Polya Process Research Articles

A bivariate optimal replacement policy for a system subject to a generalized failure and repair process

AbstractTraditionally, in the studies of the optimal maintenance policies for repairable systems, the nonhomogeneous Poisson process model, which corresponds to the minimal repair process, has been intensively applied. However, in many practical situations, the repair type is not necessarily minimal. In this article, a new repair process based on a new counting process model (so‐called the generalized Polya process) is introduced. Then, the issue of the optimal replacement problem is discussed. A bivariate preventive replacement policy is developed and the properties of the optimal policy are studied. Illustrative examples are also presented. In addition, a comparison with a conventional replacement policy is performed.

On information-based residual lifetime in survival models with delayed failures

At many practical instances, the initiating point events (e.g., shocks) affect an object not immediately, but after some random delay. These models were studied in the literature only for the case when an initial shock process is Poisson. In our paper, we generalize these results to a meaningful case of the generalized Polya process (GPP) of initial shocks that was recently introduced in the literature. Distinct from the Poisson process, the GPP possesses the property of dependent increments, which makes it much more attractive in applications. We derive the distribution of the time to failure for a system subject to the GPP with delays. Our main focus, however, is on analysis of the corresponding residual lifetime distribution that depends now on the full history (information) of the initiating shock process.

Optimal periodic replacement policy for a GPP repairable product under the free-repair warranty

This article provides an extension study on the optimal periodic replacement policy for repairable products under the free-repair warranty. A new and more general type of repair, called the ‘GPP repair’, is applied in this study, which is defined based on the Generalized Polya Process and which is more close to practical situations than the ‘minimal repair’. Under the GPP repair assumption, the future reliability performance of a product becomes worse as the number of system failures that occurred in the past increases. Cost models from the user’s perspective are developed for both a GPP repairable product with free-repair warranty and without warranty, and the corresponding optimal replacement periods are derived such that the long-run expected cost rate is minimized. The difference and relationship between GPP repairable and minimal repairable products on the optimal replacement policies are analytically presented. Numerical examples are demonstrated for illustration and verification. Finally, a practical case is provided to explain the potential application areas of this study, and managerial implications and insights are addressed.

On preventive maintenance under different assumptions on the failure/repair processes

AbstractWe discuss models of preventive maintenance (PM) under different assumptions on the failure/repair processes. The main emphasis is on the PM modeling under the Brown‐Proschan model for minimal/perfect repairs. We also present the generalization of this model for the failure/repair process governed by the generalized Polya process. Some comparisons with conventional PM models are discussed (Nonhomogeneous Poisson process, renewal process, and geometric process of failure/repairs). Several practical examples are considered.

δ-shock model based on Polya process and its optimal replacement policy

Shock models are of great interest in engineering reliability. Among the others, the δ-shock model has been widely studied in the literature. In this model, the system breaks down due to the arrivals of two successive shocks which are too close to each other. That is, the system fails when the time between two consecutive shocks falls below a fixed threshold δ. In the literature, the δ-shock model has been mostly studied by assuming that shocks arrive according to a renewal process so that the interarrival times between shocks are independent and identically distributed. In the current paper, we consider the case when the shock arrival process is described by a Polya process which has dependent interarrival times. In particular, we obtain survival function and mean lifetime of the system and study the optimal replacement policy for the δ-shock model based on Polya process.

On a New Shot Noise Process and the Induced Survival Model

Traditionally, in applications, the shot noise processes have been studied under the assumption that the underlying arrival point process (shock process) is the homogeneous (or nonhomogeneous) Poisson process. However, most of the real life shock processes do not possess the independent increments property and the Poisson assumption is made just for simplicity. Recently, in the literature, a new point process, the generalized Polya process (GPP), has been proposed and characterized. The GPP is defined via the stochastic intensity that depends on the number of events in the previous interval and, therefore, does not possess the independent increments property. In this paper, we consider the GPP as an underlying shock process for the shot noise process. The corresponding survival model is considered and the survival probability and its failure rate are derived and thoroughly analyzed. Furthermore, a new concept, the history-dependent residual life time, is defined and discussed.

