Statistical methods improve identification and estimation for nonsymmetric point processes
Identication and Estimation of Nonsymmetric Determinantal Point Processes
Machine Learning
Summary
Determinantal point processes (DPPs) help model patterns where items occur randomly, but traditional methods only handle situations where items tend to repel each other. The authors study nonsymmetric DPPs, which can capture situations where items might also attract each other, a more flexible approach. They develop conditions for these models to be valid, create ways to estimate their parameters, and show the statistical properties of these estimators. Their methods are tested through simulations and applied to real-world cybersecurity data with very different time scales.
What this means in practice
- •For cybersecurity analysts: Model and estimate patterns of security breaches and authentication events with flexible dependence structures across time.
- •For data scientists in finance: Capture and estimate complex dependencies in transactions or event data that involve both repulsion and attraction effects.
Authors
Christian Gouriéroux, Yang Lu
Abstract
Determinantal point processes (DPPs) provide tractable distributions for nite random sets, but the usual symmetric-kernel specication imposes nonpositive pairwise dependence. Statistical research on nonsymmetric DPPs, which can accommodate positive dependence, remains limited. We study identication and estimation for this class. First, we establish the equivalence of two sucient conditions for a nonsymmetric kernel to dene a valid DPP. Second, we develop the identication theory for nonsymmetric DPPs. Third, we develop cardinality-truncated maximum-likelihood estimation and estimation by averaged stochastic gradient descent. Under suitable identication and regularity conditions, we establish consistency and asymptotic normality for both estimators. Simulations examine identication, the eects of truncation, and the nite-sample performance of the estimators. We illustrate the methods with two cyber applications at markedly dierent sampling frequencies: weekly sector-by-breach data and second-by-second enterprise authentication data with more than ve million set observations.