Adversarial Data Generation of Multi-category Marked Temporal Point Processes with Sparse, Incomplete, and Small Training Samples

1 INTRODUCTION . Marked Temporal Point Processes ( MTPPs ) are widely used for modeling and analysis of asynchronous stochastic discrete events in continuous time ( Upadhyay et al. , 2018 ; Türkmen et al. , 2019 ; Yan , 2019 ) with applications in numerous domains such as homeland security , cybersecurity , consumer analytics , health care analytics , and social science . An MTPP models stochastic discrete events as marked points ( ei ) defined by its time of the occurrence ti and its category ci . Usually , point processes are characterized using the conditional intensity function , λ∗ ( t ) = λ ( t|Ht ) = P [ event ∈ [ t , t + dt ) |Ht ] , which given the past Ht = { ei = ( zi , ti ) |ti < t } specifies the probability of an event occurring at future time points . There are many popular intensity functional forms . Hawkes process ( self-exciting process ) ( Hawkes , 1971 ) is a point process used in both statistical and machine learning contexts where the intensity is a linear function of past events ( Ht ) ( Türkmen et al. , 2019 ) . In traditional parametric models , the conditional intensity functions are manually pre-specified ( Yan , 2019 ) . Recently , various neural network models ( generally called neural TPP ) have been used to learn arbitrary and unknown distributions while eliminating the manual intensity function selection . Reinforcement learning ( Zhu et al. , 2019 ; Li et al. , 2018 ) , recurrent Neural Networks ( RNN ) ( Du et al. , 2016 ) , and generative neural networks ( Xiao et al. , 2018 ) are used to approximate the intensity functions and learn complex MTPP distributions using larger datasets . Recent advances in data collection techniques allow collecting complex event data which form heterogeneous MTTPs where a marked point ( eij ) defines a time of occurrence ( ti ) and a category ( cj ) separately . Therefore , multi-category MTTPs not only concern about the time of occurrence but also the category of the next marked point . The multi-category MTTPs append extra dimensionality to the distribution which complicates the learning using existing technologies . In fact , multi-category MTPPs are greatly helpful to model the behavioral patterns of suspicious or specific individuals and groups in homeland security ( Campedelli et al. , 2019b ; a ; Hung et al. , 2018 ; 2019 ) , potential malicious network activities in cybersecurity ( Peng et al. , 2017 ) , recommendation systems in consumer analytics ( Vassøy et al. , 2019 ) , and the behavioral patterns of patients to determine certain illnesses ( Islam et al. , 2017 ; Mancini & Paganoni , 2019 ) . A number of challenges limit the collection and access to data in many fields often resulting in small and incomplete datasets . Scenarios involving social , political and crime behaviors are often incomplete due to data collection challenges such as data quality maintenance , privacy and confidentiality issues ( National Institutes of Health & Services , 2020 ) , but still a rigorous analysis with complete data is essential to produce accurate and reliable outcomes . So , there is a critical need for a technique to capture and learn from MTPP distribution , develop and apply machine learning algorithms , etc. , for a small set of data some of which may be incomplete . We present an adversarial multi-category MTPP generation technique which is capable of generating sparse , asynchronous , stochastic , multi-category , discrete events in continuous time based on a limited dataset . Adversarial training has recently evolved and is able to provide exceptional results in many data generation applications , mostly in image , audio , and video generation while precisely mimicking the features of an actual dataset . The primary GAN architecture ( Goodfellow et al. , 2014 ) only engages well for continuous and complete data distributions and GANs have not been used for learning the distribution of discrete variables ( Choi et al. , 2017 ) . Later , GAN architectures for discrete events have been introduced ( Makhzani et al. , 2015 ; Yu et al. , 2017 ) and also applied for MTTP generation using extensive training data ( Xiao et al. , 2018 ; 2017 ) . Adversarial autoencoders ( AAE ) are fluent in capturing latent discrete or continuous distributions ( Makhzani et al. , 2015 ) . In this work , we present