Multivariate Counting Processes Research Articles

An extended class of multivariate counting processes and its main properties

Abstract In this paper, a new multivariate counting process model (called Multivariate Poisson Generalized Gamma Process) is developed and its main properties are studied. Some basic stochastic properties of the number of events in the new multivariate counting process are initially derived. It is shown that this new multivariate counting process model includes the multivariate generalized Pólya process as a special case. The dependence structure of the multivariate counting process model is discussed. Some results on multivariate stochastic comparisons are also obtained.

Graphical criteria for the identification of marginal causal effects in continuous-time survival and event-history analyses

Abstract We consider continuous-time survival and event-history settings, where our aim is to graphically represent causal structures allowing us to characterize when a causal parameter is identified from observational data. This causal parameter is formalized as the effect on an outcome event of a (possibly hypothetical) intervention on the intensity of a treatment process. To establish identifiability, we propose novel graphical rules indicating whether the observed information is sufficient to obtain the desired causal effect by suitable reweighting. This requires a different type of graph than in discrete time. We formally define causal semantics for the corresponding dynamic graphs that represent local independence models for multivariate counting processes. Importantly, our work highlights that causal inference from censored data relies on subtle structural assumptions on the censoring process beyond independent censoring; these can be verified graphically. Put together, our results are the first to establish graphical rules for nonparametric causal identifiability in event processes in this generality for the continuous-time case, not relying on particular parametric survival models. We conclude with a data example on Human papillomavirus (HPV) testing for cervical cancer screening, where the assumptions are illustrated graphically and the desired effect is estimated by reweighted cumulative incidence curves.

Kalikow decomposition for counting processes with stochastic intensity and application to simulation algorithms

AbstractWe propose a new Kalikow decomposition for continuous-time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition allows us to derive simulation algorithms that hold either for stationary processes with potentially infinite network but bounded intensities, or for processes with unbounded intensities in a finite network and with empty past before zero. The Kalikow decomposition is not unique, and we discuss the choice of the decomposition in terms of algorithmic efficiency in certain cases. We apply these methods to several examples: the linear Hawkes process, the age-dependent Hawkes process, the exponential Hawkes process, and the Galves–Löcherbach process.

Birth Death Swap population in random environment and aggregation with two timescales

This paper deals with the stochastic modeling of a class of heterogeneous population in random environment, structured by discrete subgroups, and called birth–death–swap. In addition to demographic events modifying the population size, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jump measure of the population as a multivariate counting process. This process is defined as the solution of a stochastic differential system with random coefficients, driven by a multivariate Poisson random measure. The solution is obtained under weak assumptions, by the thinning of a strongly dominating point process generated by the same Poisson measure. This key construction relies on a general strong comparison result, of independent interest.The second part is dedicated to averaging results when swap events are significantly more frequent than demographic events. An important ingredient is the stable convergence, which is well-adapted to the general random environment. The pathwise construction by domination yields tightness results straightforwardly. At the limit, the demographic intensity functionals are averaged against random kernels resulting from swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a birth–death process in random environment, with non-linear intensity functionals.

Event history analysis of dynamic networks

SummaryStatistical analysis on networks has received growing attention due to demand from various emerging applications. In dynamic networks, one of the key interests is to model the event history of time-stamped interactions among nodes. We model dynamic directed networks via multivariate counting processes. A pseudo partial likelihood approach is exploited to capture the network dependence structure. Asymptotic results are established. Numerical experiments are performed to demonstrate the effectiveness of our proposal.

A new class of marginally regular multivariate counting processes generated by the mixture of multivariate Poisson processes

In this paper, a new class of marginally regular multivariate counting processes is developed and its stochastic properties are studied. The dependence of the proposed multivariate counting process is generated from two sources: by means of mixing and by sharing a common counting process. Even under a rather complex dependence structure, the stochastic properties of the multivariate process and its marginal processes are mathematically tractable. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We derive the properties of the proposed multivariate counting process and analyze the multivariate dependence structure of the class.

Construction of optimal reliability test plans for binary type multi-state strongly coherent systems

The concept of Binary Type Multi-State Strongly Coherent System (BTMSCS) or Multi-State Coherent System of Type 2 (MCS of type 2) has been defined by Natvig (1982). In this research article component based optimal reliability test plans for BTMSCS have been proposed. Here it is assumed that these systems are made up of n components, and the random variables representing lifetimes of systems and their components take values in set The construction of reliability test plans for BTMSCS is based on the use of (i) multivariate counting processes with marginal counting processes being homogeneous Poisson processes and (ii) positively upper orthant dependent property of the associated renewal processes. The applicability of the proposed reliability test plans has been demonstrated using all the three BTMSCSs made up of two components and one system consisting of three components with under the assumption that a bivariate random vector representing times spent by the ith component in states 1 and 2, follows Farlie-Gumbel-Morgenstern distribution with exponential marginals.

