The inverse problem in reducible Markov chains

1976 ◽  
Vol 13 (01) ◽  
pp. 49-56 ◽  
Author(s):  
W. D. Ray ◽  
F. Margo
Keyword(s):  
Inverse Problem ◽  
Markov Chain ◽  
Equilibrium Distribution ◽  
Probability Distributions ◽  
A Priori ◽  
Feasible Region ◽  
Numerical Examples ◽  
Transient States ◽  
Equilibrium Probability ◽  
Absorbing States

The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.

The inverse problem in reducible Markov chains

1976 ◽  
Vol 13 (1) ◽  
pp. 49-56 ◽  
Author(s):  
W. D. Ray ◽  
F. Margo
Keyword(s):  
Inverse Problem ◽  
Markov Chain ◽  
Equilibrium Distribution ◽  
Probability Distributions ◽  
A Priori ◽  
Feasible Region ◽  
Numerical Examples ◽  
Transient States ◽  
Equilibrium Probability ◽  
Absorbing States

The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.

A note on the inverse problem for reducible Markov chains

1977 ◽  
Vol 14 (03) ◽  
pp. 621-625
Author(s):  
A. O. Pittenger
Keyword(s):  
Inverse Problem ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Physical Process ◽  
Raw Materials ◽  
Transition Probability ◽  
Linear Inequalities ◽  
System Of Linear Inequalities ◽  
Absorbing States

Suppose a physical process is modelled by a Markov chain with transition probability on S 1 ∪ S 2, S 1 denoting the transient states and S 2 a set of absorbing states. If v denotes the output distribution on S 2, the question arises as to what input distributions (of raw materials) on S 1 produce v. In this note we give an alternative to the formulation of Ray and Margo [2] and reduce the problem to one system of linear inequalities. An application to random walk is given and the equiprobability case examined in detail.

scholarly journals Markov Chain Monte-Carlo Models of Starburst Clusters

2014 ◽  
Vol 9 (S307) ◽  
pp. 123-124
Author(s):  
Jorge Melnick
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Statistical Approach ◽  
Probability Distributions ◽  
A Priori ◽  
Stochastic Effects ◽  
Near Ir ◽  
Ir Photometry ◽  
The Imf

AbstractThere are a number of stochastic effects that must be considered when comparing models to observations of starburst clusters: the IMF is never fully populated; the stars can never be strictly coeval; stars rotate and their photometric properties depend on orientation; a significant fraction of massive stars are in interacting binaries; and the extinction varies from star to star. The probability distributions of each of these effects are nota prioriknown, but must be extracted from the observations. Markov Chain Monte-Carlo methods appear to provide the best statistical approach. Here I present an example of stochastic age effects upon the upper mass limit of the IMF of the Arches cluster as derived from near-IR photometry.

A note on the inverse problem for reducible Markov chains

1977 ◽  
Vol 14 (3) ◽  
pp. 621-625 ◽  
Author(s):  
A. O. Pittenger
Keyword(s):  
Inverse Problem ◽  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Physical Process ◽  
Raw Materials ◽  
Transition Probability ◽  
Linear Inequalities ◽  
System Of Linear Inequalities ◽  
Absorbing States

Suppose a physical process is modelled by a Markov chain with transition probability on S1 ∪ S2, S1 denoting the transient states and S2 a set of absorbing states.If v denotes the output distribution on S2, the question arises as to what input distributions (of raw materials) on S1 produce v. In this note we give an alternative to the formulation of Ray and Margo [2] and reduce the problem to one system of linear inequalities. An application to random walk is given and the equiprobability case examined in detail.

Iterative Bounds on the Equilibrium Distribution of a Finite Markov Chain

1987 ◽  
Vol 1 (1) ◽  
pp. 117-131 ◽  
Author(s):  
Jan Van Der Wal ◽  
Paul J. Schweitzer
Keyword(s):  
Markov Chain ◽  
Iterative Method ◽  
Lower Bounds ◽  
Equilibrium Distribution ◽  
Upper And Lower Bounds ◽  
Finite Markov Chain ◽  
Expected Number ◽  
Numerical Examples ◽  
Number Of Visits ◽  
New Iterative Method

