Neural Point Process for Forecasting Spatiotemporal Events
Authors: Zihao Zhou
Xingyi Yang
Xinyi He
Ryan Rossi
Handong Zhao
Rose Yu
Venue:  (2021)
Keywords: spatiotemporal point process
deep sequence models
time series

Abstract: Forecasting events occurring in space and time is a fundamental problem. Existing neural point process models are only temporal and are limited in spatial inference. We propose a family of deep sequence models that integrate spatiotemporal point processes with deep neural networks. Our novel Neural Spatiotemporal Point Process model is flexible, efficient,  and can accurately predict irregularly sampled events.  The key construction of our approach is based on space-time separation of temporal intensity function and time-conditioned spatial density function, which is approximated by kernel density estimation. We validate our model on the synthetic spatiotemporal Hawkes process and self-correcting process. On many benchmark spatiotemporal event forecasting datasets, our model demonstrates superior performances. To the best of our knowledge, this is the first neural point process model that can jointly predict the continuous space and time of events. 

Proceedings of Machine Learning Research vol 144:1-21,2022
Neural Point Process for Learning Spatiotemporal Event Dynamics
Zihao Zhou
ZIZ244 @ UCSD.EDU
UC San Diego
Xingyi Yang
XYANG @ U.NUS.EDU
National University of Singapore
Ryan Rossi
RYROSSI ADOBE.COM
Adobe Research
Handong Zhao
HAZHAO ADOBE.COM
Adobe Research
Rose Yu
ROSEYU @ UCSD.EDU
UC San Diego
Abstract
Learning the dynamics of spatiotemporal events is fundamental problem. Neural point processes enhance the expressivity of point process models with deep neural networks. However, most isting methods only consider temporal dynamics without spatial modeling. We propose Deep Spa- tiotemporal Point Process (DeepSTFP), deep dynamics model that integrates spatiotemporal point processes. Our method is flexible, efficient, and can accurately forecast irregularly sampled events over space and time. The key construction of our approach is the nonparametric space-time intensity function, governed by latent process. The intensity function enjoys closed form inte- gration for the density. The latent process captures the uncertainty of the event sequence. We amortized variational inference infer the latent process with deep networks. Using synthetic datasets, we validate our model can accurately learn the true intensity function. On real-world benchmark datasets, our model demonstrates superior performance over sfate-of-the-ar baselines. Keywords: spatiotemporal dynamics, neural point processes, kernll density estimation
1. Introduction
Accurate modeling of spatiotemporal event dynamics is fundamentally important for disaster re- sponse (Veen and Schoenberg, 2008), logistic optimization (Safikhani al., 2018) and social media analysis (Liang et al., 2019). Compared to other sequence data such as texts or time series, spa- tiotemporal events occur irregularly with uneven time and space intervals.
Discrete-time deep dynamics models such as recurrent neural networks (RNNs) (Hochreiter and Schmidhuber, 1997; Chung et al., 2014) assume events to be evenly sampled. Interpolating an irregular sampled sequence into regular sequence can introduce significant biases (Rehfeld et al., 2011). Furthermore, event sequences contain strong spatiotemporal dependencies. The rate of an event depends on the preceding events, as well as the events geographically correlated to it. Spatiotemporal point processes (STPP) (Daley and Vere-Jones, 2007; Reinhart al., 2018) provides the statistical framework for modeling continuous-time event dynamics. As shown in Fig- ure 1, given the history of events sequence, STPP estimates the intensity function that is evolv-
2022 Zhou, X. Yang, Rossi, Zhao R. Yu.

