Point Process Flows

1 INTRODUCTION . Data in real-life takes various forms . Event sequences , as a special form of data , are discrete events in continuous time . This type of data is prevalent in a broad spectrum of areas , for example , patient visits to hospitals , user behavior on social media , credit card transactions , etc . In this setting , each event is discrete , and the temporal dynamics of the events are complex and asynchronous . It is crucial to understand the characteristics and dynamics of such data , so that plausible future prediction , as well as other downstream applications , such as intervention or recommendation , can be performed . Despite recent success in modeling images , videos and texts with the power of deep neural networks ( DNNs ) , the asynchronous and probabilistic nature of event sequence data makes it challenging to utilize the power of off-the-shelf DNN-based models . Temporal point processes ( TPPs ; Daley & Vere-Jones 2007 ) provide us with an elegant and effective mathematical framework for modeling event sequences data . A temporal point process is defined as a stochastic process whose realizations consist of a list of events with their corresponding occurring times . These occurring times can either be real numbers from an index set ( defined from prior knowledge ) or sampled from an intensity function . While other time-series models learn temporal patterns synchronously ( with each time-step being treated as an input to the model ) , TPP-based frameworks directly model the time intervals between events as random variables . With such a setup , it allows for modeling long sequences without vanishing gradients or costly memory issues . Although the temporal point process has shown to be useful in modeling events sequences , it is usually not trivial to come up with a simple yet flexible intensity function . An intensity function encodes the rate an event occurs at a specified time-step . Poisson process ( Kingman , 1992 ) has been a popular hand-crafted design for the intensity which assumes that events are independent of each other . More sophisticated design choices are investigated in the self-exciting ( Hawkes , 1971 ) and self-correcting process ( Isham & Westcott , 1979 ) . The key contribution of these models is to find a functional form of intensity that fits data distribution well by making various parametric assumptions on the underlying generative process of the data . Although shown effective in modeling simple synthetic datasets , parametric assumptions make such frameworks lack the flexibility to model the generative process for real-life and complex data , hindering wider adoption of TPP-based frameworks . Recently , learning the intensity function using recurrent neural networks ( RNNs ) to encode the history has received an increasing amount of attention ( Du et al. , 2016 ; Zhong et al. , 2018 ; Mei & Eisner , 2017 ; Jing & Smola , 2017 ) . In this line of work , history information is encoded and exploited in learning the intensity of the point process distribution . In this case , the explicit parametric assumption on the forms of the intensity functions is relaxed . However , the maximum likelihood training criteria on these models still requires the intensity function to be simple for the likelihood to be tractable . Recent work by Mehrasa et al . ( 2019 ) proposes a probabilistic framework based on variational autoencoders for modeling point process , further facilitating the stochastic generative process . All the literature above is built upon explicit modeling of a temporal point process using the intensity function . It is not necessary to explicitly model the intensity ; a few works have tried to formulate TPP in an intensity-free manner . WGANTPP ( Xiao et al. , 2017 ; 2018 ) introduce an intensity-free framework for modeling the point process distribution using Wasserstein distance . The model is built upon a generative adversarial network ( GAN ) . RLPP ( Li et al. , 2018 ) formulates this problem in a reinforcement learning framework and treats future event predictions as actions taken by an agent . Both of these models are optimized by trying to generate sequences of samples that are indistinguishable from the ground-truth sequences ( by a discriminator in WGANTPP and policy learning in RLPP ) . Although these models are capable of generating realistic sequences , such training criteria fail to model the data distribution , resulting in intractable likelihood . In this work , We propose a novel intensity-free point process model based on continuous normalizing flow and variational autoencoders . The proposed point process flow ( PPF ) model utilizes a recurrent variational autoencoder to encode the history of a given event sequence and makes probabilistic predictions on the next event . It preserves the non-parametric characteristics of point process distributions with