Neural Spatiotemporal Point Processes: Trends and Challenges
Authors: Sumantrak Mukherjee
Mouad Elhamdi
George Mohler
David Antony Selby
Yao Xie
Sebastian Josef Vollmer
Gerrit Großmann
Venue: Accepted by TMLR ()

Abstract: Spatiotemporal point processes (STPPs) are probabilistic models for events occurring in continuous space and time. Real-world event data often exhibits intricate dependencies and heterogeneous dynamics. By incorporating modern deep learning techniques, STPPs can model these complexities more effectively than traditional approaches. Consequently, the fusion of neural methods with STPPs has become an active and rapidly evolving research area. In this review, we categorize existing approaches, unify key design choices, and explain the challenges of working with this data modality. We further highlight emerging trends and diverse application domains. Finally, we identify open challenges and gaps in the literature.

Published in Transactions on Machine Learning Research (11/2025)
Neural Spatiotemporal Point Processes: Trends and Challenges
Sumantrak Mukherjee
swmannokk.muhherjeeddff...dd
Data Science and its Applications, DFKI
Mouad Elhamdi
movad. Shwuaa.eWhammddhuumno...om
College of Computing, Mohammed VI Polytechnic Universily and Data Science and its Applications, DFKI
George Mohler
mohlerg@bc. edu
Department of Computer Science, Boston College
David A. Sclby
david seltony.selyy &dfkiidd
Data Science and its Applications, DFKI
Yao Xie
yoo. pro..iiettisye.gotee edu
Georgia Institute of Technology
Scbastian Vollmer
rebastsan.vollmerr &dfki. de
Data Science and its Applications, DFKI
Gerrit Großmann
Data Science and its Applications, DFKI
Reviewed on OpenReview: https:
// // openreview. net/ net/ forum? id= N69..YWMMM
Abstract
Spatiotemporal point processes (STPPs) are probabilistic models for events occurring in continuous space and time. Real-world event data often exhibit intricate dependencies and heterogeneous dynamics. By incorporating modern deep learning techniques, STPPs can model these complexities more effectively than traditional approaches. Consequently, the fusion of neural methods with STPPs has become an active and rapidly evolving research area. In this review, we categorize existing approaches, unify key design choices, and explain the challenges of working with this data modality. We further highlight emerging trends and diverse application domains. Finally, we identify open challenges and gaps in the literature.
1 Introduction
Real-world events, such as urban crime incidents, epidemic spread, earthquakes, and environmental changes, can be represented as sequences of discrete events with both spatial and temporal components. Studying the spatiotemporal distribution of events and discovering the relationships among diffferett types of events an increasingly important area of research for understanding the dynamics and mechanisms of the occurrence of events. One such paradigm is the spatiotemporal point process (STPP) model, defined a stochastic process that describes the spatial and temporal distribution of discrete events (Daley and Vere-Jones 2007). which is well suited to capture the complex relat ionships between events, including self-excitation, and the interactions between events and spatial covariates across time and space.
Reviews on neural point processes have focused on modeling the temporal dynamics of events using neural 2022 In contrast, reviews that address spatiotemporal
networks (NNs) (Shchur et. al.] 2021 Lin et al. 2021

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Infrastructure
News
Environment
Contextual Information
y 2/
2
X2
Encoder/ Decoder
1
X1
xo+
S0
Predictive
distribution
f1
1
density
T
Time
Time
Figure Schematic of the auloregressive construction of an STPP with two spatial dimensions. Three events are fed into an NN that predicts the likelihood of future event times and locations.
event modeling have focused on traditional statistical methods (Gnzzáeez et al. 2016 Reinhart 2018 More recently, some reviews have explored specific aspects of the use machine learning STPPs (Wikle and Cammit-Mangian 2023 Bernabeu al. 2024); these typically focus on specific aspects rather than providing comprehensive coverage. Meanwhile, work on neural STPPs has achieved remarkable progress.
Our work bridges this gap by systematically exploring design choices, methodological innovations, and key challenges in neural STPPs. To our knowledge, prior survey has comprehensively examined these aspects in this context.
1.1 Why We Need Neural STPPs
Structural differences between space and time make modeling challenging. Time is unidirectional, while spatial propagation is omnidirectional and affected by environmental factors. Traditional methods rely on strong parametric assumptions and independence, limiting flexibility. They struggle with long-range dependencies, fail capture heterogeneous dynamics across space and time, and cannot integrate multi- modal data. Additionally, single-step predictions cause error accumulation. NN-based methods overcome these limitations by encoding space and time efficiently, handling dependencies, and learning heterogeneous patterns. They automate feature extraction, integrate diverse data, and scale effectively.
1.2 Scope and Structure of the Paper
This survey reviews neural STPPs, covering models, applications, and key components in event mod- cling. We focus on studics using point processos with neural parameterization to capture spatiotemporal dynamics. Our literature search included keyword-based queries, citation tracking, and seminal works in neu- ral temporal point processes. By outlining fundamental principles and design choices, we provide a practical foundation for rescarchers. We first introduce the nccessary background and notation, then present available modeling choices, including architectures, training procedures, and metrics. The next section highlights notable applications of neural STPPs, and conclude by discussing open challenges.
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2
Background and Notation
Flavors of Eacco-Encs
Architectures
Applications
Categoricall
Nical
Komel
Natural Diosters
Euclidoan
Okavea
Thou
Erianion
Dong
Lidke
P
Okano
Chem
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2hd
Dong
Puen
Dong
ac nonoformer
Graph-bared
2017
2018
2020
2023
Donn
Guungyon
Deng
Trafic
Euelidean
Theng
RNNA STM
Fpidemioloyy
Figure 2: A timeline of the reviewed methodological and application-foeuse works, along with an overview of neural architectures and application domains. Papers are grouped based on how they encode spatial information, though in some cases these categorizations are not strictly defined.
2.1 Spatiotemporal Point Processes
STPPs are concerned with modeling sequences of random events in continuous space and time (Moller and Waagepetersen 2003 A realization or sample of an STPP is defined up to a time horizon T Rz0- is a finite, ordered event sequence containing pairs X = [(4, 81), (l2, 82)..... (In, 8n)], where t, E [0, T] denotes the time and 8; S the location of event i. Here, S C Rd represents the spatial domain, which is typically bounded region in d-dimensional Euclidean space, and typically
2. For given event sequence, H, {(t, s,) t} denotes the history of the current realization up to (but excluding) time point t (cf. Figure 1.
2.2 Likelihood
An STPP can be specified by defining the likelihood of event soquences. The framework follows the temporal priority principle (Daley and Vere-Jones 2007), which asserts that all causes must precede their effects. As a result, the likelihood of an event sequence X can be expressed in an autoregressive form (Rasmussen) 2018
f(X) II pred (t, 8; Ho.)
ppred (T Ho.
Probability of eveuts after
Likelihnod of observed
where (t, He) the predictive distribution, specifying the conditional probability density function (PDF) for the next event occurring at given timestamp and location 8, given the history of past events He. The term (T H..)) represents the cumulative distribution function (CDF) of the predictive distribution, which gives the probability that an event occurs before at time T, regardless of s. Thus, - F.reed He,)) is the probability of no events occurring after the last observed event. Conveniently, this formulation also provides a principled approach for simulation (i.e., the generation of samples from the underlying stochastic model).
