History of Stochastic and Probability Modelling - Coursework Example

History of stochastic and probability modelling Name Institution Instructor Subject Date Abstract Stochastic process gives the nature in which the random sequence may be arranged. This is in terms of time and random elements. For instance, in a scenario where  could give a representation of a process which is stochastic. This gives the probable capacity of predicting the future instance at a given unpredictable observable outcome. This gives a wholesome technique in which the nature of measurement is done, data emerged from longitudinal, and giving a wide range of opportunities that would be used in modelling. This model focuses on the discrete time, latent transition and the Markov mixtures. This paper elucidates the Markov chain model history, development and how it is used in stochastic and probability modelling. Introduction The use of statistical models is essential in the study of the qualitative status that finds its application in the Markov chain model as noted by Asparouhov & Muthen (2008). This model was put forward by Andrei Markov in 1878. Markov model utilizes the capability of ascertaining the independent of the given past provided that, the current process is known. This model utilizes the variables that have been recorded. Additionally, in engineering field, the use of stochastic aids in describing how a given system operates given certain duration of time. This means that the random variables would be represented by the failure of a given component and the number of times it has taken to repair. The method Markov chain has been in existence for as long, as the Techniques of Monte Carlo have been in use. The impacts of these techniques on statistics have not been really felt until the beginning of 1990 (Collins & Flaherty, 2002). The impacts have, however, been recognized and used in the field of image analysis as well as spatial statistics that are that is specialized. The Introduction of techniques that are based on Markov chain in Physics is a concept that is still not yet explained in the course of several researches and studies that have continue to take place. A clear consideration of Markov chain techniques, with extended references can be easily determined. The distinction between the emergence algorithms that are fond on the basis of Metropolis–Hastings and which are also related to the sampling if Gibbs, by virtue of the fact that each one of them emerges from a different origin that is radical, even though their justification through the concept of Markov chain is the same. The number of the variables is mostly classified in various classes that rely in the geospatial techniques. The application of 2-dimensional Markov in the simulation process puts together the interclass properties. As noted by (Kaplan, 2000)The developments of the Markov’s in stochastic technique was championed by the development of the functional safety that was required in the electronic field that required the use of programmable systems. This could give the possibility of failure modes so as to ensure safety aspect is adhered. Getting information concerning the developments and improvements in the techniques of Monte Carlo, it is possible to involve the revolution of Markov chain theory which is of the second generation. Begging from early to late 1990s, there occurred the improvements and developments of perfect sampling and reversible jumps, particle filters, progressive Monte Carlo, population as well as computation of standard errors. As has already been explained above, the knowledge and understanding that the concept of Markov chains could be applied in a large variety of circumstances only came to those applying statistics at the mainstream. This happened irrespective publications which were earlier made in the literature of statistics. A number of reasons could be specified which include the inadequacy of machinery used in computing, or information on the Markov chain theory, or delays in trusting the practicality of the technique. It therefore needs committed researchers to be able to provide convincing facts to the society which are supported by documents in the form of a series of applications. The techniques of Monte Carlo were first introduced in New Mexico at Los Alamos during the Second World War, eventually turning into the algorithm of Metropolis in the beginning of 1950s (Kaplan, 2000). During the time when the techniques of Monte Carlo were in use, the theory of Markov chain was taken closer to the practicality of statistics in 1970s. Essentially, the Markov model is composed of on responses categories. This model presumed the error measurement that gives the provision of error measurement and the unobserved differences in the probability transitions. In the latent mixture and the transition models give an indication of the existence of a single population that do exist in the probability transition. On the other hand there is the present of an error margin that occurs. Developments in the theory of Markov chain The fundamental reason for the development of Markov chain techniques methods was the fact that a huge number of issues that were considered to be problems of computation have now become manageable (Singer & Willett, 2003). An example considers the following basic model of random effects: Observation of the estimation of the components of variance can pose challenges to frequenters, but it was a serious challenge for a Bayesian, because of the integrals that were intractable. However, with the use of basic priors on σ2ε,σ2 θ and μ, the complete conditionals are of less importance to sample from and the issues and challenges is easy to solve with the use of Gibbs sampling. Further it is possible to raise the number of the components of variance and the solution of the Gibbs becomes less hard to implement. What is currently known as Metropolis algorithm was considered the first Markov chain and whose publication was made by Metropolis (Asparouhov & Muthen, 2008). It arises from a similar collection of scientists who made the production of technique of Monte Carlo, which is referred to as, the Los Alamos Scientist Researchers, mostly scientists studying on mathematical concepts of physics as well as the atomic bomb. Markov chain algorithms therefore traces its history back to such a is similar to the development time Techniques of Monte Carlo, which are usually found back to von Neumann and Ulam towards the end of 1940s. The original concept is associated with a combinational interaction in computation that was attempted in 1947 (making calculation of the possibility of emerging a winner in a game of playing cards). This concept was adopted in an enthusiastic manner to be used in the direct implementation of applications to the diffusion of neutron. The initial developments of Monte Carlo have got the description that provides a concise and clear history concerning the metropolis algorithm. These developments are very closely related to the appearance presentation of the earliest computer, which emerged for the first time in 1946, after undergoing construction for a period of three years. The technique of Monte Carlo adopted the use of fission problems and thermonuclear in 1947. The occurrence of Markov chain stipulates the amount of time that is occupied given the state x in an exponential value. Its development based on the fact that in a distribution where X presumes Xt+s having Ht would have a similar distribution of Xt+s given that there is a function Xt. This leads to the assertion that P1(t+s)=P1(t).[1-1+2).s] This may give an indication that a given element 1 and element 2 are both in good condition The need to ascertain the future events led to the development of this model. For instance, if the state of weather has to be determined, the state space has to be employed. Figure 1 Hierarchy of Marcov models The challenge of this model is hypothesized in the nature in which the number of estimates may be obtained. At given cases, there might be infinite numbers of the required estimates.. this requires that the identification priority to be involved in which the factor analysis would be involved in developing the structural equations. Markov processes The considerations for rules of walks that are randomly made on say {1. . . N} are observed. There are widely a lot of them all require some techniques that are random to begin. It is important establish definition of the rules that are used to select to make selections of the positions at first and second times. The Markov’s process comes into consideration when the given rules appear in a manner that the selection of the position that follows for a random walk which is considered after a certain number of steps only depends on the number of steps as well as the present level but not the positions of the earlier levels In conclusion, the recent developments of the Markov theory has made it possible to embed the two other multi-functioning models. This gives a better chance in identifying the latent variables that occur across a given group. . Any example that is used in the Markov process has to satisfy all the specified conditions. However the conditions are very much restrictive in that the condition to carry on is always dependent on the level of step as well as the number of steps. References Asparouhov, T., & Muthen, B. (2008). Multilevel mixture models. In G. R. Hancock & K. M. Samuelson (2002), Advances in latent variable mixture models. Charlotte, NC: Information Age Publishing. Collins, L. M., & Flaherty, B. P. (2002). Latent class models for longitudinal data. In. J. A. Muthe´n, B. (2008). Latent variable hybrids: Overview of old and new models. In G.R. National Center for Education Statistics. (2001). Early Childhood Longitudinal Study: Kindergarten Class of 1998–1999: Base year public-use data files user’s manual (No. NCES 2001–029). Washington, DC: U.S. Government Printing Office. Kaplan, D. (2000). Structural equation modelling: Foundations and extensions. Newbury Park, CA: Sage Publications. Vermunt, J. K., & Magidson, J. (2005). Latent GOLD 4.0 User’s Guide [Computer manual]. 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As noted by (Kaplan, 2000)The developments of the Markov’s in stochastic technique was championed by the development of the functional safety that was required in the electronic field that required the use of programmable systems. This could give the possibility of failure modes so as to ensure safety aspect is adhered. Getting information concerning the developments and improvements in the techniques of Monte Carlo, it is possible to involve the revolution of Markov chain theory which is of the second generation.