ON A MULTIVARIATE GENERALIZED POLYA PROCESS WITHOUT REGULARITY PROPERTY

Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.

Construction of optimal reliability test plans for multi-state coherent systems of type 1

This research article deals with construction of system based optimal reliability test plans for Multi-state Coherent Systems (MCSs) of Type 1 having states. The development of optimal reliability test plans for the MCS of Type 1 is based on the framework of multivariate counting process and a simple version of the multivariate central limit theorem. It may be noted that four reliability test plans using four combinations of two theoretical decision rules (TDR1, TDR2) and two data based decision rules (DDR1, DDR2), taken one from each category, have been proposed in this paper. Each of these test plans is demonstrated for and for a Farlie-Gumbel-Morgenstern (F-G-M) distribution using (i) a multi-state coherent series system, (ii) a multi-state coherent parallel system, each MCS consisting of two components, and (iii) a combination of series and parallel multi-state coherent system consisting of three components. The reliability test plan constructed using (TDR2, DDR2) combination turns out to be more economical than those constructed using (TDR1, DDR1), (TDR1, DDR2) and (TDR2, DDR1) decision rules.

A new class of multivariate counting processes and its characterization

ABSTRACTIn this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in Cha and Giorgio [On a class of multivariate counting processes, Adv. Appl. Probab. 48 (2016), pp. 443–462]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.

Modelling of Marginally Regular Bivariate Counting Process and its Application to Shock Model

In this paper, we develop a new class of bivariate counting processes that have ‘marginal regularity’ property. But, the ‘pooled processes’ in the developed class of bivariate counting processes are not regular. Therefore, the proposed class of processes allows simultaneous occurrences of two types of events, which can be applicable in practical modeling of counting events. Initially, some basic properties of the new class of bivariate counting processes will be discussed. Based on the obtained properties, the joint distributions of the numbers of events in time intervals will be derived and the dependence structure of the bivariate process will be discussed. Furthermore, the marginal and conditional processes will be studied. The application of the proposed bivariate counting process to a shock model will also be considered. In addition, the generalization to the multivariate counting processes will be discussed briefly.

On a class of multivariate counting processes

Abstract In this paper we define and study a new class of multivariate counting processes, named `multivariate generalized Pólya process'. Initially, we define and study the bivariate generalized Pólya process and briefly discuss its reliability application. In order to derive the main properties of the process, we suggest some key properties and an important characterization of the process. Due to these properties and the characterization, the main properties of the bivariate generalized Pólya process are obtained efficiently. The marginal processes of the multivariate generalized Pólya process are shown to be the univariate generalized Pólya processes studied in Cha (2014). Given the history of a marginal process, the conditional property of the other process is also discussed. The bivariate generalized Pólya process is extended to the multivariate case. We define a new dependence concept for multivariate point processes and, based on it, we analyze the dependence structure of the multivariate generalized Pólya process.

Lasso and probabilistic inequalities for multivariate point processes

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Nonasymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in (J. Amer. Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of (J. Amer. Statist. Assoc. 101 (2006) 1418-1429), our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.

Instantaneous Mean-Variance Hedging and Sharpe Ratio Pricing in a Regime-Switching Financial Model

□ We study hedging and pricing of claims in a non-Markovian regime-switching financial model. Our financial market consists of a bank account and a risky asset whose dynamics are driven by a Brownian motion and a multivariate counting process with stochastic intensities. The counting process is used to model the switching behavior for the states of the economy. We assume that the trajectory of the risky asset is continuous between the transition times for the states of the economy and that the value of the risky asset jumps at the time of the transition. We find the hedging strategy that minimizes the instantaneous mean-variance risk of the hedger’s surplus, and we set the price so that the instantaneous Sharpe ratio of the hedger’s surplus equals a predefined target. We discuss key properties of our optimal price and optimal hedging strategy.

Association between interferon use and reduced metabolic and vascular complications among patients with hepatitis C

We examined whether interferon treatment is associated with reduced metabolic/vascular complications in hepatitis C virus patients. The study had historical prospective cohort design using Maryland Medicaid administrative data (2006–2009). The end point was the incidence rate of mild, severe and combined mild/severe events from the Diabetes Complications Severity Index (DCSI). Interferon-treated and -untreated hepatitis C virus patients were matched on baseline covariates. Using multivariate counting process Cox regressions, we modeled the association between interferon receipt of at least 24 weeks and DCSI events incidence rate. Treated whites had similar rate of mild DCSI events, significantly 64% (p < 0.01) lower rate of severe DCSI events, and overall 29% (p = 0.14) lower rate of mild/severe DCSI events, compared with untreated whites. Compared with untreated blacks, treated blacks had a similar rate of DSCI events. Future studies should confirm our findings and should include important clinical variables such as viral genotype, virologic count and achieving sustained virologic response.