This article presents a new iterative method for computing the equilibrium distribution of a finite Markov chain, which has the significant advantage of providing good upper and lower bounds for the equilibrium probabilities. The method approximates the expected number of visits to each state between two successive visits to a given reference state. Numerical examples indicate that the performance of this method is quite good.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

scholarly journals A Consistent Track-to-Track Fusion Method Based on Copula Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Kelin Lu ◽  
K. C. Chang ◽  
Rui Zhou
Keyword(s):  
Probability Distributions ◽  
A Priori ◽  
Joint Probability ◽  
Network Connectivity ◽  
Consistent Estimate ◽  
Fusion Algorithm ◽  
Copula Theory ◽  
Wide Range ◽  
Primary Advantage ◽  
Joint Probability Distributions

This paper addresses the problem of distributed fusion when the conditional independence assumptions on sensor measurements or local estimates are not met. A new data fusion algorithm called Copula fusion is presented. The proposed method is grounded on Copula statistical modeling and Bayesian analysis. The primary advantage of the Copula-based methodology is that it could reveal the unknown correlation that allows one to build joint probability distributions with potentially arbitrary underlying marginals and a desired intermodal dependence. The proposed fusion algorithm requires no a priori knowledge of communications patterns or network connectivity. The simulation results show that the Copula fusion brings a consistent estimate for a wide range of process noises.

A QUEUE WITH POSITIVE AND NEGATIVE ARRIVALS GOVERNED BY A MARKOV CHAIN

2003 ◽  
Vol 17 (4) ◽  
pp. 487-501 ◽  
Author(s):  
Yang Woo Shin ◽  
Bong Dae Choi
Keyword(s):  
Markov Chain ◽  
Transient State ◽  
Single Server ◽  
Negative Customers ◽  
Service Distribution ◽  
Finite State ◽  
Sojourn Time Distribution ◽  
Absorbing States ◽  
Negative Arrivals ◽  
Selection Of

We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.

scholarly journals Inverse problem for fractional diffusion equation

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Vu Tuan
Keyword(s):  
Inverse Problem ◽  
Diffusion Coefficient ◽  
Lower Bound ◽  
Diffusion Equation ◽  
Spectral Data ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
One Dimensional ◽  
Initial Distributions

AbstractWe prove that by taking suitable initial distributions only finitely many measurements on the boundary are required to recover uniquely the diffusion coefficient of a one dimensional fractional diffusion equation. If a lower bound on the diffusion coefficient is known a priori then even only two measurements are sufficient. The technique is based on possibility of extracting the full boundary spectral data from special lateral measurements.

Dolomitization geometry and reservoir quality from supervised Bayesian classification and probabilistic neural networks: Midland Basin Leonardian Wichita and Clear Fork Formations

2021 ◽  
Vol 9 (2) ◽  
pp. T585-T598
Author(s):  
Abidin B. Caf ◽  
John D. Pigott
Keyword(s):  
Neural Networks ◽  
Acoustic Impedance ◽  
Probability Distributions ◽  
A Priori ◽  
Bayesian Classification ◽  
Reservoir Quality ◽  
3D Seismic ◽  
Probabilistic Neural Networks ◽  
Neutron Porosity ◽  
Seismic Scale

Extensive dolomitization is prevalent in the platform and periplatform carbonates in the Lower-Middle Permian strata in the Midland and greater Permian Basin. Early workers have found that the platform and shelf-top carbonates were dolomitized, whereas slope and basinal carbonates remained calcitic, proposing a reflux dolomitization model as the possible diagenetic mechanism. More importantly, they underline that this dolomitization pattern controls the porosity and forms an updip seal. These studies are predominately conducted using well logs, cores, and outcrop analogs, and although exhibiting high resolution vertically, such determinations are laterally sparse. We have used supervised Bayesian classification and probabilistic neural networks (PNN) on a 3D seismic volume to create an estimation of the most probable distribution of dolomite and limestone within a subsurface 3D volume petrophysically constrained. Combining this lithologic information with porosity, we then illuminate the diagenetic effects on a seismic scale. We started our workflow by deriving lithology classifications from well-log crossplots of neutron porosity and acoustic impedance to determine the a priori proportions of the lithology and the probability density functions calculation for each lithology type. Then, we applied these probability distributions and a priori proportions to 3D seismic volumes of the acoustic impedance and predicted neutron porosity volume to create a lithology volume and probability volumes for each lithology type. The acoustic impedance volume was obtained by model-based poststack inversion, and the neutron porosity volume was obtained by the PNN. Our results best supported a regional reflux dolomitization model, in which the porosity is increasing from shelf to slope while the dolomitization is decreasing, but with sea-level forcing. With this study, we determined that diagenesis and the corresponding reservoir quality in these platforms and periplatform strata can be directly imaged and mapped on a seismic scale by quantitative seismic interpretation and supervised classification methods.

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