2.3 Intensity Function
In practice, an STPP often described using a (conditional) intensity function (CIF) instead predictive distribution: they are mutually translatable (Chen 2021). The CIF X(t, s He), denoted by (t, s) shorthand for its dependence on He, is defined as:
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Neural STPP
Singlo-Event Prediction
Multi-Event Prediction
Kernel-Based Methods
Continuous Time Methods
Non-Parametric Modeling
Multi-Step Decoders
Full Sequence Sampling
Neural ODE TV-CNF
Diffusion-Baseed Event Modeling
Non-Stationary
Context Dependent
Neural ODE +Jump/Attn
Score-Based Event Modeling
Semi-Parametric
Figure 3: Taxonomy of neural STPP models organized by prediction lask. Single-event prediction includes kernel-based and contimuous-tine and generative approaches, while multi-event prediction covers multi-step decoders and full-sequence sct-based models. History encoders (c.g., RNNs, Transformers) not shown here, as they are architectural components that can be used across both categories.
P(Event E (B(s, A.) [1,2 A.)) H.)
(1, = lim A,,A,
(1)
BBs,, A.) At
where (ss, A.) is a d-dimensional ball (a disk if d = 2) centered at E S with radius 4. and [B(s, A.) denotes its volume.
Events can include additional information beyond location as a mark (Dalcy and Vere-Jones 2007 While the core concept remains unchanged, this approach incorporates multimodal contextual data.
3 Modeling Neural Spatiotemporal Point Processes
Neural STPPss use neural networks to model event evolution and occurrence probabilities, thereby implicitly defining probability density function over event sequences. The choice of modeling approach directly affects training efficiency, tractability of likelihood evaluation, and the quality of generated events. A common design pattern is to construct a latent representation of the event history and decode it to generate future events.
Spatial encoding strategics are a central challenge STPPs, as they strongly inffuence model performance across all modeling paradigms. The representation of space determines how heterogeneity context, and inter-event interactions captured. Temporal encoding methods are not primary focus here, as they are already well covered in prior surveys (Shchur et al. 2021
Modeling choices are often driven by the application context. For example, crime forecasting may prioritize modeling the influence of specific landmarks, earthquake prediction may require fine-grained spatial accuracy, and epidemic modcling might emphasize regional policy effects, favoring coarser spatial resolution. Some applications impose parametric constraints enable hypothesis testing, while safety-critical settings may require explicit uncertainty quantification. These considerations shape the choice of spatial encoding strategy.
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Discretization-base methods simplify modeling but can limit the ability to encode inductive biases such stronger influence from nearby events. Their performance can also be sensitive to spatial granularity, boundary effects, and local correlations. Graph-based approaches, such as graph neural networks (GNNs), alleviate these issucs by modeling interactions between discrete spatial cells.
Continuous-space methods; on the other hand, represent locations via raw coordinates (e.g., latitude- longitude) but risk ignoring anisotropic propagation patterns. To address this, models may employ location- specific parametric kernels, incorporate contextual covariates (c.g., satellite imagery geotagged text), learn spatial embeddings that capture relational structure. Attention mechanisms can implicitly model het- erogencous spatial effects, while GNNs and heterogencous kernels provide explicit representations of spatial interactions.
Section organization. Our categorization in this section reflects both architectural and task-specific considerations, integrating the spatial modeling strategies discussed above. Section 3.1 covers history event encoders, presented as architectural components that can be paired with a variety of decoding strategies. These encoders, implemented via RNNs, Transformers, other architectures, serve reusable mechanisms for representing past event sequences and can be applied in both single-event and multi-event prediction settings.
In contrast, Sections 3.2 and 3.3 are organized by prediction task. For single-event prediction, we include only modeling paradigms that are specific to this lask, namely, kernel-based approaches, continuous-tim neural ODE//CNF models, and non-parametric generative models, which require sequential likelihood evaluation for each predicted event. For multi-event prediction, we include only decoders that are explicitly designed to generate multiple events or predict entire event sequences; these may any history encoding scheme latent modeling approach, but their decoding objective is sequence-level rather than event-by-event.
3.1 History Event Encoder
Spatiotemporal event modeling requires CIF that combines temporal evolution and spatial dependencies (cf. Figure Dp. A common approach to predicting event (t,, is to first encode each historical event st), Hr, as an embedding e; = (t Some architectures directly use raw time ty and space Sj. while others transform them via w(.) (e.g., linear, trigonometric, or logarithmie mappings) and a(-) (e.g.. linear layers or one-hot encodings). history encoder then processes embeddings into a latent state h, to parameterize the CIF.
RNN variants like GRU and L.TTM are common history encoders, updating states as h441 RNN(ei ha) Du et al., 2016 Omi et al.) 2019 Shchur et al., 2020). Their sequential nature reduces storage needs but limits parallelization. Additionally, they suffer from gradient vanishing and long-tern term memory loss (Le and Zuidemna 2016).
Attention mochanisms VVaswani et al. 2017 overcome several limitations of recurrent encodors. They have demonstrated superior performance as history encoders for temporal point processes (Zhang et al. 2020a Zuo al. (020)) and lately in STPPs (Zhou al. 2022 Yuan et al. (023).. Nevertheless, the O(N)) space complexity required to construct the attention matrix can pose practical challenges when dealing with very long event sequences.
3.2 Single Event Prediction
Neural STPPs predict the time and location of the next event. We review parametric, neural, mixture, diffusion, and continuous-thn time models, emphasizing those that capture spatial heterogeneity, contextual influences, and graph-based interactions.
As described in Subsection 3.1 a common strategy is to learn a latent representation of the event history, often using recurrent networks, self-attention mechanisms, or contimuuu- embeddings, and then directly predict the timing, location, or type of the next event from this latent state. Another approach parametrize the conditional intensity function, which describes the instantancous rate of event occurrence given the history, and then this function to sample optimize for the most. likely next event. A
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third paradigm is non-parametric modeling of the intensity, which learns flexible influence functions without strong parametric assumptions, enabling the capture of complex dependencies while retaining interpretability. Finally, interaction-focee focussed methods, such as pernel-based approaches, explicitly represent how past events of different types and positions shape the likelihood of future events.
Fach paradigm offers different trade-offs. Direct latent-history prediction can be highly expressive and computationally efficient at inference time, but may sacrifice interpretability. Parametric intensity modeling preserves the probabilistic foundations of point processes and enables Htelihood-basel training, but often relies on restrictive assumptions. Non- -on-paraettric approaches relax these assumptions and flexibly capture complex dependencies, but typically require large datasets and offer limited interpretability. Interaction- focused methods, such as kernel-based models, provide clear influence functions and thus interpretabilic but face scalability challenges in high-dimensional or dense event settings. Finally, continuous time neural ODE or CNF models enable expressive spatiotemporal dynamics without discretization, but are computationally intensive due to repeated evaluations of diffcrential equations.
3.2.1 Kernel-based Methods
Kernel-based methods fall under the interaction-focustte paradigm. In this framework, past events influence the likelihood of future events through parametcrized kernel functions that capture temporal, spatial, or spatiotemporal dependencies. This design offers several benefits: it provides a transparent mechanism for attributing predictions to specific past events, supports interpretability by making event-event relationships explicit, and can be extended to incorporate flexibility using deep learning to parameterize the kernels. Kernel-based methods are also well-suited to domains where pairwise triggering patterns are of interest. such as modeling aftershocks in seismology or contagion in epidemiology.