Begging from early to late 1990s, there occurred the improvements and developments of perfect sampling and reversible jumps, particle filters, progressive Monte Carlo, population as well as computation of standard errors. As has already been explained above, the knowledge and understanding that the concept of Markov chains could be applied in a large variety of circumstances only came to those applying statistics at the mainstream. This happened irrespective publications which were earlier made in the literature of statistics.

A number of reasons could be specified which include the inadequacy of machinery used in computing, or information on the Markov chain theory, or delays in trusting the practicality of the technique. It therefore needs committed researchers to be able to provide convincing facts to the society which are supported by documents in the form of a series of applications. The techniques of Monte Carlo were first introduced in New Mexico at Los Alamos during the Second World War, eventually turning into the algorithm of Metropolis in the beginning of 1950s (Kaplan, 2000).

During the time when the techniques of Monte Carlo were in use, the theory of Markov chain was taken closer to the practicality of statistics in 1970s. Essentially, the Markov model is composed of on responses categories. This model presumed the error measurement that gives the provision of error measurement and the unobserved differences in the probability transitions. In the latent mixture and the transition models give an indication of the existence of a single population that do exist in the probability transition.

On the other hand there is the present of an error margin that occurs. Developments in the theory of Markov chain The fundamental reason for the development of Markov chain techniques methods was the fact that a huge number of issues that were considered to be problems of computation have now become manageable (Singer & Willett, 2003). An example considers the following basic model of random effects: Observation of the estimation of the components of variance can pose challenges to frequenters, but it was a serious challenge for a Bayesian, because of the integrals that were intractable.

However, with the use of basic priors on σ2ε,σ2 θ and μ, the complete conditionals are of less importance to sample from and the issues and challenges is easy to solve with the use of Gibbs sampling. Further it is possible to raise the number of the components of variance and the solution of the Gibbs becomes less hard to implement. What is currently known as Metropolis algorithm was considered the first Markov chain and whose publication was made by Metropolis (Asparouhov & Muthen, 2008).

It arises from a similar collection of scientists who made the production of technique of Monte Carlo, which is referred to as, the Los Alamos Scientist Researchers, mostly scientists studying on mathematical concepts of physics as well as the atomic bomb. Markov chain algorithms therefore traces its history back to such a is similar to the development time Techniques of Monte Carlo, which are usually found back to von Neumann and Ulam towards the end of 1940s. The original concept is associated with a combinational interaction in computation that was attempted in 1947 (making calculation of the possibility of emerging a winner in a game of playing cards).

This concept was adopted in an enthusiastic manner to be used in the direct implementation of applications to the diffusion of neutron.

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