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Author: Benjamin Avanzi Publisher: ISBN: Category : Languages : en Pages : 27
Book Description
The Markov-modulated Poisson process is utilised for count modelling in a variety of areas such as queueing, reliability, network and insurance claims analysis. In this paper, we extend the Markov-modulated Poisson process framework through the introduction of a flexible frequency perturbation measure. This contribution enables known information of observed event arrivals to be naturally incorporated in a tractable manner, while the hidden Markov chain captures the effect of unobservable drivers of the data. In addition to increases in accuracy and interpretability, this method supplements analysis of the latent factors. Further, this procedure naturally incorporates data features such as over-dispersion and autocorrelation. Additional insights can be generated to assist analysis, including a procedure for iterative model improvement. Implementation difficulties are also addressed with a focus on dealing with large data sets, where latent models are especially advantageous due the large number of observations facilitating identification of hidden factors. Namely, computational issues such as numerical underflow and high processing cost arise in this context and in this paper, we produce procedures to overcome these problems. This modelling framework is demonstrated using a large insurance data set to illustrate theoretical, practical and computational contributions and an empirical comparison to other count models highlight the advantages of the proposed approach.
Author: Benjamin Avanzi Publisher: ISBN: Category : Languages : en Pages : 27
Book Description
The Markov-modulated Poisson process is utilised for count modelling in a variety of areas such as queueing, reliability, network and insurance claims analysis. In this paper, we extend the Markov-modulated Poisson process framework through the introduction of a flexible frequency perturbation measure. This contribution enables known information of observed event arrivals to be naturally incorporated in a tractable manner, while the hidden Markov chain captures the effect of unobservable drivers of the data. In addition to increases in accuracy and interpretability, this method supplements analysis of the latent factors. Further, this procedure naturally incorporates data features such as over-dispersion and autocorrelation. Additional insights can be generated to assist analysis, including a procedure for iterative model improvement. Implementation difficulties are also addressed with a focus on dealing with large data sets, where latent models are especially advantageous due the large number of observations facilitating identification of hidden factors. Namely, computational issues such as numerical underflow and high processing cost arise in this context and in this paper, we produce procedures to overcome these problems. This modelling framework is demonstrated using a large insurance data set to illustrate theoretical, practical and computational contributions and an empirical comparison to other count models highlight the advantages of the proposed approach.
Author: Yi Lu Publisher: ISBN: Category : Markov processes Languages : en Pages : 0
Book Description
Periodic non-homogeneous Poisson processes and Poisson models under Markovian environments are studied in this thesis. By accounting for periodic seasonal variations and random fluctuations in the underlying risk, these models generalize the classical homogeneous Poison risk model. Non-homogenous Poisson processes with periodic claim intensity rates are proposed for the claim counting process of risk theory. We introduce a doubly periodic Poisson model with short and long-term trends. Beta-type intensity functions are presented as illustrations. Doubly periodic Poisson models are appropriate when the seasonality does not repeat the exact same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. The properties of the model and the statistical inference of the model parameters are discussed. An application of the model to the dataset of Atlantic Hurricanes Affecting the United States (1899-2000) is discussed in detail. Further we introduce a periodic regime-switching Cox risk model by considering both, seasonal variations and stochastic fluctuations in the claims intensity. The intensity process, governed by a periodic function with a random peak level, is proposed. The periodic intensity function follows a deterministic pattern in each short-term period, and is illustrated by a beta-type function. A finite-state Markov chain defines the level process, explaining the random effect due to different underlying risk years. The properties of this regime-switching claim counting process are discussed in detail. By properly defining the Lundberg coefficient; Lundberg-type bounds for finite time ruin probabilities in the two-state risk model case are derived. A detailed derivation of the likelihood function and the maximum likelihood estimates of the model parameters is also given. Statistical applications of the model to the Atlantic hurricanes affecting the United States dataset are discussed under two different level classifications schemes. The Markov-modulated risk model is considered to reflect a risk process or insurance business alternating between a finite number of Poisson models. Here we assume that the claim inter-arrivals, claim severities and premiums of the model are influenced by an external Markovian environment. The effect of this external environment may be characterized, at any time, by a state variable, representing for example, certain types of epidemics, a variety of weather conditions or of different states of the economy.
Author: Eliane Regina Rodrigues Publisher: Springer Science & Business Media ISBN: 1461446457 Category : Mathematics Languages : en Pages : 116
Book Description
In this brief we consider some stochastic models that may be used to study problems related to environmental matters, in particular, air pollution. The impact of exposure to air pollutants on people's health is a very clear and well documented subject. Therefore, it is very important to obtain ways to predict or explain the behaviour of pollutants in general. Depending on the type of question that one is interested in answering, there are several of ways studying that problem. Among them we may quote, analysis of the time series of the pollutants' measurements, analysis of the information obtained directly from the data, for instance, daily, weekly or monthly averages and standard deviations. Another way to study the behaviour of pollutants in general is through mathematical models. In the mathematical framework we may have for instance deterministic or stochastic models. The type of models that we are going to consider in this brief are the stochastic ones.
Author: Ant¢nio Pacheco Publisher: World Scientific ISBN: 9812793186 Category : Mathematics Languages : en Pages : 237
Book Description
The book presents a coherent treatment of Markov random walks and Markov additive processes together with their applications. Part I provides the foundations of these stochastic processes underpinned by a solid theoretical framework based on Semiregenerative phenomena. Part II presents some applications to queueing and storage systems.
Author: Alexander M Andronov Publisher: World Scientific ISBN: 9811286175 Category : Mathematics Languages : en Pages : 227
Book Description
Probabilistic models are widely used for description and an analysis of various processes in system reliability, risk, queuing, data communication, logistic and storage systems. The book contains various applications of the theory of continuous-time Markov-modulated processes in operation research. All analytical results are illustrated by numerical computations. Used algorithms allow overcoming computation difficulties successfully. For example, a calculation of transient probabilities of states for a continuous-time finite Markov chain uses eigenvalues and eigenvectors of the corresponding matrix (generator). In a more complex case of differential or integral equations, such a simple explicit form of a solution is missing. The explicit form of solution is presented by means of infinity sums of functions. For example, often we have to deal with the so-called renewal equation. Its solution is presented as an infinite sum of the renewal function. In this case, an approximation of functions of interest and iterative computation procedures are used.
Author: Oliver Ibe Publisher: Newnes ISBN: 0124078397 Category : Mathematics Languages : en Pages : 515
Book Description
Markov processes are processes that have limited memory. In particular, their dependence on the past is only through the previous state. They are used to model the behavior of many systems including communications systems, transportation networks, image segmentation and analysis, biological systems and DNA sequence analysis, random atomic motion and diffusion in physics, social mobility, population studies, epidemiology, animal and insect migration, queueing systems, resource management, dams, financial engineering, actuarial science, and decision systems. Covering a wide range of areas of application of Markov processes, this second edition is revised to highlight the most important aspects as well as the most recent trends and applications of Markov processes. The author spent over 16 years in the industry before returning to academia, and he has applied many of the principles covered in this book in multiple research projects. Therefore, this is an applications-oriented book that also includes enough theory to provide a solid ground in the subject for the reader. - Presents both the theory and applications of the different aspects of Markov processes - Includes numerous solved examples as well as detailed diagrams that make it easier to understand the principle being presented - Discusses different applications of hidden Markov models, such as DNA sequence analysis and speech analysis.
Author: Thomas J. Watson IBM Research Center Publisher: ISBN: Category : Point processes Languages : en Pages : 0
Book Description
The nonhomogeneous Poisson process is a widely used model for a series of events (stochastic point process) in which the rate or intensity of occurrence of points varies, usually with time. The process has the characteristic properties that the number of points in any finite set of nonoverlapping intervals are mutually independent random varialbes, and that the number of points in any of these intervals has a Poisson distribution. This paper first discusses several general methods for simulation of the one-dimensional nonhomogeneous Poisson process. Then a particular and very efficient method for simulation of nonhomogeneous Poisson processes is stated with log-linear rate function. The method is based on an identity relating the nonhomogeneous Poisson process to the gap statistics from a random number of exponential random variables with suitably chosen parameters. Finally, a simple and relatively efficient new method for simulation of one-dimensional and two-dimensional nonhomogeneous Poisson processes is described. The method is applicable for any given rate function and is based on controlled deletion of points in a Poisson process with a rate function that dominates the given rate function.
Author: Sidney I. Resnick Publisher: Springer Science & Business Media ISBN: 1461203872 Category : Mathematics Languages : en Pages : 640
Book Description
Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. This text offers easy access to this fundamental topic for many students of applied sciences at many levels. It includes examples, exercises, applications, and computational procedures. It is uniquely useful for beginners and non-beginners in the field. No knowledge of measure theory is presumed.
Author: Benjamin Avanzi
Author: Benjamin Avanzi
Author: Yi Lu
Author: Eliane Regina Rodrigues
Author: Ant¢nio Pacheco
Author: Kathleen S. Meier
Author: 
Author: Alexander M Andronov
Author: Oliver Ibe
Author: Thomas J. Watson IBM Research Center
Author: Sidney I. Resnick