Formally, the conditional intensity is often defined as:
(t s) = p(t, s) + Σ K(t',t,s's
(2)
where put,,) 8) the baseline rate, which can vary over time and space, and K(-) models the influence of past events. Traditional models use stationary kernels, assuming time- and space-invariant relationships ETAS, ETAS, Ogata 1998 Hawkes 1971 making strong parametric assumptions and missing heterogeneous effects.
Despite their strengths, kernel-based methods typically assume a specific functional form for the kernels (e.g., exponential, Gaussian), which can limit expressiveness and adaptability to complex data. They also tend to struggle with capturing long-range dependencics, as the influence of distant past events is often forced to decay to negligible values under standard kernel choices. Furthermore, performance may degrade in sparse data regimes, and the computational cost evaluating interactions can grow quadratically with the number of historical events unless approximation techniques are used. Recent work on deep kernel learning and non-paramttric kernel estimation aims to address some of these limitations by learning more flexible influence functions while retaining interpretability.
Non-stationary kernels. Designing the influence kernel K(+) in Equation 2 is crucial for capturing how past events trigger future occureecces. Neural parameterizations of K(-) enable non-stationary dependencies that vary across space, time, and contextual marks, moving beyond fixed parametric forms.
A representative approach is the Gaussian mixture model of Zhu al. 20221b, where K(-) is expressed as:
K(I',, 8,
(1)
10 () 11))
(3)
A neural network embeds spatial coordinates to generate locetion-specific parameters ( =!),, and mixture weights ((4) where E 1,..., L with L denoting number of mixture components. Applying constraints ensures physical interpretability and reflects heterogencous event diffusion. Visualizing learned kernels reveals region- specific influence propagation, making K(-) smooth, adaptive function.
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Alternate formulations consider heterogeneous interactions between events modeled using Mercer's theorem and neural basis functions (Zhu et al. 2022 leverage low-rank decomposition and deep non-stationary influence kernel (Dong et al. 2023a or embed events in graphs for non-Euclidean interactions (Dong et al. 2023b). These can be generalized as:
R Σ Σ
K(L, K(U,,,,8',
where (-) and (-(+) are neural feature maps modeling time displacement, spatial relations, or graph connec- tivity, with ard scaling their contributions. L and R refer to the number of basis kernels used to decompose the kernel in Equation 2
For instance, Dong et al. (2023c) and Dong and Xie (0024) use learned Gaussian bases for anisotropic geography, while Dong et al. (2024) combine stationary Gaussian kernel with a GNN mark kernel to capture network constraints and landmark effects.
Context. Event dynamics are influenced by contextual covariates like georeferenced images or text. Okawa et al. 2019) define event intensity as a spatially localized mixture of kernels:
M
(t, D) = Σ
uy),
(5)
where K(-) is compactly supported Gaussian kernel, D set of contextual features, and uj are spa- tiotemporal anchors uniformly distributed in time and space. Contextual features 2j, extracted from uj, inform mixture weights Zy; through deep network combining image and text embeddings. These weights adapt to heterogeneous conditions (e.g,, urban infrastructure, social events), enabling dynamic kernel weighting centered on Uj while restricting contextual inffuence to local neighborhoods.
Zhang et al. 2023 replace the parametric kernel with a deep kernel K 8, u) = Co(t, 8), gu)),, where g()) is a non-lines transformation by a deep NN, learning complex spatial correlations beyond standard Euclidean distance.
Okawa et al. 2022 propose method for incorporating high-dimensional contextual data into the Hawkes process. They introduce weighting term in the excitation kernel, extracting relevant features using CNNs and employing continuous kernel convolution to transform discretized features into continuous space. This enables capturing spatial heterogeneity and external influences while ensuring tractable optimization.
Semi-paramen kernels.
Zhou et al. 2022 introduce a con-aarmetrrcc sixture-based intensity given
by
n+J
(( = Σ w. (8, K, (t, Bi),
(6)
where each kernel Ka(-) and Ke(-) is a normalized radial basis function, and the parameters {wi, Ys 3c} are drawn from a latent process z. The history is encoded by a Transformer (subsection 3.1) and decoded to parameterize the latent process. The samples of the latent process are further decoded via feodforward network. This method captures uncertainty in the timing and spatial location of events. By augmenting the observed events with J randomly sampled representative points, this approach addresses global background intensity, thereby reducing the reliance strong parametric assumptions.
3.2.2 Continuous Based Methods
While the previous section examined methods that explicitly capture event-event relationships, we now turn continuous-time approaches that directly model the conditional intensity function. These methods enable inference over event distributions in space and event rates at arbitrary time points. Kernel density estimators (KDFs) have traditionally been used to model the spatial distribution of events. While simple to implement,

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this approach often fails to capture the complex and history-dependent nature of spatial patterns observed in real-world spatiotemporal processes. In parallel, kernel-based interaction models (sbsubsection 3.2.1 such as those used in Hawkes processes) focus on explicitly modeling the triggering influence between events through prodefined spatial and temporal kernels, which specify how past events affect the likelihood of future ones.
Continuous time-based CNF models take different route: rather than summing kernels or explicitly param- eterizing triggering, they directly model the spatial distribution of events and how it evolves over continuous time. Following Chen et al. 22021) the conditional intensity is factorized into temporal and spatial terms,
X (1, (1) t):
where (t) captures the overall event rate in time and pa(x t) captures where in space events occur at time t. This decomposition models space and time independently, potentially missing interaction effects. However, this is addressed by making pa t) also depend on previous events and time in extension to the basic modeling technique.
Hidden-state dynamics and ground intensity. To model history-dependence, continnous time hid- den state he summarizes all past events. This hidden state evolves smoothly between events and jumps instantancously when new events occur:
dh, fh, for + t,:
htp ho:
(8)
dt lim 14+t = gi(tes h4. Tt)) e-
(9)
= 9x))) (e.g., softplus head to enforce positivity).
(10)
Equation defines smooth dynamics via neural ODE; Equation 9 injects discrete updates at event times; and Equation 10 maps h, to the ground intensity. This lets the temporal process condition on past events in a continuous time fashion.
Static (time-independent)) CNF for space. A standard CNF learns static spatial density, the for all t, by transporting base density p(zo) through artificial flow time
dx. = f(xx);
(11)
ds
log p(z,) = logp(wo) - 1/" (f dx (Io) do.
(12)
Here, f a neural network that acts like a "velocity field" describing how each point space should move during the transformation. Equation 12 uses the instantaneous change-of-variables formula to track how the density changes along the flow. Because f does not depend on t, the learned distribution p(z) is stationary over real time.
Time- Varying CNF (TV-CNF). The TV-CNF relaxes the stationarity assumption by making the ve- locity field explicitly depend on t:
dx.
dt f(t,xe),
(13)
( (05 dr (T,22)) dr.
logp" (z, |t) = logp(zo) - tr
(14)
Now the spatial distribution can smoothly change with real time, enabling the modeling of non-stationary phenomena (c.g., hotspot that drifts across space). Because the model ignores history, trajectorics for multiple events can be computed in parallel using the reparamederivation::
(1)
t =(st),
x
=
(15)
ds
(n)
In f(sts ann))
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which integrates all event locations {a( in a single ODE solve.
Jump CNF (history-conditioned space). Jump CNFss incorporate history into the spatial model by conditioning on h, and allowing instantancous jumps in location space at event times:
dæ, fa (1, for L + li,
To p(20):
(16)
dt
lim tt,te 9a It, he.) 6-0
(17)
The log-likelihood now has continuous-Iie flow terms and discrete jump terms:
1- tr
@ffe ("o (T, (.. (T, dr - log det Ox:
Oge
(ti, Fe, mini))
logp"(xs (t) = logp(xo)
+
12-1 Ex
S.( tT tr 8x
[ (T, (T, hx)) dr.
