Spatiotemporal events occur in many disciplines, including economics, sociology, criminology, and seismology, with different patterns in space and time related to environmental characteristics, policing, and human behavior. In this paper, we propose a class of multivariate Hawkes processes with spatial covariates to consider the influence structure of spatial features in spatiotemporal events and the spatiotemporal patterns such as clustering. Baseline intensities are assumed to be a spatial Poisson regression model to explain spatial feature influence. The transfer functions are considered unknown but smooth and decreasing to explain the clustering phenomena. A semiparametric estimation method based on time discretization and local constant approximation is introduced. Transfer function estimators are shown to be consistent, and baseline intensity estimators are consistent and asymptotically normal. We examine the numerical performance of the proposed estimators with extensive simulation and illustrate the application of the proposed model to crime data obtained from Pittsburgh, Pennsylvania.

In this paper, we consider testing the martingale difference hypothesis for high- dimensional time series. Our test is built on the sum of squares of the element-wise max-norm of the proposed matrix-valued nonlinear dependence measure at different lags.

Models defined by moment conditions are at the center of structural econometric estimation, but economic theory is mostly agnostic about moment selection. While a large pool of valid moments can potentially improve estimation efficiency, in the meantime a few invalid ones may undermine consistency. This article investigates the empirical likelihood estimation of these moment-defined models in high-dimensional settings. We propose a penalized empirical likelihood (PEL) estimation and establish its oracle property with consistent detection of invalid moments. The PEL estimator is asymptotically normally distributed, and a projected PEL procedure further eliminates its asymptotic bias and provides more accurate normal approx- imation to the finite sample behavior. Simulation exercises demonstrate excellent numerical performance of these methods in estimation and inference.

We consider to model matrix time series based on a tensor canonical polyadic (CP)-decomposition. Instead of using an iterative algorithm which is the standard practice for estimating CP-decompositions, we propose a new and one-pass estimation procedure based on a generalized eigenanalysis constructed from the serial dependence structure of the underlying process. To overcome the intricacy of solving a rank-reduced generalized eigenequation, we propose a further refined approach which projects it into a lower-dimensional full-ranked eigenequation.

Modeling Hawkes process using deep learning is superior to traditional statistical methods in the goodness of fit. However, methods based on RNN or self-attention are deficient in long-time dependence and recursive induction, respectively. Universal Transformer (UT) is an advanced framework to integrate these two requirements simultaneously due to its continuous transformation of self-attention in the depth of the position. In addition, migration of the UT framework involves the problem of effectively matching Hawkes process modeling. Thus, in this paper, an iterative convolutional enhancing self-attention Hawkes process with time relative position encoding (ICAHP-TR) is proposed, which is based on improved UT. First, the embedding maps from dense layers are carried out on sequences of arrival time points and markers to enrich event representation. Second, the deep network composed of UT extracts hidden historical information from event expression with the characteristics of recursion and the global receptive field. Third, two designed mechanics, including the relative positional encoding on the time step and the convolution enhancing perceptual attention are adopted to avoid losing dependencies between relative and adjacent positions in the Hawkes process. Finally, the hidden historical information is mapped by Dense layers as parameters in Hawkes process intensity function, thereby obtaining the likelihood function as the network loss. The experimental results show that the proposed methods demonstrate the effectiveness of synthetic datasets and realworld datasets from the perspective of both the goodness of fit and predictive ability compared with other baseline methods.

Graph convolutional networks (GCN) are viewed asone of the most popular representations among thevariants of graph neural networks over graph dataand have shown powerful performance in empiricalexperiments. That l2-based graph smoothing enforces the global smoothness of GCN, while (soft)l1-based sparse graph learning tends to promotesignal sparsity to trade for discontinuity. This paper aims to quantify the trade-off of GCN betweensmoothness and sparsity, with the help of a generall, ℓp-regularized (1 < p≤ 2) stochastic learning pro.posed within. While stability-based generalizationanalyses have been given in prior work for a secondderivative objectiveness function, our ℓp-regularized learning scheme does not satisfy such a smooth con.dition. To tackle this issue, we propose a novel SGD proximal algorithm for GCNs with an inexactoperator. For a single-layer GCN, we establish anexplicit theoretical understanding of GCN with the ℓp-regularized stochastic learning by analyzing thestability of our SGD proximal algorithm. We conduct multiple empirical experiments to validate ourtheoretical findings.

We propose a new unit-root test for a stationary null hypothesis H0 against a unit-root alternative H1. Our approach is nonparametric as H0 assumes only that the process concerned is I (0), without specifying any parametric forms. The new test is based on the fact that the sample autocovariance function converges to the finite population autocovariance function for an I(0) process, but diverges to infinity for a process with unit roots.

While it is common practice in applied network analysis to report various standard network summary statistics, these numbers are rarely accompanied by uncertainty quantification. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Here we study the problem of estimating the density of an arbitrary subgraph, given a noisy version of some underlying network as data. Under a simple model of network error, we show that consistent estimation of such densities is impossible when the rates of error are unknown and only a single network is observed. Accordingly, we develop method-of- moment estimators of network subgraph densities and error rates for the case where a minimal number of network replicates are available. These estimators are shown to be asymptotically normal as the number of vertices increases to infinity. We also provide confidence intervals for quantifying the uncertainty in these estimates based on the asymptotic normality. To construct the confidence intervals, a new and nonstandard bootstrap method is proposed to compute asymptotic variances, which is infeasible otherwise. We illustrate the proposed methods in the context of gene coexpression networks. Supplementary materials for this article are available online.

Directed acyclic graph (DAG) models are widely used to represent casual relationships among random variables in many application domains. This article studies a special class of non-Gaussian DAG models, where the conditional variance of each node given its parents is a quadratic function of its conditional mean. Such a class of non-Gaussian DAG models are fairly flexible and admit many popular distributions as special cases, including Poisson, Binomial, Geometric, Exponential, and Gamma. To facilitate learning, we introduce a novel concept of topological layers, and develop an efficient DAG learning algorithm. It first reconstructs the topological layers in a hierarchical fashion and then recovers the directed edges between nodes in different layers, which requires much less computational cost than most existing algorithms in literature. Its advantage is also demonstrated in a number of simulated examples, as well as its applications to two real-life datasets, including an NBA player statistics data and a cosmetic sales data collected by Alibaba. Supplementary materials for this article are available online.

Functional linear regression is an important topic in functional data analysis. It is commonly assumed that samples of the functional predictor are independent realizations of an underlying stochastic process, and are observed over a grid of points contaminated by iid measurement errors. In practice, however, the dynamical dependence across different curves may exist and the parametric assumption on the error covariance structure could be unrealistic. In this article, we consider functional linear regression with serially dependent observations of the functional predictor, when the contamination of the predictor by the white noise is genuinely functional with fully nonparametric covariance structure. Inspired by the fact that the autocovariance function of observed functional predictors automatically filters out the impact from the unobservable noise term, we propose a novel autocovariance-based generalized method-of-moments estimate of the slope function. We also develop a nonparametric smoothing approach to handle the scenario of partially observed functional predictors. The asymptotic properties of the resulting estimators under different scenarios are established. Finally, we demonstrate that our proposed method significantly outperforms possible competing methods through an extensive set of simulations and an analysis of a public financial dataset.