Applications Of Point Processes Research Articles

Matérn cluster process with holes at the cluster centers

Inspired by recent applications of point processes to biological nanonetworks, this paper presents a novel variant of a Matérn cluster process (MCP) in which the points located within a certain distance from the cluster centers are removed. We term this new process the MCP with holes at the cluster center (MCP-H, in short). Focusing on the three-dimensional (3D) space, we first characterize the conditional distribution of the distance between an arbitrary point of a given cluster to the origin, conditioned on the location of that cluster, for both MCP and MCP-H. These distributions are shown to admit remarkably simple closed forms in the 3D space, which is not even possible in the simpler two-dimensional (2D) case. Using these distributions, the contact distance distribution and the probability generating functional (PGFL) are characterized for both MCP and MCP-H.

Multistream-Based Marked Point Process With Decomposed Cumulative Hazard Functions.

When applying a point process to a real-world problem, an appropriate intensity function model should be designed based on physical and mathematical prior knowledge. Recently, a fully trainable deep learning-based approach has been developed for temporal point processes. In this approach, a cumulative hazard function (CHF) capable of systematic computation of adaptive intensity function is modeled in a data-driven manner. However, in this approach, although many applications of point processes generate various kinds of information such as location, magnitude, and depth, the mark information of events is not considered. To overcome this limitation, we propose a fully trainable marked point process method for modeling decomposed CHFs for time and mark prediction using multistream deep neural networks. We demonstrate the effectiveness of the proposed method through experiments with synthetic and real-world event data.

Point Processes in a Metric Space and Their Applications

Point processes are important in extreme value theory due to their equivalent formulations of two popular models in various applications: the block maxima models and the peak-over-threshold model. Point processes in a metric space provide tools to analyze heavy-tailed phenomena that appear in the research of extremal behaviors of functional data. To facilitate these applications of point processes, the equivalence between the weak convergence of point processes and the MO-convergence is established in the paper.

Measuring the Rate of Information Exchange in Point-Process Data With Application to Cardiovascular Variability.

The amount of information exchanged per unit of time between two dynamic processes is an important concept for the analysis of complex systems. Theoretical formulations and data-efficient estimators have been recently introduced for this quantity, known as the mutual information rate (MIR), allowing its continuous-time computation for event-based data sets measured as realizations of coupled point processes. This work presents the implementation of MIR for point process applications in Network Physiology and cardiovascular variability, which typically feature short and noisy experimental time series. We assess the bias of MIR estimated for uncoupled point processes in the frame of surrogate data, and we compensate it by introducing a corrected MIR (cMIR) measure designed to return zero values when the two processes do not exchange information. The method is first tested extensively in synthetic point processes including a physiologically-based model of the heartbeat dynamics and the blood pressure propagation times, where we show the ability of cMIR to compensate the negative bias of MIR and return statistically significant values even for weakly coupled processes. The method is then assessed in real point-process data measured from healthy subjects during different physiological conditions, showing that cMIR between heartbeat and pressure propagation times increases significantly during postural stress, though not during mental stress. These results document that cMIR reflects physiological mechanisms of cardiovascular variability related to the joint neural autonomic modulation of heart rate and arterial compliance.

A Point Processes Approach to Bicycle Sharing Systems’ Design and Management

Public bicycle systems have attracted a great deal of attention in recent years. The success of this service heavily depends on the topology of the city and on the locations of bike stations in relation to potential demand. In fact, it is primarily important that users find bike stations in convenient locations, sufficiently close both to the origins of their trips and to their destinations, and that each rental station guarantees the availability both of enough bicycles and of enough empty docking slots. This paper proposes a point processes approach to the study of bike-sharing systems, allowing us to quantify and control parameters having a key role in decisions both of strategic and operational type. Differently from previous studies, the point processes approach catches both the interdependence among the stations and the links between spatial and time aspects of the problem. The application of point processes, in particular spatial mixed Poisson processes, to this field requires the statement and proof of an in-variance property of such processes under stochastic dependent transformations, that may be of interest also from a theoretical point of view.

The Accumulated Persistence Function, a New Useful Functional Summary Statistic for Topological Data Analysis, With a View to Brain Artery Trees and Spatial Point Process Applications

We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The supplementary materials include additional methods and examples, technical details, and the R code used for all examples.