O5e (0fe
(18)
The second term in the equation models the change in density up to the last event, and the last term models the change in density from the last event. The integrals account for gradual deformation between events, and the log-determinants capture instantaneous changes at jumps. Because jumps require restarting the ODE solver, computation scales linearly with the number of past events.
Attentive CNF (parallel, history-conddtioned space). The Attentive CNF uses attention to couple all event trajectories and their histories inside a single parallel ODE system:
r
d
X,,
(19)
ds
where fAum is Lipschitz multihead-attenttan network over past events and H denotes their hidden states. This retains history-dependence but avoids the sequential cost of Jump CNTs by evolving all trajectories together.
In contrast to kernel-based methods (sbsubscction 3.2.1 which rely on explicitly defined influence fune- tions, CNF-based approaches adopt a generative viewpoint by parameterizing the continuous evolution of event distributions through neural differential equations. Rather than prescribing parametric kernels, these models learn transformations from simple base distributions into complex spatiotemporal event patterns. The conditional intensity function then derived from the latent trajectory defined by the flow, enabling highly flexible modeling of non-stationary and heterogeneous event. dynamics.
The key advantage of CNFs lies in their expressiveness: they capture intricate dependencios in continuous time without the need for handcrafted kernel forms. This flexibility makes them particularly suitable for domains where interactions are irregular, long-range, or anisotropic. At the same time, CNFs unify density estimation and generative modeling, providing a principled framework for both likelihood- based inference and sample generation.
However, this expressiveness comes with trade-offs. Training CNF-based STPPs typically involves solving ordinary differential equations with neural network dynamics, which introduces significant computational overhead compared to kernel methods. Likelihood estimation requires backpropagating through ODE solvers and computing trace terms, which can suffer from numerical instability. Moreover, while kernel-based ap- proaches yield interpretable influence functions, the latent reprosentations learned by CNFs are implicit, making it harder to attribute event dependencies or quantify influence an interpretable manner.
3.2.3 Non-parametric modeling
Neural STPP models often rely on restrictive assumptions, such as conditional independence or unilateral dependence between the distributions of temporal and spatial events. These assumptions limit their ability accurately predict events in real-world scenarios, where events exhibit complex interdependencies in both time and space.
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Diffusion-based event modeling. To overcome the limitations of conditional independence or unilat- eral dependence in neural STPPs, Yuan et al. (2023) introduce Spatiotemporal Diffusion Point Processes (DSTPP), which directly learn the joint distribution of temporal and spatial events via a denoising diffu- sion probabilistic model (Ho et al.) 2020) without structural constraints. A spatiotemporal self-attention encoder separately embeds time and space; fuses them into history representations, and conditions the dif- fusion process. The forward process gradually perturbs event coordinates with Gaussian noise, while the reverse process iteratively denoises them using a co-attention network that dynamically captures mutual spa- tiotemporal dependencies, enabling joint distribution learning without assuming independence or requiring integrable intensity functions. At each reverse step k, the model predicts
po x
(x) xi x nn--1) = po (111t- | 8 1 ,
(R) (k) n1-1) po (s.s 1) |T! , .h-1);
(K)
(20)
(k) denotes the noisy state at step k. Model parameters 0 are learned by minimizing the denoising where z; objective
[ C= Egggg c [||e- co co xf + vT- an e, h hi-1, +)1"
(21)
where 2f the clean event, ~ NNO, is Gaussian noise, ag 1 3g the variance schodule, and co(+) is the learned denoising network. For spatial decoding, the model directly predicts continuous coordinates or can apply rounding step for discrete locations. This method eliminates the need for approximation during sampling and supports continuous and discrete spatial domains.
Score- based spatiotemporal modeling. While diffusion- STPPs avoid restrictive assumptions, they require multi-step denoising with carefully tuned noise schedules. To provide a simpler yet expressive alternative, Li et al. (2024) introduce SMASH, which directly models future events via their conditional score functions. Instead of parameterizing intensities or perturbing trajectories with diffusion noise, SMASH represents the temporal distribution through the score vi((L k) 0 log k) and the spatial distribution through wi (z t,k) = V p'' (x t,k). Marks are modeled separately with a categorical distribution (r f) x (e(t, This decomposition allows the model to capture complex dependencies across time, space, and marks without assuming conditional independence or integrability of intensities.
New events are generated via Langevin dynamics,
x (n+1) 1) ((() + & ((olln)) + VeRn:
(22)
where v-) is the relevant score function, e is the step size, and on (O, I) is Gaussian noise. This iterative refinement moves noisy samples toward regions of high model density, yiclding continuous event times and locations as well as calibrated mark probabilities. As a result, SMASH provides not only point predictions but also principled uncertainty quantification, producing confidence intervals for event times and confidence regions for locations. Compared to Yuan et al. (2023), who model perturbed distributions via denoising diffusion with multiple noise scales, SMASH achieves comparable flexibility through direct score-based modeling, offering a more efficient and uncertainty-aware generative framework.
Despite their flexibility, diffiusion- and score-based approaches to STPPs have notable trawbacks compared to more classical kernel-based and vontimuuss-imm formulations. First, both diffusion and score matching rely on iterative sampling procedures (multi-step denoising or Langevin dynamics), which are substantially more expensive than the closed-form sampling available in kernel-based models such as Hawkes processes. This limits their scalability when simulating long event sequences. Second, unlike continuous intensity models, which provide explicit hazard rates and interpretable dynamics, basee generative models treat event prediction as implicit density estimation. While this allows them to capture complex dependencies, it also makes interpretability and direct likelihood evaluation more difficult. Finally, these methods require careful tuning of noise schedules (diffusion) or step sizes (Langevin dynamics), and their performance may degrade under distributional shifts where kernel-based methods retain robustness through their parametric structure. As a result, diffusion- and score-basod models trade analytical tractability and efficiency for greater modeling flexibility and uncertainty quantification.
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3.3 Multi-Event Prediction
Single-event prediction methods estimate the CIF one step at time. However, predicting multiple future events becomes computationally expensive because it requires repeated integration over time and space. This process also involves updating the history after each prediction, which can lead to errors that accumulate over longer sequences. Two recent approaches tackle these problems from different angles. The first approach employs multi-step decoders that predict several future events simultancously, reducing both computational overhead and the risk of error accumulation. The second approach bypasses the CIF entirely by modcling intire sequences through diffusion, which allows for parallel sampling and flexible conditioning on partial observations.
3.3.1 Multi-Step Decoders
Rather than predicting one event at a time, Erfanian et al. (2022) introduce a Transformer-based model that generates the next L events in a single forward pass. The model takes as input a history of n past events, each with a timestamp and spatial location, and encodes this into hidden representations that capture spatiotemporal dependencics. The decoder then produces probability distributions for the next events. At each future step / [m+II...., n-L,, it predicts both the inter-event time 4 given the history and the location 21 conditioned on the history and the sampled t.. Time is modeled with an exponential base distribution and space with multivariate Gaussian, both made more flexible through normalizing flows: a softsign bijector for time and RealNVP flow (Dinh et al. 2017) for space. The outputs are full probability distributions over when and where events may occur. Predicting them in parallel avoids the repeated integrations of classical point processes and reduces the step-by-step error accumulation of autoregressive rollouts.