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DEEP SPATIOTEMPORAL. POINT PROCESS
ing in space and time. However, traditional statistical methods for estimating STPPs often re- quire strong modeling assumptions, feature engineering, and can be computationally expensive.
Machine learning community is observing growing interest in continuous-time deep dy- namics models that can handle irregular time intervals. For example, Neural ODE (Chen et al., 2018) parametrizes the hidden states in an RNN with an ODE. Shukla and Marlin (2018) uses a separate network to interpolates between reference time points. Neural temporal point process (TPP) (Mei and Eisner, 2017: Zhang et al., 2020; Zuo et al., 2020) is an exciting that combines fundamental concepts from temporal point processes with deep learning to model continuous-time event sequences, see recent review on neural TPP (Shchur et al., 2021). However, most of the existing models only focus on temporal dynamics without con- sidering spatial modeling.
Histotical events ((9-))
Upcoming event (8., (c)
User's emjectory
Reprseetive
Figure 1: Illustration of learning spatiotemporal point process. We aim to learn the space-time intensity function given the historical event se- quence and representative points background.
In the real world, while time is a unidirectional process (arrow of time), space extends in multi- ple directions. This fundamental difference from TPP makes it nontrivial to design unified STPP model. The naive approach to approximate the intensity function by deep neural network would lead to intractable integral computation for likelihood. Prior research such as Du et al. (2016) dis- cretizes the space as "markers" and use marked TPP classify the events. This approach cannot produce the space-time intensity function. Okawa et al. (2019) models the spatiotemporal density using mixture of symmetric kernels, which ignores the unidirectional property of time. Chen et al. (2021) proposes to model temporal intensity and spatial density separately with neural ODE, which is computational expensive.
We propose a simple yet efficient approach learn STPP. Our model, Deep Spatiotemporal Point Process (DeepSTPP) marries the principles of spatiotemporal point processes with deep learning. We take non-parametric approach and model the space-time intensity function as mixture of kernels. The parameters of the intensity function are governed by latent stochastic process no sampling which captures the uncertainty the event sequence. The latent process is then inferred via amortized variational inference. That is, we draw sample from the variational distribution for every event. We use a Transformer network to parametrize the variational distribution conditioned on the previous events.
Compared with existing approaches, our model is non-parametric, hence does not make as- sumptions on the parametric form of the distribution. Our approach learns the space-time intensity function jointly without requiring separate models for time-intensity function and spatial density as in Chen et al. (2021). Our model probabilistic by nature and can describe various uncertainties in the data. More importantly, our model enjoys closed form integration, making it feasible for pro- cessing large-scale event datasets. To summarize, our work makes the following key contributions:
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DEEP SPATIOTEMPORAL. POINT PROCESS
Deep Spatiotemporal Point Process. We propose novel Deep Point Process model for forecasting unevenly sampled spatiotemporal events. It integrates deep learning with spa- tiotemporal point processes to learn continuous space-time dynamics.
Neural Latent Process. We model the space-time intensity function using a nonparametric approach, governed by latent stochastic process. We use amortized variational inference to perform inference on the latent process conditioned on the previous events.
Effectiveness, We demonstrate our model using many synthetic and real-workd world spatiotempo- ral event forecasting tasks, where it achieves superior performance accuracy and efficiency. We also derive and implement efficient algorithms for simulating STPPs.
2. Methodology
We first introduce the background of spatiotemporal point process, and then describe our approach to learn the underlying spatiotemporal event dynamics.
2.1. Background on Spatiotemporal Point Process
Spatiotemporal Point Process. Spatiotemporal point process (STPP) models the number of events N(S x (a,b) b)) that occurred in the Cartesian product of the spatial domain S S IR.2 and the time interval (a, b]. It is characterized by non-negative space-time intensity function given the history He = {s11 fI.,..., (Sn, stns
E[N(B(s, As) x (t,t + At))
(ss = lim A8-10A11-0
((s, As)
which is the probability of finding an event in an infinitesimal time interval (t, + At) and an infinitesimal spatial ball S Bs,, centered at location
Example I: Spatiotemporal Hawkes process (STH). Spatiotemporal Hawkes (or self-exciting) process assumes every past event has additive, positive, decaying, and spatially local influence over future events. Such pattern resembles neuronal firing and earthquakes. It is characterized by the following intensity function (Reinhart et al., 2018):
X'(s, t) == (s) =g0((s))))))))))
E (t) : 0
(2)
where go(s) is the probability density of a distribution over S, gi is the triggering kemnel and is often implemented the exponential decay function, 91 (At) a exp(-BAtt) a, B > 0, and 92(s, s:) is the density of an unimodal distribution over centered at s;- Example 2: Spatiotemporal Self-Correcting process (STSC). Self-correcting spatiotemporal point process Isham and Westcott (1979) assumes that the background intensity increases with varying speed at different locations, and the arrival of each event reduces the intensity nearby. STSC can model certain regular event sequences, such alternating home-to-work travel sequence. It has the following intensity function:
(s, pxe (go(s)
Σ Egg2(s si)) : a, B, #
(3)
Here (s) is the density of a distribution over S, and (2 s:) is the density of an unimodal distri- bution over centered at location s,.
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DEEP SPATIOTEMPORAL. POINT PROCESS
Maximum likelihood Estimation. Given a history of events Ht, the joint log-likelihood func- tion of the observed events for STPP as follows:
S/./ Xu,, r)dudr
logp(H)) Σ log (s;,
(4)
Here, the space-time intensity function (*,s t) plays a central role. Maximum likelihood estimation seeks the optimal (* t) from data that optimizes Eqn. (4).
Predictive distribution. Denote the probability density function (PDF) for STPP as f(s, t(H,) which represents the conditional probability that next event will occur at location s and time I, given the history. The PDF is closely related to the intensity function:
X (s(,t) t)
f(s, =
(s, exp (-Js Se, X X" T)drdu)
(5)
F (s, t/H4))
where the cumulative distribution function (CDF), see derivations Appendix A. This means the intensity function specifies the expected number of events in region conditional on the past. The predicted time of the next event is the expected value of the predictive distribution for time (t) in the entire spatial domain:
E [7,]] - - L 'Se ("s
0o
[eep Lxxp
(-/"x(()ddr X² (1)dsdl
tn
Similarly, the predicted location of the next event evaluates to:
// ("(ss / s tn
= foa exp (- (-f"x(e)dr)
/ sX* l)dsdl
E S [7o]
tn
Unfortunately, Eqn. (4) is generally intractable. It requires either strong modeling assumptions or expensive Monte Carlo sampling. We propose the Deep STPP model to simplify the learning.
2.2. Deep Spatiotemporal Point Process (DSTPP)
We propose DeepSTPP, a simple and efficient approach for learning the space-time event dynam- ics. Our model (1) introduces a latent process to capture the uncertainty (2) parametrizes the latent process with deep neural networks to increase model expressivity and (3) approximates the intensity function with a set of spatial and temporal kernel functions.
Neural latent process. Given a sequence of n event, we wish to model the conditional density of observing the next event given the history f(s, H/H)). We introduce latent process to capture the uncertainty of the event history and infer the latent process with armotized variational inference. The latent process dictates the parameters the space-time intensity function. We sample from the latent process using the pe-aaaameterizttion trick Kingma and Welling (2013).
As shown in Figure 2, given the event sequence H, (11,,4), ... (sn, (n) Jt, we encode the entire sequence into the high-dimensional embedding. We use positional encoding to encode the sequence order. To capture the stochasticity in the temporal dynamics, we introduce latent process
4

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DEEP SPATIOTEMPORAL POINT PROCESS
Pample
Zz
decoder
4 (z)
po(s,t|2)
transformer
Wi
Vi
B
1'(s,t) Σ, (.
H {(St,, t.)..., (Sn tens
Figure 2: Design of our DeepSTPP model. For historical event sequence, we encode it with transformer network and map to the latent process (Z1, 2n). We use decoder generate the parameters (w,, Ni, 3c) for each event given the latent process. The estimate intensity is calculated using kernel functions hi and kt and the decoded parameters.
(21, 2n) for the entire sequence. We assume the latent process follows a multivariate Gaussian at each time step:
24 ~ (0(2 = N N(u, Diag(o))
(6)
where the mean f and covariance Diag(o) are the outputs of the embedding neural network. In our implementation, we found using a Transformer Vaswani et al. (2017) with sinusoidal positional en- coding be beneficial. The positions be encoded the normalized event time instead of the in- dex number, to account for the unequal time interval. Recently, Zuo et al. (2020) also demonstrated that Transformer enjoys better performance for learning the intensity in temporal point processes.
Non-parametric model. function (s, t) as:
We take a non-parameteric approach to model the space-time intensity
ntJ Σ
(s,
(s, (kk t4: B:)
Here wi(2), 7e(2), Bz(z) are the parameters for each event that conditioned on the latent process. Specifically, wy represents the non-negative intensity magnitude, implemented with soft-plus ac- tivation function. --))-)) and (x(-,)) are the spatial and temporal kernel functions, respectively. For both kernel functions, we parametrize them as a normalized RBF kernel:
(s, si) = al exp (- w/ls sil)), (t, t:) = exp (- [-Billt- tzll)
(8)
where the bandwidth parameter 7/ controls an event's influence over the spatial domain. The param- eter 3 is the decay rate that represents the event's influence over time. Js exp (-wlls -sil) ds is the normalization constant.
We use a decoder network to generate the parameters {wi, 7. B.} given separately, shown in Figure 2. Each decoder is 4-layer feed-forward network. We use softplus activation function to
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DEEP SPATIOTEMPORAL. POINT PROCESS
ensure w; and % positive. The decay rate 3 can be any number, such that event could have constant or increasing triggering intensity over time.
In addition to n historical events, we also randomly sample J representative points from the spatial domain to approximate the background intensity. This is account for the influence from unobserved events in the background, with varying rates at different absolution locations. The inclusion of these representative points can approximate this background distribution. The model design (7) enjoys a closed form integration, which gives the conditional PDF as:
n+J
f(s (² exp
(t): kettt, t.)]
(9)
3
See the derivation details in Appendix A.2. DeepSTPP circumvents the integration of the intensity function and enjoys fast inference in forecasting future events. In contrast, NSTPP Chen et al. (2021) is relatively inefficient as its ODE solver also requires additional numerical integration.
Parameter learning. Due to the latent process, the posterior becomes intractable. Instead, we use amortized inference by optimizing the evidence lower bound (ELBO) of the likelihood. In particular, given event history Hr, the conditional log-likelihood of the next event is:
log p(s, (HH)) 2 log (s,, (72, 2) (L(ga(()HH)ll((a))
(10)
[ x²())dT (L(g)pp)
=log X* (s,
(11)
where Φ represents the parameters of the encoder network and 0 are the parameters of the decoder network. p(z) is the prior distribution, which we assume to be Gaussian. KL(-II-) is the Kull- back-Leibler divergence between two distributions. We can optimize the objective function in Eqn. (11) w.rt. the parameters & and 0 using back mack-propagation.
3. Related Work
Spatiotemporal Dynamics Learning. Modeling the spatiotemporal dynamics of a system in or- der to forecast the future is a fundamental task in many fields. Most work on spatiotemporal dynamics has been focused on spatiotemporal data measured regular space-time interval, e.g., (Xingjian et al., 2015; Li et al., 2018; Yao et al., 2019; Fang et al., 2019; Geng et al., 2019). For dis- crete spatiotemporal events, statistical methods include space-time point process, see (Moller and Waagepetersen, 2003; Mohler al., 2011). (Zhao et al., 2015) propose multi-task feature learning whereas (Yang et al., 2018) propose RNN-based model to predict spatiotemporal check-in events. These discrete-time models assume data are sampled evenly, thus are unsuitable for our task.
Continous Time Sequence Models. Continuous time sequence models provide an elegant ap- proach for describing irregular sampled time series. For example, (Chen et al., 2018; Jia and Benson, 2019; Dupont et al., 2019; Gholami et al., 2019; Finlay et al., 2020; Kidger et al., 2020; Norcliffe et al., 2021) assumes the latent dynamics are continuous and can be modeled by an ODE. But for high-dimensional spatiotemporal processes, this approach can be computationally expensive. Che et al. (2018); Shukla and Marlin (2018) modifies the hidden states with exponential decay, GRU- ODE-Bayes proposed by De Brouwer et al. (2019) introduces continuous-time version of GRU and Bayesian update network capable of handling sporadic observations. However, Mozer al.