normalizing flow . The predicted non-parametric point process distribution is capable of capturing complex time distributions of arbitrary shape , leading to more accurate modeling of event sequences . Extensive experiments are conducted on synthetic and real datasets to evaluate the performance of the proposed model . Experimental results show that our model is capable of capturing complex point process distributions as well as performing accurate stochastic forecast . The contributions are summarized as follows : ( 1 ) A novel intensity-free point process model built upon continuous normalizing flow . The proposed PPF is capable of capturing highly complex temporal distributions and does not rely on restrictive parametric forms ; ( 2 ) PPF can be optimized by maximizing the exact likelihood using change of variable formula , relaxing the constraint in previous works that likelihood has to be tractable ; ( 3 ) Evaluation on both synthetic and challenging real-life datasets shows improvement over the state-of-the-art point process models . 2 PRELIMINARIES . 2.1 TEMPORAL POINT PROCESS . A temporal point process ( TPP ; Daley & Vere-Jones 2007 ) is a mathematical framework for modeling asynchronous sequences of actions . It is a stochastic process whose realization is a sequence of discrete events in time t1 : n = { t1 , t2 , ... , tn } where tn is the time when the nth event occurred . A temporal point process distribution is modeled by specifying the probability density function of the time of the next event : f ( t|Ht ) = λ ( t|Ht ) exp { − ∫ t tn−1 λ ( u|Hu ) du } , ( 1 ) where the intensity function λ ( t|Ht ) is the conditional intensity function . It encodes the expected rate of event happening in a small area around t andHt = { t1 , t2 , ... , tn−1|tn−1 < t } is the sequence of event times up to time t. Many works explored different design choices of intensity function to capture the phenomena of interest . Here we review two popular hand-crafted design choices : Poisson Process . Poisson process ( Kingman , 1992 ) is based on the assumption that events happen independent of each other where the intensity is a fixed positive constant λ ( t ) = λ and λ > 0 . In a more general case , λ could be a function of time λ ( t|Ht ) = λ ( t ) but still independent of other events , which is called inhomogeneous Poisson process . Self-exciting Process ( Hawkes process ) . Self-exciting process ( Hawkes , 1971 ) assumes that occurrence of an event increases the probability of other events happening in near future . Its intensity function has the functional form of λ ( t|Ht ) = µ + α ∑ ti < t exp ( − ( t − ti ) ) , where µ and α are positive constants and ti < t are all the events happening before time t . 2.2 NORMALIZING FLOW . Normalizing flows are generative models that allow both density estimation and sampling . They map simple distributions to complex ones using bijective functions . Specifically , if our interest is to estimate the density function pX of a random vector X ∈ Rd , then normalizing flows assume X = gθ ( Z ) , where gθ : Rd → Rd is a bijective function , and Z ∈ Rd is a random vector with a tractable density function pZ . We further denote the inverse of gθ by fθ . On one hand , the probability density function can be evaluated using the change of variables formula : pX ( x ) = pZ ( fθ ( x ) ) ∣∣∣∣det ( ∂fθ∂x ) ∣∣∣∣ , ( 2 ) where ∂fθ/∂x denotes the Jacobian matrix of fθ . On the other hand , sampling from pX can be done by first drawing a sample from the simple distribution z ∼ pZ , and then apply the bijection x = gθ ( z ) . Given the expressive power of deep neural networks , it is natural to construct gθ as a neural network . However , it requires the bijection gθ to be invertible , and the determinant of the Jacobian matrix should be efficient to compute . Several methods have been proposed along this research direction ( Rezende & Mohamed , 2015 ; Dinh et al. , 2014 ; 2017 ; Kingma et al. , 2016 ; Kingma & Dhariwal , 2018 ; Papamakarios et al. , 2017 ) . An extensive overview of normalizing flow models is given by Kobyzev et al . ( 2019 ) . 2.3 CONTINUOUS NORMALIZING FLOW . From a dynamical systems perspective , the residual network can be regarded as the discretization of an ordinary differential equation ( ODE ; Haber & Ruthotto 2017 ; Chang et al . 2018 ; Lu et al . 2018 ) . Inspired by that , Chen et al . ( 2018 ) propose neural ODE , where the continuous dynamics of hidden units is parameterized using an ordinary differential equation specified by a neural network : dz ( t ) dt = h ( z ( t ) , t , θ ) . ( 3 ) The neural ODE can be used to construct a continuous normalizing flow . The invertibility is naturally guaranteed by the theorem of the existence and uniqueness of the solution of the ODE . Furthermore , using the instantaneous change of variables formula , similar to Equation 2 , the log-density can be evaluated by solving the following ODE : ∂ log p ( z ( t ) ) ∂t = −Tr ( ∂h ∂z ( t ) ) . ( 4 ) Grathwohl et al . ( 2019 ) propose an improved version of neural ODE , named FFJORD , which has lower computational cost by using an unbiased stochastic estimation of the trace of a matrix . 