Joint Modelling of Goals and Bookings in Association Football

SummaryA multivariate counting process formulation is developed for the quantification of association football event interdependences which permits dynamic prediction as events unfold. We model data from English Premier League and Championship games from the 2009–2010 and 2010–2011 football seasons and assess predictive capacity by using a model-based betting strategy, applied prospectively to available live spread betting prices. Both the scoreline and the bookings status were predictive of match outcome. In particular, the award of a red card led to increased goal rates for the non-penalized team and the home team scoring rate decreased once they were ahead. Overall the betting strategy profited with gains made in the bookings markets.

PIN6 - Association Between Interferon Use And Reduced Metabolic And Vascular Complications Among Patients With Hepatitis C

We examined whether interferon treatment is associated with reduced metabolic/vascular complications in hepatitis C virus patients. The study had historical prospective cohort design using Maryland Medicaid administrative data (2006–2009). The end point was the incidence rate of mild, severe and combined mild/severe events from the Diabetes Complications Severity Index (DCSI). Interferon-treated and -untreated hepatitis C virus patients were matched on baseline covariates. Using multivariate counting process Cox regressions, we modeled the association between interferon receipt of at least 24 weeks and DCSI events incidence rate. Treated whites had similar rate of mild DCSI events, significantly 64% (p < 0.01) lower rate of severe DCSI events, and overall 29% (p = 0.14) lower rate of mild/severe DCSI events, compared with untreated whites. Compared with untreated blacks, treated blacks had a similar rate of DSCI events. Future studies should confirm our findings and should include important clinical varia...

DYNAMIC ANALYSIS OF A MULTIVARIATE REWARD PROCESS DEFINED ON THE UMCP WITH APPLICATION TO OPTIMAL PREVENTIVE MAINTENANCE POLICY PROBLEMS IN MANUFACTURING

The unified multivariate counting process (UMCP), previously studied by the same authors, enables one to describe most of the existing counting processes in terms of its components, thereby providing a comprehensive view for such processes often defined separately and differently. The purpose of this paper is to study a multivariate reward process defined on the UMCP. By examining the probabilistic flow in its state space, various transform results are obtained. The asymptotic behavior, as t→∞, of the expected univariate reward process in a form of a product of components of the multivariate reward process is studied. As an application, a manufacturing system is considered, where the cumulative profit given a preventive maintenance policy is described as a univariate reward process defined on the UMCP. The optimal preventive maintenance policy is derived numerically by maximizing the cumulative profit over the time interval [0, T].

Additive‐Multiplicative Regression Models for Spatio‐Temporal Epidemics

An extension of the stochastic susceptible-infectious-recovered (SIR) model is proposed in order to accommodate a regression context for modelling infectious disease data. The proposal is based on a multivariate counting process specified by conditional intensities, which contain an additive epidemic component and a multiplicative endemic component. This allows the analysis of endemic infectious diseases by quantifying risk factors for infection by external sources in addition to infective contacts. Inference can be performed by considering the full likelihood of the stochastic process with additional parameter restrictions to ensure non-negative conditional intensities. Simulation from the model can be performed by Ogata's modified thinning algorithm. As an illustrative example, we analyse data provided by the Federal Research Centre for Virus Diseases of Animals, Wusterhausen, Germany, on the incidence of the classical swine fever virus in Germany during 1993-2004.

Dynamic Analysis of a Unified Multivariate Counting Process and Its Asymptotic Behavior

The class of counting processes constitutes a significant part of applied probability. The classic counting processes include Poisson processes, nonhomogeneous Poisson processes, and renewal processes. More sophisticated counting processes, including Markov renewal processes, Markov modulated Poisson processes, age-dependent counting processes, and the like, have been developed for accommodating a wider range of applications. These counting processes seem to be quite different on the surface, forcing one to understand each of them separately. The purpose of this paper is to develop a unified multivariate counting process, enabling one to express all of the above examples using its components, and to introduce new counting processes. The dynamic behavior of the unified multivariate counting process is analyzed, and its asymptotic behavior ast→∞is established. As an application, a manufacturing system with certain maintenance policies is considered, where the optimal maintenance policy for minimizing the total cost is obtained numerically.