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DEEP SPATIOTEMPORAL. POINT PROCESS
(2017) shows that there is no significant benefit of using continuous-time RNN for discrete event data. Special treatment is still needed for modeling unevenly sampled events.
Deep Point Process. Point process is well-studied in statisties (Moller and Waagepetersen, 2003; Daley and Vere-Jones, 2007; Reinhart et al., 2018). Deep point process couples deep learning with point process and has received considerable attention. For example, neural Hawkes process applies RNNs to approximate the temporal intensity function (Du et al., 2016; Mei and Eisner, 2017; Xiao et al., 2017: Zhang et al., 2020), and (Zuo et al., 2020) employs Transformers. (Shang and Sun, 2019) integrates graph convolution structure. However, all existing works focus on temporal point processes without spatial modeling. For datasets with spatial information, they discretize the space and treat them discrete "markers". Okawa et al. (2019) extends Du et al. (2016) for spatiotemporal event prediction but they only predict the density instead of the next location and time of the event. Zhu et al. (2019) parameterizes the spatial kernel with a neural network embedding without consider the temporal sequence. Recently, Chen et al. (2021) propose neural spatiotemporal point process (NSTPP) which combines continuous-time neural networks with continuous-tin normalizing flows to parameterize spatiotemporal point processes. However, this approach is quite computationally expensive, which requires evaluating the ODE solver for multiple time steps.
4. Experiments
We evaluate DeepSTPP for spatiotemporal prediction using both synthetic and real-world data.
Baselines
We compare DeepSTPP with the statc-of-the-ar models, including
Spatiotemporal Hawkes Process (MLE) (Reinhart et al., 2018): it learns a spatiotemporal para- metric intensity function using maximum likelihood estimation, see derivation in Appendix A.3.
Recurrent Marked Temporal Point Process (RMTPP) (Du et al., 2016): it uses GRU to model the temporal intensity function. We modify this model to take spatial location as marks.
Neural Spatiotemporal Point Process (NSTPP) Chen et al. (2021): a neural point process model that parameterizes the spatial PDF and temporal intensity with continuous normalizing flows. Specifically, we use Jump CNF as it is better fit for Hawkes processes.
All models are implemented PyTorch, trained using the Adam optimizer. We set the number of representative points to be 100. The details of the implementation are deferred to the Appendix C. For the baselines, use the authors' original repositories whenever possible.
Datasets. We simulated two types of STPPs: spatiotemporal Hawkes process (STH) and spa- tiotemporal self-correcting process (STSC) For both STPPs, we generate three synthetic datasets, each with a different parameter setting, denoted as DS1, DS2, and DS3 in the tables. We also de- rive and implement efficient algorithms for simulating STPPs based on Ogata's thinning algorithm Ogata (1981). We view the simulator construction as an independent contribution from this work. The details of the simulation can be found in Appendix B. We use two real-world spatiotemporal event datasets from NSTPP Chen et al. (2021) to benchmark the performance.
Earthquakes Japan: catalog earthquakes data including the location and time of all earth- quakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from the U.S. Geological
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DEEP SPATIOTEMPORAL. POINT PROCESS
Spatial Temporal Self-Correcting
Spatial Temporal Hawkes
00
02
0.1
GT
ADeep
82
STPP
0.3
02
0.1 NSTPP
03
150
170
time
75
80
85
90
time
Figure 3: Ground-truth and learned intensity on two synthetic data. Top: ground-truth; Middle:
learned intensity by our DeepSTPP model. Bottom: learned conditional intensity by NSTPP. 'X's refer to event history, where smaller 'X' refers to larger time difference.
Survey. There are in total 1,050 sequences. The number of events per sequences ranges between 19 to 545
COVID-19: daily county level COVID-19 cases data in New Jersey state published by The New York Times. There are 1,650 sequences and the number of events per sequences ranges between 7 to 305.
For both synthetic data and real-world data, we partition long event sequences into non-overlapping subsequences according to fixed time range T. The targets the last event, and the input is the rest of the events. The number of input events varies across subsequences. For each dataset, we split each into train/val/test sets with the ratio of 8:1:1. All results are the average of 3 runs.
Table 1:
Test log likelihood (LL) and Hellinger distance of distribution (HD) on synthetic data (LL higher better, HD lower is better). Comparison between ours and NSTPP on synthetic datasets from two type of spatiotemporal point processes.
Spatiotemporal Hawkes process
Spatiotemporal Self' Correcting process
DSI
DS2
DS3
DSI
DS2
DS3
LL.
HD
LL
HD
LL
HD
LL
HD
LL
HD
LL
HD 0.2075
DeepSTeP (ours)
-3,8420
0,0033
-31142
0.4920
-3.6327
0.0908
-1.2248
0.2348
-1.4915
0.1813
-1.3927
NSTPP
-5,3110
0,5341
-485644
0,5849
-3,7366
0.1498
-2.0759
0.5426
-2.3612
0.3933
-3.0599
0.3097
4.1. Synthetic Experiment Results
For synthetic data, we know the ground truth intensity function. We compare our method with the best possible estimator: maximum likelihood estimator (MLE), as well as the NSTPP model. The MLE is learned by optimizing the log-likelihood using the BFGS algorithm. RMTPP can only learn the temporal intensity thus is not included in this comparison.
Predictive log-likelihood. Table shows the comparison of the predictive distribution for space and time. We report Log Likelihood (LL) of (s, t)Hi) and the Hellinger Distance (HD) between the predictive distributions and the ground truth averaged over time.
The statistics differ slightly from the original paper due updates in the data source.

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DEEP SPATIOTEMPORAL POINT PROCESS
On both the STH and STSC datasets with different parameter set- tings, DeepSTPP outperform the baseline NSTPP in terms of LL and HD. It shows that DeepSTP? can estimate the spatiotemporal intensity more accurately for point processes with unknown parameters.
Table 2: Estimated (t) MAPE on synthetic data
STH
STSC
DS1
DS2
DS3
DSI
DS2
DS3
DeepSTPP
3.33
369.44
11:30
7.84
3.22
20.98
NSTPP
53.41
17.69
3.85
99.99
39.33
37.39
RMTPP
263.83
729.78
0.62
45.55
21.26
37.46
MLE
2.98
11.30
4.38
27.38
18.20
20.01
Temporal intensity estimate. Ta-
ble 2 shows the mean absolute percentage error (MAPE) between the models' estimated temporal intensity and the ground truth X (t) over short sampled range. On the STH datasets, since MLE has the correct parametric form, it is the theoretical optimum. Compared to baselines, DeepSTPP generally obtained the same or lower MAPE. It shows that joint spatiotemporal modeling also im- prove the performance of temporal prediction.
Intensity visualization. Figure 3 visualizes the learned space-time intensity and the ground truth for STH and STSC, providing strong evidence that DeepSTPP can correctly learn the underlying dynamics of the spatiotempo- ral events. Especially, NSTPP has difficulty in modeling the complex dynamics of the mul- timodal distribution such as the spatiotempo- ral Hawkes process. NSTPP sometimes pro- duces overly smooth intensity surfaces, and lost most of the details at the peak. In contrast, our DeepSTPP can better fit the multimodal distri- bution through the form of kernel summation and obtain more accurate intensity functions.
10000
Training Tame Bacoon3))
100
COVIDNYEO-JP
Figure 4: Log train time comparison on all datasets
Computational efficiency. Figure 4 provides the run time comparison for the training between DeepSTPP and NSTPP for 100 epochs. To ensure fair comparison, all experiments are conducted on 1 GTX 1080 Ti with Intel Core i7-4770 and GB RAM. Our method is 100 times faster than NSTPP in training. It is mainly because our spatiotemporal kernel formulation has close form of integration, which bypasses the complex and cumbersome numerical integration.
4.2. Real- Rerl-World Experiment Results
For real-world data evaluation, re- port the conditional spatial and tem- poral log-likelihoods, i.e., log (s|l) and log (t), of the final event given the input events, respectively. The to- tal log-likelihood, log f* (s, is the summation of the two values.
Table 3: Test log likelihood (LL) comparison for space and time on real-world data over 3 runs.
LL
COVID-19 NY
Earthquake JP
Space
Time
Space
Time
DecpSTP3
-0.1150gnou
2.4583-0.00m
4.402550002s
0.4173gawma
NSTPP
0.079820048
2.6364_n.0.11
-4.814120.116
Predictive performances. As our model is probabilistic, we compare
RMTPP
0.371620007

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DEEP SPATIOTEMPORAL. POINT PROCESS
against baselines models on the test predictive LL for space and time separately in Table 3. RMTPP can only produce temporal intensity thus we only include the time likelihood. We observe that DeepSTFP outperforms NSTPP most of the time in terms of accuracy. It takes only half of the time to train, shown in Figure 4. Furthermore, we see that STPP models (first three rows) achieve higher LL compared with only modeling the time (RMTPP). It suggests the additional benefit of joint spatiotemporal modeling to increases the time prediction ability.
Ablation study We conduct abla- tion studies on the model design. Our model assumes a global latent pro- cess that governs the parameters {w;, 8, 7h with separate decoders. We examine other alternative designs experimentally. (1) Shared decoders: We use one shared decoder to gener- ate model parameters. Shared decoders input the sampled to one decoder and partition its output to generate model parameters.(? Separate process: We assume that each of the {w; 8j, % follows a separate latent process and we sample them separately. Separate processes use three sets of means and variances to sample {wi, Bi, 753 separately. (3) LSTM encoder: We replace the Transformer encoder with LSTM module.
Table 4: Test LL for alternative model designs over 3 runs
(higher better)
COVID-19
Space
Time
Space
Time
Shared decoders
0.11122.001
200001
2.4307
-0 60600.0
Seporate processes
-0.1057 20014
2.4561 20.0045
-2,4200 a00022
-.7022200050
0.1162200112
2.456420.cos
2.4031 200174
0.68455au2222
DeepSTep
0.1150g00:00
2.4583
2.4289gnoica
0.695320.014
As shown in Table 4, we see that (1) Shared decoders decreases the number of parameters but reduces the performance. (2) Separate process largely increases the number of parameters but has negligible influences in test log-likelihood.
(3) LSTM encoder: changing the encoder from Transformer to LSTM also results in slightly worse performance. Therefore, we validate the design of DeepNSTPP: we assume all distribution parameters are governed by one single hidden stochastic process with separate decoders and a Transformer as encoder.
5. Conclusion
We propose a family of deep dynamics models for irregularly sampled spatiotemporal events. Our model, Deep Spatiotemporal Point Process (DeepSTPP), integrates principled spatiotemporal point process with deep neural networks. We derive tractable inference procedure by modeling the space-time intensity function as a composition of kernel functions and a latent stochastic process. We infer the latent process with neural networks following the variational inference procedure. Using synthetic data from the spatiotemporal Hawkes process and self-correcting process, we show that our model can learn the spatiotemporal intensity accurately and efficiently. We demonstrate superior forecasting performance many real-world benchmark spatiotemporal event datasets. Future work include further considering the mutual-exciting structure the intensity function, as well as modeling multiple heterogencous spatiotemporal processes simultancously.
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References
Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. Scientific reports, 8(1):1-12, 2018.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. In Advances in neural information processing systems, pages 657 6571-6583, 2018.
Ricky TQ Chen, Brandon Amos, and Maximilian Nickel. Neural spatio-temporal point processes. ICLR, 2021.
Junyoung Chung, Çaglar Gülçehre, KyungHyun Cho, and Yoshua Bengio. Empirical evaluation of 2014.
gated recurrent neural networks on sequence modeling. CoRR,
Daryl Daley and David Vere-Jones. An introduction to the theory of point processes: volume II: general theory and structure. Springer Science & Business Media, 2007.
Edward De Brouwer, Jaak Simm, Adam Arany, and Yves Moreau. Gru-ode-bayes, Continuous modeling of sporadically-obsereed time series. In Advances in Neural Information Processing Systems, pages 7379-7390, 2019.
Nan Du, Hanjun Dai, Rakshit Trivedi, Utkarsh Upadhyay, Manuel Gomez-Rodriguez,, and Le Song. Recurrent marked temporal point processes: Embedding event history to vector. KDD, pages 1555-1564, 2016.
Emilien Dupont, Arnaud Doucet, and Yee Whye Teh. Augmented neural odes. In Advances in Neural Information Processing Systems, pages 3140-3150, 2019.
Shen Fang, Qi Zhang, Gaofeng Meng, Shiming Xiang, and Chunhong Pan. Gstnet: Global spatial- temporal network for traffic flow prediction. In IJCAI, pages 2286-2293, 2019.
Chris Finlay, Jörn-Henrik Jacobsen, Levon Nurbekyan, and Adam M Oberman. How to train your neural ode. arXiv preprint arXiv:2002.02798, 2020.
Xu Geng, Yaguang Li, Leye Wang, Lingyu Zhang, Qiang Yang, Jieping Ye, and Yan Liu. Spatiotem- poral multi-graph convolution network for ride-hailing demand forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages
2019.
Amir Gholami, Kurt Keutzer, and George Biros. Anode: Unconditionally accurate memory-efficient gradients for neural odes. arXiv preprint arXiv: 10298, 2019.
Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation, 9(8): 1735-1780, 1997.
Valerie Isham and Mark Westcott. A self-correcting point process. Stochastic processes and their applications, 8(3):335-347, 1979.
Junteng Jia and Austin R Benson. Neural jump stochastic differential equations. In NeurIPS, pages 9847-9858, 2019.
11

---

DEEP SPATIOTEMPORAL. POINT PROCESS
Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural controlled differential equa- tions for irregular time series. NeurIPS, 2020.
Diederik P Kingma and Max Welling. arXiv:1312.6114, 2013.
Auto-encoding variational bayes.
arXiv preprint
Yaguang Li, Rose Yu, Cyrus Shahabi, and Yan Liu. Diffusion convolutional recurrent neural net- work: Data-driven traffic forecasting. In ICLR, 2018.
Wenwei Liang, Wei Zhang, and Xiaoling Wang. Deep sequential multi-task modeling for next check-in time and location prediction. In International Conference on Database Systems for Advanced Applications, pages 353-357. Springer, 2019.
Hongyuan Mei and Jason Eisner. The neural hawkes process: A neurally self-modulating multivari- ate point process. NeurIPS, 2017.
George O Mohler, Martin B Short, P Jeffrey Brantingham, Frederic Paik Schoenberg, and George E Tita. Self-exciting point process modeling of crime. Journal of the American Statistical Associ- ation, (66(493):100-100 2011.
Jesper Moller and Rasmus Plenge Waagepetersen. Statistical inference and simulation for spatial point processes. CRC Press, 2003.
Michael Mozer, Denis Kazakov, and Robert V Lindsey. Discrete event, continuous time rnns. arXiv:1710.04110, 2017. 2017.
Alexander Norcliffe, Cristian Bodnar, Ben Day, Jacob Moss, and Pietro Lio. Neural ode processes. ICLR, 2021.
Yosihiko Ogata. On lewis' simulation method for point processes. IEEE transactions on information theory, 27(1):23-31, 1981.
Maya Okawa, Tomoharu Iwata, Takeshi Kurashima, Yusuke Tanaka, Hiroyuki Toda, and Naonori Ueda. Deep mixture point processes: Spatio-temporal event prediction with rich contextual in- formation. In KDD, pages 373-383, 2019.
Kira Rehfeld, Norbert Marwan, Jobst Heitzig, and Jürgen Kurths. Comparison of correlation anal- ysis techniques for irregularly sampled time series. Nonlinear Processes in Geophysics, 18(3): 389-404, 2011.
Alex Reinhart et al. A review of self-exciting spatio-temporal point processes and their applications. Statistical Science, 33(3):299-318, 2018.
Abolfazl Safikhani, Camille Kamga, Sandeep Mudigonda, Sabiheh Sadat Faghih, and Bahman Moghimi. Spatio-temporal modeling of yellow taxi demands in new york city using general- ized star models. International Journal of Forecasting, 2018.
Jin Shang and Mingxuan Sun. Geometric hawkes processes with graph convolutional recurrent neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 4878 4878-4885, 2019.
12

---

DEEP SPATIOTEMPORAL. POINT PROCESS
Oleksandr Shehur, Ali Caner Türkmen, Tim Januschowski, and Stephan Günnemann. Neural tem- poral point processes: A review. arXiv preprint arXiv:2104403528, 2021.
Satya Narayan Shukla and Benjamin Marlin, Interpolation-prediction networks for irregularly sam- pled time series. In ICLR, 2018.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NIPS, 2017.
Alejandro Veen and Frederic P Schoenberg. Estimation of space time branching process models seismology using an em-type algorithm. Journal of the American Statistical Association, 103 (482):614-624, 2008.
Shuai Xiao, Mehrdad Farajtabar, Xiaojing Ye, Junchi Yan, Le Song, and Hongyuan Zha. Wasser- stein learning of deep generative point process models. In NeurlPS, pages 3247-3257, 2017.
Shi Xingjian, Zhourong Chen, Hao Wang, Dit- Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional Istm network: A machine learning approach for precipitation nowcasting. In NeurIPS, pages 802-810, 2015.
Guolei Yang, Ying Cai, and Chandan K. Reddy. Recurrent spatio-temporal point process for check- in time prediction. In CIKM, pages 2203-2211, ACM, 2018.
Huaxiu Yao, Xianfeng Tang, Hua Wei, Guanjie Zheng, and Zhenhui Li. Revisiting spatial-tempords similarity: A deep learning framework for traffic prediction. In Proceedings of the AAAI Confer- ence on Artificial Intelligence, volume 33, pages 5668-5675, 2019.
Qiang Zhang, Aldo Lipani, Omer Kirnap, and Emine Yilmaz. Self-attentive hawkes processes. In International Conference on Machine Learning (ICML), 2020.
Liang Zhao, Qian Sun, Jieping Ye, Feng Chen, Chang- -heng-Tien Lu, and Naren Ramakrishnan. Multi- task learning for spatio-temporal event forecasting. In KDD, pages 1503-1512, 2015.
Shixiang Zhu, Shuang Li, Zhigang Peng, and Yao Xie. Imitation learning of neural spatio-tempora point processes. arXiv preprint arXiv:1906.0544 2019.
Simiao Zuo, Haoming Jiang, Zichong Li, Tuo Zhao, and Hongyuan Zha. Transformer hawkes process. International Conference on Machine Learning (ICML), 2020.
13