A semiparametric additive rate model for a modulated renewal process.

Recurrent event data from a long single realization are widely encountered in point process applications. Modeling and analyzing such data are different from those for independent and identical short sequences, and the development of statistical methods requires careful consideration of the underlying dependence structure of the long single sequence. In this paper, we propose a semiparametric additive rate model for a modulated renewal process, and develop an estimating equation approach for the model parameters. The asymptotic properties of the resulting estimators are established by applying the limit theory for stationary mixing sequences. A block-based bootstrap procedure is presented for the variance estimation. Simulation studies are conducted to assess the finite-sample performance of the proposed estimators. An application to a data set from a cardiovascular mortality study is provided.

Approximating the -permanent

SUMMARY The standard matrix permanent is the solution to a number of combinatorial and graph-theoretic problems, and the α-weighted permanent is the density function for a class of Cox processes called boson processes. The exact computation of the ordinary permanent is known to be #P-complete, and the same appears to be the case for the α-permanent for most values of α. At present, the lack of a satisfactory algorithm for approximating the α-permanent is a formidable obstacle to the use of boson processes in applied work. This paper proposes an importance-sampling estimator using nonuniform random permutations generated in a cycle format. Empirical investigation reveals that the estimator works well for the sorts of matrices that arise in point-process applications, involving up to a few hundred points. We conclude with a numerical illustration of the Bayes estimate of the intensity function of a boson point process, which is a ratio of α-permanents.

An Implementation of Bayesian Adaptive Regression Splines (BARS) inCwithSandRWrappers

BARS (DiMatteo, Genovese, and Kass 2001) uses the powerful reversible-jump MCMC engine to perform spline-based generalized nonparametric regression. It has been shown to work well in terms of having small mean-squared error in many examples (smaller than known competitors), as well as producing visually-appealing fits that are smooth (filtering out high-frequency noise) while adapting to sudden changes (retaining high-frequency signal). However, BARS is computationally intensive. The original implementation in S was too slow to be practical in certain situations, and was found to handle some data sets incorrectly. We have implemented BARS in C for the normal and Poisson cases, the latter being important in neurophysiological and other point-process applications. The C implementation includes all needed subroutines for fitting Poisson regression, manipulating B-splines (using code created by Bates and Venables), and finding starting values for Poisson regression (using code for density estimation created by Kooperberg). The code utilizes only freely-available external libraries (LAPACK and BLAS) and is otherwise self-contained. We have also provided wrappers so that BARS can be used easily within S or R.

How to couple from the past using a read-once source of randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 85–113, 2000

Einige bemerkungen zur verwendung der palmschen verteilung in der bedienungstheorie

In this paper the investigation concerns the interpretation of stationery and PALM distribution for stochastic point processes as a description of statistical equilibrium of a system in continuous time or from the point of view of certain events, which frequently used in the applications of point processes in the queneign theory.

Some applications of point processes to geology

Of late, there has been a considerable show of interest in outlining quantitative models for geological processes, usually in connection with computer applications. Harbaugh and Bonham-Carter (1970) have recently published an interesting book on the computer-'oriented simulation of several geological processes. Almost all of the applications made are nonstatistical (i.e., determinative models) or semistatistical and little attention has been paid to testing how well a model fits the facts, and in many instances, there seems to be an unawareness of how to go about the derivation of a working model and how to judge the degree of success achieved by it. This has led to a tendency for some quantitative geologists to be satisfied with a dangerously simplified model just because it may seem to fit some set of trial data. By so doing the geologist falls victim to his commonly voiced criticism of physicists for attempting to explain geological processes by vastly oversimplified models. Seismology (s.str.) is an important area for the application of point processes but, owing to the special nature of the problems involved, I have not considered it here. The newly developed theory of point processes, particularly in the manner in which it has been presented by Cox and Lewis (1966), offers good possibilities of providing a useful accessory to geological model building because it is able to analyze complicated processes in a logical and ordered fashion. From the point of view of data deriving from the earth sciences, point processes, however, do have a weak point and that is that in order to carry out a detailed and fully satisfactory analysis, such long series of data are required as to be an almost impossible prerequisite.