The nonlinear Polya process of entrepreneurial agglomeration

The geographically converged entrepreneurship, or the entrepreneurial agglomeration has been widely researched in recent years. As for the formation mechanism, traditional researches emphasize regional “economic advantage” and “social advantage” in the enhancing of agglomeration, ignoring the effects of new agent’s entrepreneurial decision and existing regional characteristics. In order to explore the dynamics of entrepreneurial agglomeration formation, this paper focuses on factors that affect individual decisions such as regional initial entrepreneurial ratio, new agent’s risk compensation characteristics and regional economic characteristics. Firstly, this paper constructs the entrepreneurship decision model based on new agents choice preference; then, from the dynamic perspective, a non-linear Polya process that reflects the formation of entrepreneurial agglomeration is constructed. According to Polya process, the ratio of entrepreneur population among whole population in a given region will probably converge to a stable point if new entrants make continuous entrepreneurial decisions. Entrepreneurial agglomeration can be explained in light with the existence and the converging speed of this point. Finally, a numerical simulation is conducted to testify the entrepreneurial agglomeration formation process.

Justifying the Gompertz curve of mortality via the generalized Polya process of shocks

A new probabilistic model of aging that can be applied to organisms is suggested and analyzed. Organisms are subject to shocks that follow the generalized Polya process (GPP), which has been recently introduced and characterized in the literature. Distinct from the nonhomogeneous Poisson process that has been widely used in applications, the important feature of this process is the dependence of its future behavior on the number of previous events (shocks). The corresponding survival and the mortality rate functions are derived and analyzed. The general approach is used for justification of the Gompertz law of human mortality.

Bin sizes in time-inhomogeneous infinite Polya processes

At the nth step of a time-inhomogeneous infinite Polya process, either a new bin is created and a ball is put in that bin, or a ball is put into an existing bin, in which case the bin is chosen according to a preferential attachment type rule. We introduce a new class of such processes and find the asymptotics of the expected number of bins of size k, for k fixed, as n→∞.

New shock models based on the generalized Polya process

Various shock models have been extensively studied in the literature, mostly under the assumption of the Poisson process of shocks. In the current paper, we study shock models under the generalized Polya process (GPP) of shocks, which has been recently introduced and characterized in the literature (see Konno (2010) and Cha, 2014). Distinct from the widely used nonhomogeneous Poisson process, the important feature of this process is the dependence of its stochastic intensity on the number of previous shocks. We consider the extreme shock model, where each shock is catastrophic for a system with probability p(t) and is harmless with the complementary probability q(t)=1−p(t). The corresponding survival and the failure rate functions are derived and analyzed. These results can be used in various applications including engineering, survival analysis, finance, biology and so forth. The cumulative shock model, where each shock results in the increment of wear and a system's failure occurs when the accumulated wear reaches some boundary is also considered. A new general concept describing the dependent increments property of a stochastic process is suggested and discussed with respect to the GPP.

Single Nucleotide Polymorphisms Can Create Alternative Polyadenylation Signals and Affect Gene Expression through Loss of MicroRNA-Regulation

Alternative polyadenylation (APA) can for example occur when a protein-coding gene has several polyadenylation (polyA) signals in its last exon, resulting in messenger RNAs (mRNAs) with different 3′ untranslated region (UTR) lengths. Different 3′UTR lengths can give different microRNA (miRNA) regulation such that shortened transcripts have increased expression. The APA process is part of human cells' natural regulatory processes, but APA also seems to play an important role in many human diseases. Although altered APA in disease can have many causes, we reasoned that mutations in DNA elements that are important for the polyA process, such as the polyA signal and the downstream GU-rich region, can be one important mechanism. To test this hypothesis, we identified single nucleotide polymorphisms (SNPs) that can create or disrupt APA signals (APA-SNPs). By using a data-integrative approach, we show that APA-SNPs can affect 3′UTR length, miRNA regulation, and mRNA expression—both between homozygote individuals and within heterozygote individuals. Furthermore, we show that a significant fraction of the alleles that cause APA are strongly and positively linked with alleles found by genome-wide studies to be associated with disease. Our results confirm that APA-SNPs can give altered gene regulation and that APA alleles that give shortened transcripts and increased gene expression can be important hereditary causes for disease.

Revised-Path Dependence

In this article, we define a class of revised path—dependent processes and characterize their basic properties. A process exhibits revised-path dependence if the current outcome can revise the value of a past outcome. A revision could be a change to that outcome or a reinterpretation. We first define a revised path—dependent process called the accumulation process: in each period, a randomly chosen past outcome is changed to match the current outcome and show that it converges to identical outcomes. We then construct a general class of models that includes the Bernoulli process, the Polya process, and the accumulation process as special cases. For this general class, we show that, apart from knife-edge cases, all processes converge either to homogeneous equilibria or to an equal probability distribution over types. We also show that if random draws advantage one outcome over the other, then the process has a unique equilibrium.