---

DEEP SPATIOTEMPORAL. POINT PROCESS
Appendix A. Model Details
A.1. Spatiotemporal Point Process Derivation
Conditional Density. The intensity function and probability density function of STPP is related:
X (*(s t)
f(s, t/Hb)) = 1- F*,
(-1s I \* r)drds
= =s, t) exp
(s, (- (-f(" [(x((T)dt)
tt
The last equation uses the relation that (s = ((())((s))) according Daley and Vere-Jones (2007) Chapter 2.3 (4). Here X (f) is the time intensity and f*((s)) f(slt, H4) is the spatial PDF that the next event will at location s given time According to Daley and Vere-Jones (2007) Chapter 15.4, we can also view STPP as a type of TPP with continuous (spatial) marks.
Likelihood. Given a STPP, the ligelikeld of observing sequence Hc = {(51, &1), (52, t2),,..).... (.. tn is given by:
C(H(m. log
II
/ (s;, 44 H4; (s
li=1
log X2 (si, ti) 10-1//1 1 (T(r)ddds + +log(1 - Fs, t))
Σ
i=1
log 10 (3; / tn
X² T)dr f [. (²(
/ 1-" (²
Σ log X3 (Si (i)
Σ log (silt)) - S"d (²(
n
= Σ log (t,) +
Inference. With a trained STPP and a sequence of history events, we can predict the next event timing and location using their expectations, which evaluate to
f. 1 $/8 f(s, 1/7,, )dsdt = Jen en
f no
texp (-/"x(()at) / X* {*(sst)dsdt,
E [ =
(-f" (((())rr) X² (²)d)
texp
(12)
In
The predicted location for the next event is:
E Eisn+11744,1 8 5/S x"ss,t) [/N(6.t)COD//// Xt, (()) dsdt
=
tn
(- (((()(rr) / sX ss" t)dstt
exp
(13)
tn
14

---

DEEP SPATIOTEMPORAL. POINT PROCESS
Computational Complexity. It is worth noting that both learning and inference require condi- tional intensity. If the conditional intensity has no analytic formula, then we need to compute numerical integration over S. Then, evaluating the likelihood or either expectation requires at least triple integral. Note that [[t][H4,-1] and [[s,[H,,-1]] actually are sextuple integrais, but we can memorize all (s, 1) from = 4 to > 4 to avoid re-compute the intensities. However, memorization leads to high space complexity. As result, generally want avoid an intractable conditional intensity in the model.
A.2. Deep Spatiotemporal Point process (DeepSTPP) Derivation
PDF Derivation The model design of DeepSTPP enjoys closed form formula for the PDF. First recall that
(*tt)= = (t)) exp [-[[x(t)a)
f*())
Also notice that (()) - and X* 1 Ft)
(*))t))**()
Therefore
(s, = (' (t)
= f (t) exp (- [-f("x(taat
exx exp (-f" X((F)ar) lm
= t)
For DeepSTPP, the spatiotemporal intensity is
ts, = Σ wxx (s si)
The temporal intensity simply removes the ki (which integrates one). The bandwidth doesn' 't matter.
x²(t)= Σ wy exp(=B -t))
Integrate (T) yields
I (x(T)dT== -- Σ exp(-3i(T 4.) +
Wi B
Note that deriving the exp would multiply the coefficient -3z. The definite integral is
0 (T
=
-
Σ
Wi 7 Bi
(exp( eexp(-Bi(t (tt)) exp(-Bi(tn tz))]
15

---

DEEP SPATIOTEMPORAL. POINT PROCESS
Then replacing the integral in the original formula yields
((s,t)= X² exp exp
(())dd)
Wi 3
(s, exp
(exp(-Brl) 44) exp(-Ba(Lrr 4))1
The temporal kernel function ke(L, 44)
exp(- 44),, we reach the closed form formula.
Inference
The expectation of the next event time is
( (-fe" (T(T)dr dt
/ 14-1
1 t (t*(t) exp
E
(f)
where the inner integral has closed form. It requires ID numerical integration. Given the predicted time &,, the expectation of the space can be efficiently approximated by
E ES
Σ a P'
lwpke tr)s
where a Wekk Lp) is a normalize coefficient.
A.3. Spatiotemporal Hawkes Process Derivation
Spatiotemporal Hawkes process (STHP). Spatiotemporal Hawkes (or self-exciting) process is one of the most well-known STPPs. It assumes every past event has additive, positive, decay- ing, and spatially local influence over future events. Such a pattern resembles neuronal firing and earthquakes.
Spatiotemporal Hawkes is characterized by the following intensity function (Reinhart al., 2018):
(*ss,) = ==gosss))))
Σ (t p 0 itt
(14)
where go(s) is the probability density of distribution over S, 91 is the triggering kernel and is often implemented as the exponential decay function, 91 (At) == a cxp(- (BBt)) a, B 0, and g2(s,, s;) is the density of an unimodal distribution over S centered at s;.
Maximum Likelihood. For spatiotemporal Hawkes process, we pre-specified the model kernels go and g2(s, Sj) to be Gaussian:
1 -- sn 5 sy)²
go(s)
Egol
exp
(15)
2TT
g2(s, 8;)
==
Egel
exp
- 8, E (s 5n)²
(16)
2t
16

---

DEEP SPATIOTEMPORAL POINT PROCESS
Specifically for the STHP, the second term in the STPP likelihood evaluates to
/-] [ c
T
(T = Uf + a
WN (u)dT
0 5
(0S S 7, S T S T) (usrs
1' i
=TT +a
w drdN(u)
UT-
a
BT -w) 1 dN(u))
3
J
N
HT a
-14 1]
3 1=0
Finally, the STHP likeliielihood is
N
log (s; - μT + a
-11) -1]
L =
3
This model has 11 scalar parameters: 2 for Sy, 3 for Eg,, 3 for Eg2, a, B, and μ. We directly estimate Sy as the mean of {s and then estimate the other parameters by minimizing the negative log-likelihood using the BFGS algorithm. T in the likelihood function is treated as tn
Inference
Based on the general formulas in Appendix A.1, and also note that for an STHP,
(T)) = // x((r)dT-
f.
(T))
4--1 lit
a
i
e (((-11)) -1]
}
a
i1
84-11-1)
-00---1---1
3 j=0
1=
i-1
= uul 14-1)=
a
Σ
e 311- -((001-1th-1111)
E - -1(4--1-1)
j=0
i-1
= =u(t- ti-1)
a
(c-8(1-1-1)) - 1)
Σ
8((-1111) and
3
f=0
/ (ugg(ss,s)dde us
g1 (1, 4)g2(5, s,)ds En((,,t)) / s92(s,s;)ds = Σ g1 (1, 4));
S
==0
sX" μs + Σ (4,44))
0=0
17

---

DEEP SPATIOTEMPORAL, POINT PROCESS
have
i-1
Elt,
=
t
+
a
Σ
8(1 8(1-(z)
1:--
5=0
-
1
Σ
p(t ti-1) dt and
exp
3
1=0
60
E(s/
E(s
1n84
+
a
((tt-t)ss)
4-1
==
exp
c
8(t
4-11)
1
Σ
e
8(4-11 1)) - u(t 1--1)
dt
B
5=
Both require only ID numerical integration.
Spatiotemporal Self-Correcting process (STSCP). A lesser-known example is self-correcting spatiotemporal point process Isham and Westcott (1979). It assumes that the background inten- sity increases with a varying speed at different locations, and the arrival of each event reduces the intensity nearby. The next event is likely to be in a high-intensity region with no recent events. Spatiotemporal self-correcting process capable of modeling some regular event sequences, such as an alternating home-to-work travel sequence. It has the following intensity function:
(s, exe (g0(s)
Σ agg(s, si)) a. B, itt:
(17)
Here (s) is the density of a distribution over S, and g2 s:) is the density of an unimodal distri- bution over S centered at Sj.
Appendix B. Simulation Details
In this appendix, we discuss a general algorithm for simulating any STPP, and specialized algo- rithm for simulating STHP. Both are based an algorithm for simulating any TPP.
B.1. TPP Simulation
The most widely used technique to simulate a temporal point process is Ogata's 's modified thinning algorithm, as shown in Algorithm Daley and Vere-Jones (2007) It is a rejection technique; it samples points from a stationary Poisson process whose intensity is always higher than the ground truth intensity, and then randomly discards some samples get back to the ground truth intensity.
The algorithm requires picking the forms of M (t) and L (t) such that
sup(X* + At), At E [0, L(t))) S (t).
In other words, M*(t) is an upper bound of the actual intensity in [t. t + L(t)]. It is noteworthy that if M (t) is chosen to be too high, most sampled points would be rejected and would lead to inefficient simulation.
18

---

DEEP SPATIOTEMPORAL. POINT PROCESS
When simulating process with decreasing inter-event intensity, such the Hawkes process, (t) and (t) can be simply chosen to be (t) and oc. When simulating a process with in- creasing inter-event intensity, such as the self-correcting process, L (L) is often empirically chosen to be (t), since the next event is very likely to arrive before twice the mean interval length at the beginning of the interval. M (t) is therefore (t + (t)).
Algorithm 1 Ogata Modified Thinning Algorithm for Simulating TPP
Input: Interval [0, T), model parameters
2: < 0, < 0
3: while true do
4: Compute m 4- (( - L({H)
5: Draw At ~ Exp(m) (exponential distribution with mean 1/m)
if At > T then
7: return H
8: end if
if At 1 then
10: L ++1+1 1+1
11:
else
12:
+ ++ AL
13:
Compute À = (t)
14:
Draw 0 1)
15:
if 1/m m > a then
16:
H = HUt
17:
end if
18:
end if
19:
end while=0
B.2. STPP Simulation
It has been mentioned in Section 2.1 that an STPP can be seen as attaching the locations sampled from (') to the events generated by TPP. Simulating an STPP basically adding one step Algorithm 1: sample a new location from (s(t) after retaining new event at t.
As for spatiotemporal self-correcting process, neither f"(s, t) nor (²) has closed form, so the process's spatial domain has to be discretized for simulation. (* can be approximated by ExEs (ss,t))(S) where S is the set of discretized coordinates. L+(t) and M*(t) are chosen to be 2/x*(t) and (*(t= + (t)).. Since f*(s)t is proportional to (*,, sampling location from (s((t is implemented as sampling from a multinomial distribution whose probability mass function is the normalized A ts,
B.3. STHP Simulation
To simulate spatiotemporal Hawkes process with Gaussian kernel, we mainly followed an efficient procedure proposed by Zhuang (2004), that makes use the clustering structure of the Hawkes process and thus does not require repeated calculations of X* (s,tt).
19

---

DEEP SPATIOTEMPORAL. POINT PROCESS
Algorithm 2 Simulating spatiotemporal Hawkes process with Gaussian kernel
I: Generate the background events ((O) with the intensity (s. t) pgo(s), i.e., simulate ho- mogenous Poisson process (pis(H) and sample each event's location from a bivariate Gaussian distribution (S E)
S, G(
3: while G A 0 do
4: for i E G do
Simulate event i's offsprings 0 with the intensity (* = gi s;), i.e., sim- ulate non-homogenous stationary Poisson process Pois(gi(tt (t) by Algorithm 1 and sample each event's location from bivariate Gaussian distribution (s; Σ)
G(++1) U $ SU (
6:
7: end for
8: end while
return S =0
Table 5: Parameter settings for the synthetic dataset
8
1
Eg0
Σg2
ST-Hawkes
DS1
.5
.2
[.2 0; 0.2]
[0.5 O; 0.5.5
DS2
.6
.15
[550
[.1 0.0]
DS3
.3
2
1
01
[.1 01 0.0.1
ST-Self Correcting
DS1
2
[1000
[0.85 0; 0.855
DS2
.3
2
1
04 .4]
[.3 0; 0 .3]
DS3
.4
22
1
[.25 0; 0 25]
[.2 0: 0000.2
B.4. Parameter Settings
For the synthetic dataset, we pre-specified both the STSCP's and the STHP's kernels go and g2 g2(s, sy) to be Gaussian:
go(s) = 1
Egol - exp
-s [0, [0,0))= (0,0))
27
2
(2( s,) = Egg 1/2 cxp
1
1
(s
s)) -,)
2m
2
The STSCP is defined on S = [0, 1| x [0, 1]. while the STHP is defined on S R2. The STSCP's kernel functions normalized according to their cumulative probability S. Table 5 shows the simulation parameters. The STSCP's spatial domain is discretized as an 101 x 101 grid during the simulation.
Appendix C. Experiment Details
In this section, we include experiment configurations and some additional experiment results.
20

---

DEEP SPATIOTEMPORAL POINT PROCESS
C.1. Model Setup Details
For a better understanding of DeepSTPP, we list out the detailed hyperparameter settings in Table 6. We use the same set of hyperparameters across all datasets.
Name
Value
Description
Optimizer Learning rate
Adam
Optimizer of Transformer- VAE is to Adam
0.01 (.0 0.015(Real World)
Momentum
0.9
Epoch
200
Train the VAE for 200 epochs for step prediction
Batch size
128
Encoder: n nlayers
Encoder is composed a stack 3 identical Transformer layers
Encoder: nheads
Number of attention heads each Transformer layer
Encoder: dmmodel
128
3-tuple history is embedded to 128-dimension before fed into encoder
Encoder: dnidden
128
Dimension the feed-for nordafo network model in cach Transformer layer
Positional Encoding
Sinusoidal
Default encoding scheme in Vaswani al. (2017)
Decoder: dhidden
128
Decoders for wi, Sr andy; are all MLPs with 2 hidden layers whose dim 128
de
128
Dimension of the latent variable 2 as shown in Figure
50
Number of representative points described in Section 2.2: 100 during
3
le-3
Scale factor multiplied to the log-likelihood in VAE loss
Table 6: Hyperparameter settings for training DeepSTPP on all datasets.
21


Reviews:
[{"review_id": "hKj5rFzGyb9", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "### Summary\nThis paper introduces a mechanism to learn spatio-temporal point processes using RNNs (for time estimation) and Kernel density estimation for spatial dependencies. They perform experiments on synthetic data generated by famous point processes such as Hawkes process. \n\n### Strong/Weak points\n- The paper is well-written (with a few exceptions)\n- It is very easy to read and understand\n- The model is very easy to understand, but very hard to reason about and to give guarantees\n- It is only tested on synthetic data and no real-world example is tested upon\n\nI have a tendency to reject this paper, as the idea is straightforward: if one is asked to mix RNNs with spatial data, the way to combine these two is very clear. There is nothing new about the theory of point processes either, hence, no new understanding of these types of processes is provided. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "brhmHwmblEc", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposed a neural point process that learns to predict time and location of an event. The temporal component uses the parametrization of Du et al 2016 and the spatial component uses a kernel density function. \n\nPros: \n\nThe model design is reasonable and it performs well on multiple synthetic and real datasets, compared against several appropriate baseline models. \n\nThe paper presentation is fairly clear, with thoroughly documented appendices. \n\nCons: \n\nIt is a lack of novelty. Precisely, I don’t mean the model is not novel at all; it is indeed new. But its novelty and significance is not enough. \n\nThe key design is to factor $\\lambda(s, t)$ as $f(s|t) \\lambda(t)$, but this design is really not well-motivated. In general, coupling $s$ and $t$ inside $\\lambda$ would definitely increase the model expressiveness and, for a neural model, it often won’t increase the number of parameters. E.g., one might simply move $s$ into the power of exp in eqn-5 (and possibly allow $t$ and $s$ to interact with each other by feeding $h$ and $s$ through a neural net)---why and in what sense is the proposed factorization better than (say) this simple one? \n\nI am sure there must be reasons to favor the proposed design, but they need to be carefully thought through and clearly presented---that is missing in the current paper. \n\nThe regularization for $w$ (suppressing the largest past $w$) is a little hacky: if one would like to balance between past and background points, considering their weights sum to 1, the most straightforward way would be to smooth the $w$ distribution, namely aggregating past $w$ and background $w$ and then increasing the entropy of the resulting binary distribution.  \n\nThere are things the authors need to clarify: \n(1) citations for spatiotemporal Hawkes and self-correcting processes; \n(2) $\\lambda$ of eqn-5 is inherited from Du et al 2016 and the authors didn’t mention it. \n\nIt is a little too bold to claim as \"the first neural point process model that can jointly predict both the space and time of events\": many past models can, they are just not great (e.g., they may need discretization). So I suggest the authors revise their claim: if they’d like to be a little more specific (e.g., continuous space), then their claim might be more convincing. \n\nQuestions: \n\nThe list of baseline models has Mei & Eisner 2017 but I didn’t find this model in any presented table or figure. Is this a typo or is it missing? (The authors are not obligated to compare to every single published model but they should keep the paper consistent.)\n\nHow could multiple types of events be handled in this framework? The authors use the parametrization of Du et al where they use Du’s event-type-marker to handle the event-location---this is a pity: the authors propose a new model to handle some new features, but lose the capacity of handling old features. How could they do both? E.g., in the Taxi data, in principle, one might want to model (time, location, pickup/dropoff), but the current model only handles (time, location). \n", "extra_scores": {}, "extra_text": {}}, {"review_id": "G8UvVgxv7FA", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper sets up a spatiotemporal neural network, claiming it hasn't been done before. It may however be similar to a spatiotemporal graph network (using a regular lattice graph connectivity structure). The authors should clarify this. \nThe paper is well written however and sound otherwise. \nSome edits are required:\n- Page 2, ln 1: 'hidden states'? Language needs to be improved here. \n- Section 2, ln 6: Zhao et al (2015) rather. \n- Section 3 refers to 'markers'. In spatial statistics these are marks. I suggest sticking to the terminology of the field? \n- Page 2, ln -4: cannot rather than can not \n- Where is Figure 1 referred to in the text? \n- Page 3, ln -4: What is 'firing'? \n- Section 4.1, ln 1: fix the language\n- Section 4.1: be consisten: 2(a) or 2 (a) - not both\n- Page 6, lns -7, -8: spaces are missing\n- Caption of Figure 4 needs editing. \n- Figure 5: remove the names from the graphics at the top - put into captions. \n- There are many capitals missing in the references and journal names that should be in full. \n- Park (2019) has an et al in it? \n- Is there a published version of Mozer (2017)?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "2OW_FmyeOHe", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This work studies the DNN-based spatiotemporal point process model. It points out the drawback of most existing DNN-based point process models: incapability to incorporate the spatio information. Although in statistics, the spatiotemporal point process is capable of capturing events in continuous space and time, such methods are computation expensive. The theoretical analysis is provided, and experimental comparisons are conducted on synthetic and real data. \n\nStrengths: The model setup is reasonable, and the paper writing is easy to follow. The work successfully includes the space information into the DNN-based point process model with a KDE method. \n\nBut I recommend rejection of the paper for the reasons below.\n\nWeakness: The major concern is the contribution of the work is not significant enough for publishing on ICLR, because most of the work (in fact, almost all the work on the temporal part) is same as the work “Recurrent marked temporal point processes: Embedding event history to vector. KDD, 2016.” as cited by the submission, e.g. the conditional intensity in Eq.(5). The improvement of the model on spatial information is implemented by introducing a KDE method to learn the spatial pdf. The KDE method has a major drawback that the current spatial pdf does not depend on the current time (eq.(3)) which is counter-intuitive, as in most cases it is natural to decide when it happens first and then locate where it happens, just as the definition of \\lambda_(s, t) = f_(s|t)*\\lambda__(t) in the submission. \n\nSome specific concerns: in abstract, the author stated “time-conditioned spatial density function”. This is not accurate as it is not conditioned on current time as mentioned above.\nThe example in the end of sec.3.1 is not appropriate. Because the earthquake observation in the example only contains time and magnitude (no location information), of course it has no advantage over RNN for location prediction. \nThe kernel for KDE is chosen to be Gaussian kernel, so how to choose the hyperparameters, e.g. \\Sigma, for the kernel function? I only saw it is fixed to identity matrix in the experiments, so why choose identity? Any validation experiments?\nIn sec.4.2, why use the ‘max’ regularizer to avoid w_j>>w_k? why not use the L2 regularizer? is the max regularizer better than L2? why set the hyperparameter \\gamma to 1e-4 in experiments? Any validation experiments?\nIn experiments, why fix w^t and b to 0.1, respectively? If my understanding is right, these two parameters should be learned in the training process. \nIn experiments, why are only 1 or 2 baseline models considered in Table 1 and 2? What is the performance of other baselines?\nIn experiments, what is the definition of joint RMSE? \n\n\nTypo: in “Relationship to TPP”, f^*(s|t)=f(s|t,H_t)\nIn Eq.(5), h_{i-1}-->h_{i}    H_{t_{i-1}}-->H_{t_{i}}   \nIn eq.(7), E[t_i|H_t_{i-1}]-->E[t_{i+1}|H_t_i]     E[s_i|H_t_{i-1}]-->E[s_{i+1}| H_t_i]\nSec.4.2, S={s_i-j,t_i-j}_{j=1}^L-->S={s_i,t_i}_{i=1}^N  the RNN input is J but the whole observation is N.\nIn “Baselines”, ofspatiotem --> of spatiotemporal ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: There was a consensus among all the reviewers that the methodological contribution is not significant enough for publication at ICLR. In short, the main contribution of the paper is to include spatial modeling into a deep temporal point process model. However, to do that, they just use a well-known method (KDE) on top of a methodology that is very similar to Du et al., KDD 2016. 

In addition, in the original submission, the specific functional form for KDE was independent on time, as highlighted by the reviewers, which basically separates spatial and temporal modeling. Unfortunately, further experiments performed by the authors failed to show performance benefits on making it dependent on time.

The authors also claim that an additional contribution is the sampling method, however, this seems to thin of a contribution for a full paper.
Decision: Reject