Multi-event forecasting approaches of this kind illustrate the trade-offs of moving beyond singlo-step pre- diction. Parallel decoding offers efficiency and reduces long-horizon drift, but also raises challenges. Fach step has its own probabilistic head, conditioned modeled implicitly but weakens direct coupling across horizons. This can lead to less coherent trajectories when rolled out iveer. Performance also depends on how well the base distributions align with the data, and mismatches must be absorbed by the flows, which can reduce stability. Evaluation adds further complexity. Erfanian et al. (2022) for example, condition spatial likelihoods on the true future times during testing, which provides stronger supervision than would be available in a fully generative rollout. Finally, multi-event predictors often show higher sensitivity to hyperparameters and regularization than single-step models, limiting robustness across datasets.
shared hidden states, which allows interactions to be
Single-step models, by contrast, are straightforward and often easier to interpret, since each prediction is conditioned directly on the observed past. They tend to perform reliably at short horizons but suffer from error accumulation when rolled out over many steps. Multi-event models mitigate this compounding of errors and improve elliciency, but their joint forecasts are less transparent. and sometimes harder to interpret. The choice between them therefore a balance between short-term stability and interpretability on one side, and longer-range efficiency on the other.
3.3.2 Sampling Full Sequences
A different line of work models entire event sequences rather than predicting them one at a time. Lüdke let al. (2023) introduce diffusion-based framework where the basic modeling unit the set of events. Tet a soquence be represented as a set S = ((t,, where t, denotes the timestamp and x, the spatial location (or more general element in a metric space). The forward process gradually corrupts this set by perturbing event coordinates and randomly removing points, yielding sequence of noisy sets { The reverse model is then trained to reconstruct So from Sr by approximating the conditional distributions Po(S,-11 S,,, t = I....,,,
This formulation parametrizes events collectively rather than individually. Instead of specifying an intensity function A(t, H₂) or predicting the next event (tn+1, 1)),, the model learns transformation that maps noisy sets back to realistic point patterns. In practice, events are represented as continuous densities, for example, by approximating each point mass d(,,) with Gaussian kernel. The diffusion dynamics
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Table 1: Parameter estimation methods for Neural Spatiotemporal Point Processes
Method
Description
Strengths
Limitations / Trade-offs
Maximum Likelihood
Fits parameters by maximiz- ing likelihood (or minimiz- ing NLL); requires evaluating the CTF and its integral over space-time.
Statistically efficient; well- understood;
Integral often intractable; needs numerical/Me ap- proximations; expensive in high-dimensional STPPs; sensitive to misspecification.
direct proba- bilistic interpretation; widely used baseline.
Estimation
(MLE)
Amortized Variational
Learns an approximate pos- terior by maximizing an ELBO; often paired with kernel-base based intensities for tractable integration.
Scales to irregular contimoous-imme data; sup-
Quality depends on the variational family; risk of posterior collapse, and added modeling/impp /implemmntttiinn complexity.
Inference
ports latent variables; sometimes admits closed- form integrals.
(AVI)
Score-
Matches gradients of log- density (score) between model and data; DSM vari- ant adds noise for stability and avoids second deriva- lives.
Avoids explicit normalization and integrals; flexible, non- parametric; naturally sup- ports uncertainty quantifica- tion.
Requires differentiability; may be less sample-effcient than MLE; DSM noise re- quires careful tuning.
Matching (SM)
Score-
Decomposes the joint into conditionals; applies score matching for times/ locations and conditional likelihood for marks.
Handles intractable normal- izers; mitigates over/under- confidence via calibrated posteriors; supports marked STPPs.
Needs iterative sampling (e-g., Langevin); compu- tationally heavy at scale; sensitive to the chosen decomposition.
Matching Pseudolikeli- hood
Reinforcement / Imitation Learning
Trains generative policy to match data sequences using a distance (e.g., MMD) instead of likelihood.
No Ikelihood/integrall putation; robust to model mismatch; can optimize task- specific rewards.
Requires well-chosen dis- tance rrewad:: may underuse probabilistic structure; po- tential instability without careful tuning.
then operate directly on these densities, allowing the model to capture both temporal and spatial structure in unified way. Lüdke et. al. (2025) extend this formulation to general metric spaces by decomposing the dynamics into thimning (removing points) and superposition (adding points), which provides a more interpretable view of how events evolve through the diffusion process. Conditioning is implemented with binary masks, making it straightforward to restrict sampling to specific windows or regions without retraining the model.
This parameterization offers notable benefits: events are modeled jointly a set, long-range dependencies are naturally captured, and sampling is parallel rather than sequential. However, there are also drawbacks. First, the representation of point masses via smooth kernels introduces an approximation that may limit precision, particularly in high-resolution or high-dimensional settings. Second, because the model acts on the entire set, interpretability is reduced: the influence of specific past events on future outcomes is less transparent than single-step or intensity-basee models. Finally, while sampling time scales more favorably than autoregressive rollouts, the need for multiple reverse steps still makes the procedure less efficient practice than direct one-step prediction.
Compared to single-step models, which explicitly predict (tx+1, Tx+1) conditioned on the observed history H., diffusion models trade local precision and interpretability for the ability to model entire futures jointly. They often provide more coherent long-range forecasts, but al the cost of less Iransparency and weaker short-term accuracy.
4 Parameter Estimation and Inference
The key objectives of event prediction include enhancing predictive performance, improving generalization and robustness, understanding event dynamics through learned parameters, accounting for event behavior
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heterogeneity, and capturing the inffuence of external factors. Predicting future events from learned model requires sampling from its intensity function. Traditional statistical methods often rely on strong parametric assumptions for modeling event intensities, using techniques such as likelihood-based methods, partial likelihood, the EM algorithm, or Bayesian approaches.
In statistical inference, Maximum Likelihood Estimation (MLF) is most commonly used to fit classical and neural STPPs, typically by maximizing the likelihood function or, equivalently, minimizing the negative log- ligllikli (NLL). For an observed sequence of N events, the NLL is given by (Daley et al. 2003
LNLL
Σ log X 8k)
> (7, du. [0,7] 5
Log- likelhhood of observed events
Expected number of events
When using neural networks to parameterize the CIF, evaluating the integral term is typically intractable. This complexity often requires numerical methods (Chen et al. 2021) or Monte Carlo methods (Mei and Eisner 2017) for likelihood evaluation. However, these strategies can be computationally expensive and prone to numerical errors, particularly in high-dimensionall spatiotemporal domains. While certain simplifying assumptions, such exponential decay (Du et al. 2016) and linear interpolation (Zuo et al. 2020), can lead to closed-form solutions or faster approximations, they often restrict the expressiveness of the model.
In the purely temporal setting, Zhou and Yu (2023) introduce a paradigm for efficient and non-parametric inference of TPPs. They approximate the influence function (cf. Equation 2 via a NN,, using automatic integration to compute its integral. A monotonically increasing intogral network is trainod; its partial derivative defines the CIF. This approach directly yields the CIF and its antiderivative from the network parameters, avoiding functional form restrictions. Building on this foundation, Zhou and Yu (0224) address the computational challenge of integrating the intensity function in 3D spatiotemporal domains by employ- ing automatic integration. This approach learns an integral network, whose partial derivatives with respect to spatial and temporal inputs yield the intensity, ensuring an exact antiderivative without restricting the model's functional form. Furthermore, a ProdNet factorization of the influence function into 1D compo- nents enforces non-negativity while capturing spatiotemporal interactions. Maximizing the log- ligeiikl of observed events learns a highly expressive CIF, improving spatiotemporal event prediction.
