Mapping the Intensity Function of a Non-stationary Point Process in Unobserved Areas

Seismic networks provide data that are used as basis both for public safety decisions and for scientific research. Their configuration affects the data completeness, which in turn, critically affects several seismological scientific targets (e.g., earthquake prediction, seismic hazard...). In this context, a key aspect is how to map earthquakes density in seismogenic areas from censored data or even in areas that are not covered by the network. We propose to predict the spatial distribution of earthquakes from the knowledge of presence locations and geological relationships, taking into account any interactions between records. Namely, in a more general setting, we aim to estimate the intensity function of a point process, conditional to its censored realization, as in geostatistics for continuous processes. We define a predictor as the best linear unbiased combination of the observed point pattern. We show that the weight function associated to the predictor is the solution of a Fredholm equation of second kind. Both the kernel and the source term of the Fredholm equation are related to the first- and second-order characteristics of the point process through the intensity and the pair correlation function. Results are presented and illustrated on simulated non-stationary point processes and real data for mapping Greek Hellenic seismicity in a region with unreliable and incomplete records.

Erlang mixture modeling for Poisson process intensities

We develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function which is modeled nonparametrically with a gamma process prior. Such model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting. The prior model structure supports general shapes for the point process intensity function, and it also enables effective handling of the Poisson process likelihood normalizing term resulting in efficient posterior simulation. The Erlang mixture modeling approach is further elaborated to develop an inference method for spatial Poisson processes. The methodology is examined relative to existing Bayesian nonparametric modeling approaches, including empirical comparison with Gaussian process prior based models, and is illustrated with synthetic and real data examples.

The Confidence Interval of the Estimator of the Periodic Intensity Function in the Presence of Power Function Trend on the Nonhomogeneous Poisson Process

The nonhomogeneous Poisson process is one of the most widely applied stochastic processes. In this article, we provide a confidence interval of the intensity estimator in the presence of a periodic multiplied by trend power function. This estimator's confidence interval is an application of the formulation of the estimator asymptotic distribution that has been given in previous studies. In addition, constructive proof of the convergent in probability has been provided for all power functions.

Neural Relation Inference for Multi-dimensional Temporal Point Processes via Message Passing Graph

Relation discovery for multi-dimensional temporal point processes (MTPP) has received increasing interest for its importance in prediction and interpretability of the underlying dynamics. Traditional statistical MTPP models like Hawkes Process have difficulty in capturing complex relation due to their limited parametric form of the intensity function. While recent neural-network-based models suffer poor interpretability. In this paper, we propose a neural relation inference model namely TPP-NRI. Given MTPP data, it adopts a variational inference framework to model the posterior relation of MTPP data for probabilistic estimation. Specifically, assuming the prior of the relation is known, the conditional probability of the MTPP conditional on a sampled relation is captured by a message passing graph neural network (GNN) based MTPP model. A variational distribution is introduced to approximate the true posterior. Experiments on synthetic and real-world data show that our model outperforms baseline methods on both inference capability and scalability for high-dimensional data.

Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

A time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift $$\alpha (t)x+\beta (t)$$ α ( t ) x + β ( t ) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function $$\beta (t)$$ β ( t ) and the noise intensity function r(t) are connected via the relation $$\beta (t)=\xi \,r(t)$$ β ( t ) = ξ r ( t ) , with $$0\le \xi <1$$ 0 ≤ ξ < 1 . Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that $$\alpha (t)$$ α ( t ) , $$\beta (t)$$ β ( t ) or both of these functions exhibit some kind of periodicity.

Feature Extraction of Sequence of Keystrokes in Fixed Text Using the Multivariate Hawkes Process

In this paper, we propose a new method of extracting the features of keystrokes. The Hawkes process based on exponential excitation kernel was used to model the sequence of keystrokes in fixed text, and the intensity function vector and adjacency matrix of the model obtained through training were regarded as the characteristics of the keystrokes. A visual analysis was carried out on the CMU keystroke raw data and the feature data extracted using the proposed method. We used one-class classifier to compare the classification effect of CMU keystroke raw data and the feature data extracted by the Hawkes process model and POHMM model. The experimental results show that the feature data extracted using the proposed method contains rich information to distinguish users. In addition, the feature data extracted using the proposed method has a slightly better classification performance than the original CMU keystroke data for some users who are not easy to distinguish.

Penduga Konsisten dari Fungsi Sebaran dan Fungsi Kepekatan Peluang Waktu Tunggu Proses Poisson Periodik

ABSTRAKPenduga yang konsisten dari fungsi distribusi dan fungsi kepekatan peluang waktu tunggu dari proses Poisson periodik dibahas dalam artikel ini. Tidak ada asumsi bentuk parametrik tertentu dari fungsi intensitas proses Poisson periodik. Situasi dipertimbangkan ketika hanya ada realisasi tunggal dari proses Poisson periodik yang teramati dalam interval terbatas [0,n]. Hasil pembuktian menunjukkan bahwa penduga yang diusulkan konsisten ketika n-??. ABSTRACTThe consistent estimator of the distribution and the density functions of the waiting time of a cyclic Poisson process is considered and investigated. We do not assume any particular parametric form of the intensity function of the cyclic Poisson process. We consider the situation when there is only a single realization of the cyclic Poisson process is spotted in a bounded interval [0,n]. We proved that the propose estimators are consistent as n-??.

A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach

In this paper, we investigate the problem of the local linear estimation of the conditional ageing intensity function, when the variable of interest is subject to random right-censored. We establish under appropriate conditions the asymptotic normality of this estimator.

A Stress Intensity Function for FCG Analyses in Metals

This paper presents a damaging stress intensity function K for analyses of R-ratio effects on fatigue crack growth (FCG) in metals. The proposed formulation is based on the sum of strain and complementary energy and its role in FCG rate behavior in threshold and Paris region at R-ratios ranging from -2 to 0.97. It doesn’t invoke a crack closure assumption or fitting parameters for R<0.5-0.6. For a high R>0.7 it utilizes an experimentally determine correction factor, which accounts for excessive plastic dissipation in the monotonic plastic zone (MPZ).