WEIGHTED CLOCK LOGIC POINT PROCESS

Abstract

Datasets involving multivariate event streams are prevalent in numerous applications. We present a novel framework for modeling temporal point processes called clock logic neural networks (CLNN) which learn weighted clock logic (wCL) formulas as interpretable temporal rules by which some events promote or inhibit other events. Specifically, CLNN models temporal relations between events using conditional intensity rates informed by a set of wCL formulas, which are more expressive than related prior work. Unlike conventional approaches of searching for generative rules through expensive combinatorial optimization, we design smooth activation functions for components of wCL formulas that enable a continuous relaxation of the discrete search space and efficient learning of wCL formulas using gradient-based methods. Experiments on synthetic datasets manifest our model's ability to recover the ground-truth rules and improve computational efficiency. In addition, experiments on real-world datasets show that our models perform competitively when compared with state-of-the-art models.

1. INTRODUCTION AND RELATED WORK

Multivariate event streams are emerging types of data that involve occurrences of different types of events in continuous time. Event streams are observed in a wide range of applications, including but not limited to finance (Bacry et al., 2015) , politics (O'Brien, 2010) , system maintenance (Gunawardana et al., 2011 ), healthcare (Weiss & Page, 2013) , and social networks (Farajtabar et al., 2015) . As opposed to time series data that typically comprises continuous-valued variables evolving in regular discrete time stamps, event streams involve events occurring irregularly and asynchronously in continuous time. Modeling the dynamics in event streams is important for a wide range of scientific and industrial processes, such as predicting the occurrence of events of interest or understanding why some deleterious events occur so as to possibly prevent their occurrence. A (multivariate) temporal point process (TPP) provides a formal mathematical framework for representing event streams, where a conditional intensity rate for each event measures its occurrence rate at any time given the historical events in the stream (Daley & Vere-Jones, 2003; Aalen et al., 2008) . There has been a proliferation of research around TPPs in recent years, particularly around the use of neural networks for modeling conditional intensity rates as a function of historical occurrences (Du et al., 2016; Mei & Eisner, 2017; Xiao et al., 2017; Xu et al., 2017; Gao et al., 2020; Zhang et al., 2020; Zuo et al., 2020) . One stream of research studies graphical event models (GEMs) as a compact and interpretable graphical representation for TPPs, where the conditional intensity rate for any particular event depends only on the history of a subset of the events (Didelez, 2008; Gunawardana & Meek, 2016) . While any TPP can be represented as a GEM, various models make assumptions about the parametric form of conditional intensity rates for the sake of learnability, for instance that rates are piece-wise constant with respect to occurrences within historical windows (Gunawardana et al., 2011; Bhattacharjya et al., 2018) . Ordinal GEMs(OGEM) (Bhattacharjya et al., 2020; 2021) are a recent model from this family where a conditional intensity rate depends on the order in which parent events occur within the most recent historical time period. A temporal logic point process (TLPP) framework was proposed as an alternate way to lend some interpretability to TPPs by modeling intensity rates using temporal logic rules (Li et al., 2020) . Although the initial work pre-specified temporal logic rules, recent work has introduced a temporal logic rule learner (TELLER) for automatically discovering rules (Li et al., 2021) . There is however 1