Zhou et al. 2 integrate fiexible non-parametric modcling with amortized variational inference. This model captures events in continuous time, capturing irregular sampling dynamics and unifying spatial and temporal dependencies. By employing a kernel-based intensity function, the approach allows for closed-form integration, addressing previously intractable likelihood computations. This design avoids computationally expensive numerical integration inherent in neural ODE-based approaches (Chen et al., 2021). Furthermore, this non-parametric approach avoids restrictive parametric assumptions. Training via amortized variational inference maximizes the evidence lower bound (ELBO) of the likelihood while balancing reconstruction accu- racy and posterior regularization. The framework uses the kernel- based intensity for gradient computation, lacilitating end-to-end optimization of both encoder and decoder parameters.
Zhang et al. (2023) use score matching, which minimizes the Fisher divergence between the model's log- density gradient and the data's log-density gradient, thus bypassing the need to calculate the intractable integral. A denoising score matching (DSM) method is used, which improves stability by introducing a small amount of noise to the data, which avoids the computation of second derivatives. In contrast, [i al. (2024) utilize saatre-mathiinn--br based pseudolikelihood objective, which eliminates the need for explicit caleulation of the normalizing term that makes the likelihood integral intractable. The model decomposes the joint intensity function into a product of conditional distributions, allowing for the application of scoro-matching techniques for event times and locations, while using conditional likelihood for event marks. This approach is designed to overcome overconfidence and underconfidence by learning a posterior distribution that matches the actual data distribution, which also requires score-based sampling with Langevin dynamics.
Reinforcement learning (RL) frameworks provide a training approach that does not rely on likelihood caleulations. Zhu et al. 2021b) employ imitation learning framework to train their model. The learner
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policy is defined by a PDF associated with the CIF of the point process and parameterized by the model's parameters. The goal is for this learner policy to replicate the expert policy reflected in the training data. The training process maximizes the expected reward, determined by the Maximum Mean Discrepancy (MMD) between empirical distributions of the training data and data generated by the learner's policy. This data- driven MMD reward function offers ness to model mismatch as it compares data distributions instead of relying on a predefined likelihood. Additionally, the closod-form representation of the reward function enables computationally efficient optimization via analytical gradient calculation, avoiding intensive inverse reinforcement learning.
Each inference approach presents distinct trade-offs between computational cost, flexibility, and interpretabil- ity. A concise comparison of these methods, highlighting their respective advantages, limitations is provided in Table
5 Evaluation Metrics
Evaluating STPPs requires task-specific metrics, summarised in Table 2 Model fit often assessed using NLL, which measures how well predicted distributions align with observed data, However, NLL prioritizes overall distributional fit over individual event accuracy and can be biased due computational approxima- tions like Monte Carlo integration. Moreover, since NLL conditions ground truth history, it is limited in evaluating true generative capacity.
Alternative distributional metrics address these shorteomings. HD Zhou et al. 2022 directly compares learned and true distributions but requires ground truth intensity and spatial discretization, limiting real- world use, but useful for testing the ability of models to recover known patterns. MMD (Zhu et al. 2021b) avoids strong distributional assumptions by comparing generated and observed sequences but is computationally expensive and sensitive to kernel selection. MMD can also be used with CD (Lüdke et al,, 2025 as the distance measure. This improves distributional comparisons and is useful for evaluating the generative performance of STPPs. SL, mcasured via Wasserstein distance (WD) between two categorical distributions, provides additional insight into model performance by comparing event sequence lengths, especially useful when the task involves measuring case counts, such as in epidemiology or crime modeling.
For point prediction accuracy, MAE, MSE, and RMSE are commonly used. MAE is robust uniform errors, while RMSF is more sensitive to large deviations. Aggregated event predictions rely on normalized MAE (NMAE) (Okawa et al. 2022 and MAPE (Okawa et al. 2019 While NMAE enables cross-scale comparisons, it depends on predefined spatiotemporal regions. MAPE, though intuilive, can be unstable near zero values. Mean relative error (MRE) (Dong et al.) 2023a) assesses intensity differences but also suffers from instability near zero. Prediction accuracy (ACC) measures the correctness of event counts but does not capture spatial or temporal precision.
Uncertainty quantification is crucial for robust evaluation. CS and ECE (Li al., 2024 assess how well predicted confidence intervals align with observed distributions, though ECE's binning requires adaptive methods for reliability.
Selecting metrics depends on the task and dataset. A comprehensive evaluation combines prediction accuracy, distributional fit, and uncertainty quantification.
While NLL has been the standard objective for training and evaluating STPPs, it has notable shortcom- ings (Shchur et al. 2021) when used as the sole evaluation criterion. First, computing the NLL requires tractable access to the conditional intensity function and its integral. For many neural STPPs, this integral is intractable and must be approximated through Monte Carlo sampling, numerical quadrature, tional bounds, which complicates reproducibility and makes comparisons across models less reliable. Second, NLL primarily reflects how well model predicts individual events conditioned on past observations but it does not necessarily capture how faithfully the model reproduces the joint distribution of sequences or higher-order spatio-temporal dependencies.
varia-
To address these limitations, recent work increasingly adopts sample-based distributional metrics such as the Wasserstein distance (WD) and MMD. Unlike NLL. these metrics operate directly on sets of generated
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Table 2: Common Evaluation Metrics for Neural Spatiotemporal Point Processes
Name
Definition
Advantages
NLL
Negative Lig-Likeli (likelihood of ob- served data given the predicted distribution). Hellinger Distance (a measure of distance be- tween two probability distributions).
Directly tied to MLE-based training; widely adopted for distributional fit.
HD
Fine-graines assessment of similarity; often used with known ground-truth distributions.
MMD
Maximun Mean Discrepancy beernel-based comparison of two sample sets).
Distribution-agnene aggiosti,;; captures higher-order differences in generative quality.
MAE MSE
Mean/ Baaam/AAbouutt/Suureeee Ciliquared/Rooo Root Errors for point predictions (time or space).
Straightforward and interpretable; highlight large errors (MSE/RMSE) or typical errors (MAE).
/ RMSE
MAPE
Mean Absolute Percentage Error (ratio- based prediction error).
Scale-independent, intuitive for relative er- across different magnitudes.
SL
Sequence Length (Wasserstein Distance be- tween the categorical distribution of event se- quence lengths).
Suitable for applications where the event count within an interval outweighs spatial and temporal accuracy.
CD
Counting Distance (A generalization of Wasserstein distance for order TPPs to STPPs using L' distance)
Useful for evaluating the generative perfor- mance of a model, especially for multi-step predictions
CS ECE
Calibration Score / Expected Calibration Error (comparison of predicted vs. observed confidence).
Evaluates how well probability estimates re- flect true event frequencies (calibration).
and observed sequences, providing a more faithful assessment of generative quality. WD leverages the underlying geometry of space and time, making well-suited evaluating event distributions with structured dependencies, while MMD provides flexible kernel-based test that is computationally efficient and model- agnostic. Together, these metrics complement NLL by shifting the focus from pointwise predictive fit toward the overall realism and fidelity of generated event sequences.
That said, NLL remains valuable in contexts where likelihood-base inference is required, such as hypothesis testing, model comparison under shared parametric assumptions, or applications where interpretability the intensity function central. Thus, a balanced evaluation strategy should combine NLL with distributional metrics like WD and MMD to provide fuller picture of model performance.
6 Applications
The existing literature on neural STPPs mainly underscores their applications in settings where forecasts of when and where events occur support operational decisions. We organize this section by domain: natural disasters, crime, traffic, and epidemiology, because each area requires different modeling choices concerning space, time, and contextual factors. In practice, many models combine a background rate that carries covariates or marks with a self-cxciting term that encodes event history. Spatial structure represented either in continuous coordinates along road and street networks, and temporal dynamics are modeled in continuous time. Contextual information, such environmental conditions, demographic attributes, traffic states, policy interventions, enters through the background or the marks.
6.1 Natural Disasters
Earthquakes. Nicolis et al. (2021) study daily forecasting for Chile, aiming to predict the maximum seismic rate for the following day and the macro-zone in which this maximum occurs. Their approach begins by estimating daily ETAS intensity maps on regular grid. It then employs recurrent network to predict the scismic rate based on recent ETAS summaries and a convolutional network classify the macro-zone using recent FTAS images. Performance is reported using R2 for rate prediction, accuracy for
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zone classification, and error metrics such as MAE, MSE, and RMSE; the neural components are compared against. ETAS using these measures. Zhang et al. forecasts within & ConvLSTM framework during both training and inference. They compare it with ETAS and ConvLSTM. They evaluate using the CSEP testing suite (number, magnitude, spatial, and likelihood tests) that is standard earthquake forecasting, and report improvements under these lests relative to the basclines.
(2024) propose CL-FETAS, a model that integrates ETAS
Wildfires. Xu al. (2023) develop a marked, mutually exciting spatiotemporal Hawkes model designed for assessing binary ignition risk. The marks combine multimodal covariates, including static and dynamic environmental information, with both discrete and continuous features. Model parameters are estimated by an alternating optimization scheme, and evaluation is carried out on large California dataset. Because ignitions are rare, they summarize detection with the F1 score. Additionally, they report conformal prediction sets for fire magnitude.
6.2 Crime
Dong and Xie (2024 model gunshot incidents in Atlanta with non-stationary spatiotemporal Hawkes process. The temporal triggering is kept simple, while the spatial kernel is allowed to vary over the city and is represented through neural construction that learns location-depnndent shapes. Exogenous fac- tors are represented by a location-dependent background intensity based on census covariates, leading to a conditional intensity that combines contagion and covariate effects. Parameters are estimated by MLE with gradient-based optimization, together with tailored approximations for the spaco-time intograls. The evaluation assesses goodness-of-fi. using log-likelihood and AIC, and lorecasting accuracy through MAE. Results shown for both in in-sample estimation of monthly counts and out-of oit-oo-sapplle weekly predictions, well as comparisons to time series and stationary Hawkes baselines. The paper also inspects the learned spatial kernel and the fitted covariate coefficients to explain spatial heterogeneity in the intensity. Dong et al. (2024) analyze theft and robbery events in Valeneia, focusing on incidents that occur along the city's street network. The model defines a self exciting point process on the network, measuring spatial interaction by street- network distance rather than Euclidean distance, and using an exponential temporal kernel. To encode urban context, each event carries a mark that pairs the crime category with the functional zone inferred from nearby landmarks. The interactions among these marks are learned and regularized through graph attention architecture. Estimation proceeds by MLE using stochastic gradient descent over event subsequences. The study reports training and testing log likelihood, ATC, and MAE for weekly event-coum count prediction, alongside qualitative maps of predicted network intensity. The learned mark interaction struc- ture is used to interpret how the type of crime and the surrouning landmark context relate to triggering patterns.
Both studies use self exciting STPPs with MLE and gradient based training, and both assess models using log-likelihood and AIC for fit and MAE for count prediction. The Atlanta study incorporates non-stationary spatial kernels with continuous-space covariates, whereas the Valencia analysis constrains interactions to street network topology and models crime-landmark relationships through learned mark interactions. To- gether, they show how modeling choices about space (continuous vs. network) and context (covariates vs. marks) shape evaluation under common metrics and provide interpretable summaries alongside predictive performance.
6.3 Traffic
Zhu et al. (202ia) model congestion events in Atlanta using attention-based STPP that combines three components in the conditional intensity: a background term, an exogenous term linked to police 911 traffic incidents, and an endogenous self-excitation term learned by an attention mechanism. Directional spatial dependence is handled with a 'tail-up' formulation that uses stream distance on the highway network. Parameters are estimated by MLE with gradient-based optimization. The study reports goodness-of-fi with log- li-elikiihood and assesses next-event prediction with location accuracy and MAE for event time, comparing to point-process and neural baselines the same data.
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Jin et al. 2023 represent the road system graph and integrates traffic states with congestion events in spatiotemporal graph neural point process. Temporal context is encoded with a link-wise Transformer, spatial dependence with graph convolutions, and event history with a continuous GRU layer. Periodic effects, such as peak hours, are included through a gated term in the intensity function. The model jointly predicts the timing of the next. congestion event and its duration. Training optimizes the NLL for inter-event. times together with an absolute-crror loss for duration. Evaluation on two city datasets uses NLL and MAE for occurrence time, and MAE for duration, alongside comparisons graph-based and neural point-process baselines.
6.4 Epidemiology
Li et al. 2021bb treat confirmed COVID 19 cases as events and introduces intensity free generative STPP learned via imitation learning. The policy-based generator uses recurrent history encoder and nonlinear transformations to sample the next event, bypassing direct likelihood evaluation and avoiding a fixed parametric intensity. Population, lockdown timing, and other covariates are incorporated as features to study how interventions may alter event dynamics. The evaluation focuses on recovering spaliolemporal patterns and forecasting next events, and includes uncertainty quantification for predicted case counts via Poisson confidence intervals. The study also contrasts the imitation learning generator with an MLE trained baseline and uses the learned model to explore counterfactual lockdown scenarios.
Dong et al. (202cc) analyze COVID-19 dataset from Cali, Colombia that records the exact time and location of individual cases. They propose a non-stationary STPP in which the spatial triggering kernel varies by location and is parameterized through a simple neural mapping, allowing spatial anisotropy and heterogeneity while keeping the model interpretable. Exogenous effects from city landmarks (e.g., churches, schools, town halls) are added to the background intensity to capture setting specific promotion. Parameters are learned by MLE with an efficient. stralegy for the likelihood integral. Model fit and forecasting are summarized with log- likelihood and MAE for one week ahead case counts, along with maps illustrating the predicted conditional intensity.
7 Open Challenges
7.1 Reproducibility
A significant barrier to advancing neural STPP research is the lack of standardized experimental setups and consistent bascline comparisons. In contrast to temporal point processos (TPPs), which benefit from unified libraries such as Xue al. (2024), there is currently no widely adopted framework that jointly supports spatial and temporal modeling for STPPs. This limits the ability to incorporate and compare diverse neural architectures, evaluation metrics, and training strategies within controlled environment, making rigorous ablation studies and reliable benchmarking more difficult.
Reproducibility (Bouthillier et al. 2019 Pham et al. 2020) in neural STPPs is particularly sensitive to issues arising from random sampling and stochastic training procedures. Variability induced by sampling in event sequence generation, Monte Carlo integration for likelihood estimation, and non-deterministic GPU operations (Wang et al., (0118) can introduce subtle discrepancies between runs. These effects may be amplificd by the long-range temporal dependencics and coupled spatial-temporal dynamics of STPPs, where small variations in early predictions can propagate through subsequent events. Inconsistencies in data preprocessing, such spatial coordinate normalization, event time quantization, or sub-sequence extraction strategies, can further complicate comparisons between studies.
Furthermore, most methods comprise multiple interchangeable components, such as different temporal en- coders, spatial encoders, and interaction modeling mechanisms. Since new approaches often modify various components, becomes difficult to attribute observed performance gains to specific design choice. The in- terplay between modeling space, modeling time, and capturing spatiotemporal interaction effects introduces additional confounding factors, making it challenging to isolate the source of improvements.
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A unified, open-source library with standardized data handling, reproducibility controls, modular architec- tural components, and configurable experimental protocols could facilitate more transparent, rigorous, and scalable research in neural STPPs.
7.2 Benchmarking
Understanding and evaluating new neural STPP methods requires standardized benchmarking datascts. Ex- isting datasets often suller from selection bias, missing data, and varying granularity, making integration and cross-study comparisons difficult. They also capture diverse spatiotemporal phenomena such as spatial heterogeneity, long-range dependencies, and entangled dynamics, with some showing self-exciting behavior and others self-correcting patterns. The lack of unified, consistently tested dataset collection limits bench- marking to specific attributes or domains, while the absence of comprehensive contextual datasets hinders foundational model development and the study of event dynamics in data-scarce settings.
Key guidelines for dataset curation and splitting can facilitate fair and reproducible evaluation. Standardized schemas improve compatibility, with event-level data including timestamps, spatial coordinates, and optional categorical marks, alongside metadata specifying regions, time zones, and units. Covariates such as weather or temporal indicators can improve modeling, and having standard spatial and temporal resolutions enables meaningful comparison.
The literature points to several effective data-splitting strategies for testing generalization: temporal splits (e.g., roll-forward training on past events, testing on future periods), spatial splits (holding out entire re- gions), combined spatiotemporal splits (unseen regions and future periods), and cross-region splits (evaluating transferability across distinct geographic contexts). These approaches colloctively offer a robust framework for assessing model performance across scenarios.
Synthetic datasets play an important role in neural STPP research by providing controlled environments for understanding model behavior under known conditions. Common examples include Pinwheel (Chen et al. 2021 for spatial heterogeneity and multimodal patterns, spatiotemporal Hawkes and self-correcting processes for self-exciting and self-correcting behaviors (Zhou et al., 2022,, and HawkesGMM (Yuan et al. 2023 for suster-based triggering with long-range dependencies. These benchmarks capture real-world phenomena such as entangled spatiotemporal dynamics, seasonal patterns, and varying background rates. Such controlled datasets allow for isolating specific model behaviors before applying methods to real-world data with unknown ground truth. We see need for the creation of simulation testbed for spatiotemporal point processes that captures key challenges in STPP modeling. The testbed should incorporate spatiolemporal entanglement, heterogeneous spatial and temporal patterns, multiple event types with interdependencies, and contextual effocts that include both spurious and genuine causal prodictors. Tunable paramcters would allow the complexity of these factors to be varied, enabling controlled experiments across a range of modeling scenarios.
Real-world datasets in neural STPP research capture diverse spatiotemporal phenomena that challenge current methods. Earthquake catalogs from Japan (U.S. Geological Survey 2020) and California (NCEDC)
Ross et al. 2019) exhibit long-range dependencies with continuous processes and varying background rates. Epidemic datasets using COVTD-19 data from New Jersey (Chen et al. 2021). ACLED conflict records, and EMPRES-i disease (Okawa et al., (022) outbreak reports reveal self-exciting behavior with seasonal effects and entangled spatiotemporal dynamics. Urban mobility datasets, including Citibike NYC (Chen et al. 2 and taxi records, capture spatial heterogeneity through high-volume flows with clustering patterns. Public safety datasets involving Atlanta 911 calls (Zhu et al. ) Chicago crime records (Okawa et al. 2 Vancouver crime data, and NYC collision reports (Zhang et al. (023)) showcase both self-exciting and self- off-correeting patterns with heterogeneous event types across different urban environments.
2014
Contextual datascts are becoming incrcasingly important as they provide environmental, demographic, and infrastructure information. For example, National Geophysical Data Center nightlights capture patterns of human activity, while Natural Earth land cover and GeoNetwork weather data explain spatial and temporal event variation. Additionally, population maps and gROADS contribute demographic and infrastructure context (Okawa et al., (2022). Many of these datasets now include multimodal information, such as text, categorical attributes, and numerical features, all aligned in both space and time. A unified repository that
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links each event dataset to its associated contextual sources, annotated with geotags and collection periods, would be valuable resource for advancing the field.
Building on this, the neural STPP literature points toward the broader need for standardized dataset repos- itories that combine synthetic benchmarks, real-world datasets, and contextual information under unified schemas. Such resources would feature consistent preprocessing pipelines, standardized data formats, and comprehensive documentation with versioned releases and fixed splits that support reproducible research.
7.3 Architectures
Generative strategies and alternative training objectives represent promising yet underexplored avenues for STPP research. While GAN-based architectures and Wasserstein objectives have shown promise in temporal point processes (Xiao et al. 2017), their adaptation to sosppaiotemporal applications remains limited. Similarly, methods moving beyond MLE, particularly those loveraging Transformer-based models instead of RNNS, could enhance the flexibility and robustness of event modeling. Continuous- neural TPPs, such as those proposed by Biloš et al. (021) and Chen et al. (2024), have demonstrated potential for temporal modeling but have yet to be sufficiently applied to spatial event modeling. Many existing neural STPPs treat space and time independently, missing opportunities to improve temporal predictions by capturing spatial relationships. Integrating GNNs to enhance spatial modeling remains an open challenge.
7.4 Interpretability
While interpretability in temporal point processes has received considerable attention, with advances in event attribution (Zhang et al.] 2020b; Cüppers et al., 2024), counterfactual reasoning (Noorbakhsh and Rodriguez 2022 Zhang et al. 2022; Großmann et al.) 2024). and rule-based interpretability techniques (Li et al.. 2021a Kuang et al. 2024 Yang et al.] 2024), research in the spatiotemporal domain remains relatively nascent. Extending interpretability to STPPs introduces unique challenges, such as disentangling spatial and temporal influences, capturing heterogencous covariates, and accounting for interaction effects across modalities. Kernel-based methods provide a promising starting point, as they offer explicit representat ions of event-event relationships and triggering patterns. However, they often struggle to model background rates unrelated to observed events, suggesting opportunities for hybrid formulations. Recent progress in deep kernel (Dong et al.] 2023b Zhu al.] 2022) approaches and kernel-dependent designs indicates that is possible to retain interpretability while improving flexibility.
Recent works (Okawa et al. 2022 2019 Dong et al. 2024 have explored the use of covariates, such as satellite imagery and local landmarks, to inform event rates and propagation dynamics. These approaches leverage contextual data to retain interpretability through kernel-based formulations, while offering greater flexibility by moving beyond standard parametric kernel assumptions. This research direction remains promising and warrants further investigation, particularly by extending beyond parametric forms. Potential avenues include continuous-ting formulations combined with covariates to reveal how specific factors shape event rates and propagation over space and time without discretization artifacts, as well as fully neural diffusion-based methods to more deeply investigate the effects of contextual variables.
Semi- palametriaec models Zhou et al. 2022 Zhou and Yu 2023) continue to offer valuable interpretability by decomposing the conditional intensity into components such as baseline rates, self and cross excitation, co- variate effects, and spatial terms. