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Register a new userEstimation of entropy for Poisson marked point processes
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In this paper, a kernel estimator of the differential entropy of the mark distribution of a homogeneous Poisson marked point process is proposed. The marks have an absolutely continuous distribution on a compact Riemannian manifold without boundary. $L^2$ and almost surely consistency of this estimator as well as its asymptotic normality are investigated.
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We are interested in estimating the location of what we call smooth change-point from $n$ independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length $delta_n$ is considered to be decreasing to $0$ as $nto+infty$. We show that if $delta_n$ goes to zero slower than $1/n$, our model is locally asymptotically normal (with a rather unusual rate $sqrt{delta_n/n}$), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary, $delta_n$ goes to zero faster than $1/n$, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate $1/n$, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where $delta_n$ goes to zero faster than $1/n$, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.
Self- and mutually-exciting point processes are popular models in machine learning and statistics for dependent discrete event data. To date, most existing models assume stationary kernels (including the classical Hawkes processes) and simple parametric models. Modern applications with complex event data require more general point process models that can incorporate contextual information of the events, called marks, besides the temporal and location information. Moreover, such applications often require non-stationary models to capture more complex spatio-temporal dependence. To tackle these challenges, a key question is to devise a versatile influence kernel in the point process model. In this paper, we introduce a novel and general neural network-based non-stationary influence kernel with high expressiveness for handling complex discrete events data while providing theoretical performance guarantees. We demonstrate the superior performance of our proposed method compared with the state-of-the-art on synthetic and real data.
Let $U_1,U_2,ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $ntoinfty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $|x|^{-(d+gamma)}$, where $gamma=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $gamma>0$.
Individual mobility prediction is an essential task for transportation demand management and traffic system operation. There exist a large body of works on modeling location sequence and predicting the next location of users; however, little attention is paid to the prediction of the next trip, which is governed by the strong spatiotemporal dependencies between diverse attributes, including trip start time $t$, origin $o$, and destination $d$. To fill this gap, in this paper we propose a novel point process-based model -- Attentive Marked temporal point processes (AMTPP) -- to model human mobility and predict the whole trip $(t,o,d)$ in a joint manner. To encode the influence of history trips, AMTPP employs the self-attention mechanism with a carefully designed positional embedding to capture the daily/weekly periodicity and regularity in individual travel behavior. Given the unique peaked nature of inter-event time in human behavior, we use an asymmetric log-Laplace mixture distribution to precisely model the distribution of trip start time $t$. Furthermore, an origin-destination (OD) matrix learning block is developed to model the relationship between every origin and destination pair. Experimental results on two large metro trip datasets demonstrate the superior performance of AMTPP.
The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about non-central moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.
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