3 PROPOSED FRAMEWORK . We propose an intensity-free flow framework to model the timing of events in point process sequences . More specifically , we learn a non-parametric distribution over the timing of asynchronous event sequences by transforming a simple base probability density through continuous normalizing flow , i.e. , a series of invertible transformations . With our proposed framework , we are able to model complex point process distributions without making any assumption on the functional form of the distribution while being able to evaluate the likelihood of sequences under our model . 3.1 NON-PARAMETRIC POINT PROCESS FLOWS . Let the input be a sequence of asynchronous events { t1 , t2 , ... } , where ti ∈ R+ represents the starting time of the i-th event . We define the inter-arrival time τn as the time difference between the starting time of events tn−1 and tn . Our goal is to model the distribution over inter-arrival time τn given the past history of events inter-arrival times τ1 : n−1 , i.e. , learning to model the conditional distribution p ( τn|τ1 : n−1 ) . Our approach is to construct the distribution over inter-arrival time τn by transforming a simple base distribution through normalizing flow transformations . At time-step n , we assume that inter-arrival time τn was generated by first sampling from a simple distribution p ( zn ) and then transforming the drawn sample zn through an invertible transformation gθ : R→ R+ parametrized by θ : zn ∼ p ( zn ) , τn = gθ ( zn ) . ( 5 ) With this assumption and the change of variable formula discussed in subsection 2.2 , we can write the distribution over inter-arrival time τn as : pθ ( τn ) = p ( zn ) ∣∣∣∣∂fθ∂τn ∣∣∣∣ , ( 6 ) where fθ ( τn ) = g−1θ ( τn ) = zn , and the scalar Jacobin value dfθ/dτn shows the changes in the density when moving from τn to zn . We dropped the determinant term in the change of variable formula , because in our case , the inter-arrival time τn is a one-dimensional variable . With this formulation , we are able to model the inter-arrival time distribution , without any specific assumption on the functional form of the distribution . Exact samples of the inter-arrival time distribution can be obtained by sampling from the base distribution zn ∼ p ( zn ) and transforming it through flow transformation τn = gθ ( zn ) . We are also able to compute the exact likelihood of τn by computing the likelihood of fθ ( τn ) and multiplying it with the associated Jacobian term ∣∣∣ ∂fθ∂τn ∣∣∣ . Current formulation models the inter-arrival distribution of each time-step independent of past history . The future event timing might depend on previous events in a very complex way , so its important to take the history information into consideration while modeling future events . To capture this dependency , we adapt our formulation to construct the point process distribution by learning normalizing flow parameters conditioned on the history . More specifically , we learn the parameters of flow base distribution by a time-dependent model parametrized by ρ that encodes history and provides the conditional base distribution pρ ( zn|τ1 : n−1 ) at each time-step . The overal procedure can be seen in Figure 1 . In our framework , the base distribution is assumed to follow a Gaussian distribution : pρ ( zn|τ1 : n−1 ) = N ( µρn , σ2ρn ) , ( 7 ) where ( µρn , σ 2 ρn ) are the parameters of the base distribution . These parameters can be obtained by encoding the history τ1 : n−1 using various approaches . In this work , we propose two approaches to construct the base distribution of the flow : 1 ) Base distribution with deterministic parameters . 2 ) Base distribution with stochastic parameters . In the following sections , we describe each approach in more details .

Summary: The authors propose a method for learning models for discrete events happening in continuous time by modelling the process as a temporal point process. Instead of learning the conditional intensity for the point process, as is usually the case, the authors instead propose an elegant method based on Normalizing Flows to directly learn the probability distribution of the next time step. To further increase the expressive power of the normalizing flow, they propose using a VAE to learn the underlying input to the "Flow Module". They show by means of extensive experiments on real as well as synthetic data that their approach is able to attain and often surpass state of the art predictive models which rely on parametric modelling of the intensity function. The writers have put their contributions in context well and the presentation of the paper itself is very clear.