Chain ◽

One Step ◽

Alternative Approaches

In this paper, alternative approaches for synthetically generating a wind speed time series are discussed. These approaches include: (1) the use of independent values from a specific probability distribution; (2) the use of an algorithm based on the statistical behavior of a one-step Markov chain; (3) the use of an algorithm based on the behavior of a transition probability matrix that describes the next wind speed value statistically as a function of the current wind speed value and the previous wind speed value; (4) the use of Box-Jenkins models; (5) the use of the Shinozuka algorithm; and (6) the use of an embedded Markov chain. The ability of each approach to capture the statistical properties of the desired wind speed time series is discussed. In this context the statistical properties of interest are the probability distribution of the wind speed values, the autocorrelation function of the wind speed values, and the spectral density of the wind speed values.

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On the Convergence Rate of Bonus-Malus Systems

Astin Bulletin ◽

10.2143/ast.22.2.2005116 ◽

1992 ◽

Vol 22 (2) ◽

pp. 217-223 ◽

Cited By ~ 6

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.

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Finite dams with inputs forming a Markov chain

Journal of Applied Probability ◽

10.1017/s0021900200034884 ◽

1970 ◽

Vol 7 (02) ◽

pp. 291-303 ◽

Cited By ~ 5

Author(s):

M.S. Ali Khan

Keyword(s):

Markov Chain ◽

Probability Distribution ◽

Generating Functions ◽

State Transition ◽

Transition Probability ◽

Transition Probability Matrix ◽

Time Dependent ◽

Probability Generating Functions ◽

State Transition Probability ◽

Finite Dam

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

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Finite dams with inputs forming a Markov chain

Journal of Applied Probability ◽

10.2307/3211965 ◽

1970 ◽

Vol 7 (2) ◽

pp. 291-303 ◽

Cited By ~ 13

Author(s):

M.S. Ali Khan

Keyword(s):

Markov Chain ◽

Probability Distribution ◽

Generating Functions ◽

State Transition ◽

Transition Probability ◽

Transition Probability Matrix ◽

Time Dependent ◽

Probability Generating Functions ◽

State Transition Probability ◽

Finite Dam

This paper considers a finite dam fed by inputs forming a Markov chain. Relations for the probability of first emptiness before overflow and with overflow are obtained and their probability generating functions are derived; expressions are obtained in the case of a three state transition probability matrix. An equation for the probability that the dam ever dries up before overflow is derived and it is shown that the ratio of these probabilities is independent of the size of the dam. A time dependent formula for the probability distribution of the dam content is also obtained.

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A note on cumulative sums of markovian variables

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700026847 ◽

1965 ◽

Vol 5 (2) ◽

pp. 285-287 ◽

Cited By ~ 4

Author(s):

R. M. Phatarfod

Keyword(s):

Markov Chain ◽

Probability Distribution ◽

Finite Number ◽

Transition Probability ◽

Transition Probability Matrix ◽

The Matrix ◽

Initial Probability Distribution ◽

Regular Markov Chain ◽

Image Position ◽

Cumulative Sums

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .

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

Austrian Journal of Statistics ◽

10.17713/ajs.v46i3-4.669 ◽

2017 ◽

Vol 46 (3-4) ◽

pp. 37-45 ◽

Cited By ~ 1

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.

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Computer aided solving the high-order transition probability matrix of the finite Markov chain

Applied Mathematics and Computation ◽

10.1016/j.amc.2005.02.003 ◽

2006 ◽

Vol 172 (1) ◽

pp. 267-285 ◽

Cited By ~ 5

Author(s):

Zhenqing Li ◽

Weiming Wang

Keyword(s):

Markov Chain ◽

Transition Probability ◽

Transition Probability Matrix ◽

High Order ◽

Order Transition ◽

Finite Markov Chain ◽

Computer Aided

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

International Journal of Modern Physics B ◽

10.1142/s0217979210055755 ◽

2010 ◽

Vol 24 (14) ◽

pp. 2175-2188 ◽

Cited By ~ 3

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.

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