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.

On the Exact Solution of a Generalized Polya Process

There are two types of master equations in describing nonequilibrium phenomena with memory effect: (i) the memory function type and (ii) the nonstationary type. A generalized Polya process is studied within the framework of a non‐stationary type master equation approach. For a transition‐rate with an arbitrary time‐dependent relaxation function, the exact solution of a generalized Polya process is obtained. The characteristic features of temporal variation of the solution are displayed for some typical time‐dependent relaxation functions reflecting memory in the systems.

Mathematical Habits of the Mind for Preservice Teachers

This article provides an overview of mathematical habits of the mind and discusses how the concept relates to Polya's problem solving principles as well as exemplification. Specific problems are discussed as a means to assist preservice elementary school teachers' in their development of mathematical habits of the mind. Without a technique to begin solving these rich problems, preservice teachers may have difficulty getting started. The process of preservice teachers outlining their thinking as they progress through Polya's process is discussed. Students' reflections from this technique are discussed to explore the outcomes that may be expected from establishing an environment where students are encouraged to develop mathematical habits of the mind.

Martingale characterization of Pólya processes and sequences

It is proved that a mixed Poisson process ξt is a Polya process if and only if there exists a nondegenerate linear transform ξt → ηt = a(t)ξt + b(t) such that ηt is a martingale. A similar result is valid for Polya sequences.

Exact and limiting distributions in diagonal Pólya processes

We investigate the polya process, which underlies an urn of white and blue balls growing in real time. A partial differential equation governs the evolution of the process. For urns with (forward or backward) diagonal ball addition matrix the partial differential equation is amenable to asymptotic solution. In the case of forward diagonal we find a solution via the method of characteristics; the numbers of white and blue balls, when scaled appropriately, converge in distribution to independent Gamma random variables. The method of characteristics becomes a bit too involved for the backward diagonal process, except in degenerate cases, where we have Poisson behavior. In nondegenerate cases, limits characterized implicitly by their recursive sequence of moments are found, via matrix formulation involving a Leonard pair.

Concentration Inequalities and Martingale Inequalities: A Survey

We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs.

Random sprouts as internet models, and P�lya processes

The sprout, a tree growing in real time is introduced. The sprout is a generalization and an embedding in time of the standard recursive tree. The sprout is proposed as a model for the growth of the Internet. The tree size is analyzed via an associated two-color stochastic process (the sprout process), which is a special case of the Polya process. Owing to its potential as a modeling tool, the more general Polya process is analyzed on average. In addition to the usage of the Polya process in evaluating sprouts, we also give a heuristic interpretation of the result for Polya urns, which might be a first step toward understanding several nonclassic urn models, as those with nonconstant row sum and those with multiple eigenvalues.

Bayesian Forecasting of Economic Time Series

A model is suggested to forecast economic time series. This model incorporates some innovative ideas of Harrison and Stevens [20] for building into the forecasting process important external shocks to the systems. Thus the occurrence of possibly significant real-world events may cause a fundamental change in the time series in question. The Jeffreys-Savage (JS) Bayesian theory of hypothesis testing is used to test the hypothesis that a particular event has been such as to free the series from its immediate past behavior. When the event frees the series in this way, then we model the sequence of observations following such an event (until the next such event) as an exchangeable sequence. In the simplest case of 0–1 valued data, such as in recording the ups and downs of the value of a particular commodity or stock, our alternative hypothesis is a Pólya process, and the null hypothesis is a simple random walk (unit roots model) with p = .50. Any exchangeable sequence is strictly stationary, and the observations in the Polya process are positively correlated, which can give rise to “explosive” behavior of the series at isolated time points. We then use the JS theory to predict future observations by taking a weighted average of the optimal predictions for each model, with weights given by the posterior probabilities of the hypotheses. Results of simulation studies are presented which compare the predictive performance of the fully Bayesian method based upon the JS theory with those based upon the “p-value” or pre-test method. The de Finetti method for scoring predictions is used to assess their empirical performance. A theoretical methodology, which extends the “evaluation game” of Hill [28,37], is developed for comparing predictors.