feature mapping modules for accommodating incomplete data and make AAE capable of capturing the MTPP distributions of incomplete and small datasets . The incompleteness of the data points can be occurred in following ways . The marked points have been not collected or actors did not originally expose some marked points due to the dynamicity of these stochastic processes , which is the case especially in social and behavioral domains . Main contribution of the paper is a novel technique to synthetically generate high-fidelity multi-category MTPPs using adversarial autoencoders and feature mapping techniques by leveraging sparse , incomplete , and small datasets . To the best of our knowledge , there is no technique available for multi-category MTTP generation using such a dataset which is significantly more challenging than the existing generation scenarios . Section 2 reviews related literature on MTTPs and AAEs . Section 3 presents the definition of multicategory MTTPs and Section 4 discusses the usage of AAEs for incomplete , multi-category MTTP generation . Then Section 5 presents the unique preprocessing and postprocessing techniques include in the feature mapping encoder and the decoder . Section 6 discusses the results of the experiment , and Section 7 summarises the conclusion and future work . 2 RELATED WORK . MTPPs are widely used for modeling of asynchronous stochastic discrete events in continuous time ( Upadhyay et al. , 2018 ; Du et al. , 2016 ; Li et al. , 2018 ; Türkmen et al. , 2019 ) . Usually , an MTTP is defined using a conditional intensity function ( Türkmen et al. , 2019 ) which provides the instantaneous rate of events given previous points . Intensity functions are often approximated by various processes such as the Poisson process , Hawkes process ( self-exciting process ) ( Hawkes , 1971 ) , and self-correcting process ( Isham & Westcott , 1979 ) . In traditional MTPPs , the intensity function has to be explicitly defined ; however any mismatch between the manually defined and the underlying intensity function of a process can have a significant adverse impact on the accuracy of models and outcomes . Deep generative networks avoid the requirement of manually identifying the intensity and thus allows the use of arbitrary and complex distributions . Recurrent Neural Networks ( RNNs ) with reinforcement learning have been widely used in recent years ( Du et al. , 2016 ; Li et al. , 2018 ) as well as several hybrid and extended models are also presented . A stochastic sequential model is proposed in ( Sharma et al. , 2019 ) as a combination of a deep state space model and deterministic RNN for modeling MTPPs . FastPoint ( Türkmen et al. , 2019 ) uses deep RNNs to capture complex temporal patterns and self-excitation dynamics within each mark are modeled using Hawkes processes . A semi-parametric generative model is introduced in ( Zhu et al. , 2019 ) for spatio-temporal event data by combining spatial statistical models with reinforcement learning . The advanced data collection techniques and online social media platforms produce complex event data and thus social network analysis can now be used to inform solutions to many societal issues ( Bonchi et al. , 2011 ) . Many such processes of heterogeneous and complex events require multi-category MTPP-based representation . The data integrity is also a major concern in social networks as many fake and misleading data is not uncommon ( Muramudalige et al. , 2019 ) . In many disciplines , such as economics , biological and social sciences , removal of non-verifiable entries is crucial for maintaining the required data integrity , which in turn leads to incomplete and small datasets . Various techniques are introduced to handle missing data in different contexts ( Folch-Fortuny et al. , 2015 ; MacNeil Vroomen et al. , 2016 ) . Generative adversarial networks ( Goodfellow et al. , 2014 ) have become an alternative for data generation without extensive problem specific theoretical foundation or empirical verification ( Yan , 2019 ) . The initial GAN architecture ( Goodfellow et al. , 2014 ) is capable of capturing the exact distribution of continuous and complete data but can not be used for learning the distribution of discrete variables ( Choi et al. , 2017 ) . The recent improvement in the form of Wasserstein GAN ( Arjovsky et al. , 2017 ) is used to implement generative TPP models ( Xiao et al. , 2018 ) . medGAN is designed to learn the distribution of discrete features , such as diagnosis or medication codes , via a combination of an autoencoder and the adversarial framework ( Choi et al. , 2017 ) . Adversarial Autoencoder ( AAE ) ( Makhzani et al. , 2015 ) is a probabilistic autoencoder which uses the GAN framework as a variational inference algorithm for both discrete and continuous latent variables . An aggregated posterior distribution of q ( z ) on the latent code is defined with the encoding function q ( z|x ) and the data distribution pd ( x ) as follows where x denotes a input sample set . q ( z ) = ∫ x q ( z|x ) pd ( x ) dx ( 1 ) In general usage of an AAE , x represents consistent , discrete or continuous data samples where almost all data points are captured or completed in a given context . The challenge addressed in our paper is to apply an AAE for scattered and incomplete multi-category MTPPs generation using our proposed feature mapping techniques with a data approximation method . Details of such an AAE for sparse , incomplete , and multi-category MTPPs generation and feature mapping techniques are presented in Sections 4 and 5 respectively . 3 MULTI-CATEGORY MARKED TEMPORAL POINT PROCESSES . A marked temporal point process ( MTPP ) represents a certain set of asynchronous stochastic discrete actions/events in continuous time ( Upadhyay et al. , 2018 ; Li et al. , 2018 ) . Due to the immense availability of heterogeneous and complex event data in recent years , it is significant to model such complex events using multi-category MTPPs . A multi-category marked point is denoted as follows . A marked point eij is an event of category cj occurring at time ti as shown in Figure 1 . Tables in Appendix A illustrate categories ( i.e. , cj ) in datasets , where these heterogeneous events exhibits complex dependencies and correlations . Usually , a MTPP analysis deals with an ensemble of marked temporal point processes ( MTPPs ) . Multi-category MTPPs are described as follows . If the kth MTPP is Hk and its marked points are denoted as ekij ∈ Hk . Consider n events ( marked points ) and m categories in the kth MTPP , then marked points are characterized as ekij = ( ti , cj ) where i ∈ [ 1 , n ] and j ∈ [ 1 , m ] . Then , H = { ekij = ( ti , cj ) ∈ Hk ; k ∈ [ 1 , N ] } , ( 2 ) whereH represents N number of multi-category MTPPs . However , without loss of generality , we denote ekij as eij in the following discussion . In some problem domains , i ( time of occurrences ) or j ( categories ) values may change rapidly and some categories may not be recorded frequently . More importantly , with many problems in domains such as social and behavioral sciences , not all marked points ( eij ) of a MTPP are known or observable due to the limitations of information gathering process , confidentiality constraints , unverifiability , deception , etc . A notable aspect of our work , is the use of sparse , incomplete , and multi-category MTPPs where an actor ( an MTTP ) either did not carry out activities corresponding to a certain event categories and marked points or they carried them out , but such activities were not reported in the reliable or permissible sources . To address these challenges , we propose a feature mapping encoder and decoder which are capable of capturing the sparseness and incompleteness of the data . The proposed feature mapping techniques consist of multiple steps including the calculation of cumulative probabilities for each category and a data approximation technique for incomplete data ( briefly described in Algorithm 1 ) . More details of the feature mapping encoder and the decoder are discussed in Section 5 .

Summary: The paper is concentrated at dealing with the data missing problem of MTPP and applies an AAE for the “incomplete multi-categorical MTPPs”.  First, the problem description appears questionable. Point processes are a class of stochastic processes for modelling discrete event sequences in a continuous time domain. They are statistical models and have a well-defined mathematical meaning. The expression of “incomplete point process” is quite confusing. One possible reason is that the authors fail to distinguish between the model and the data. One can say “incomplete data” or “incomplete observations of a model”, but “incomplete model” is not acceptable unless properly defined. Therefore, the proposed method seems not specifically for point processes but for the sequential data.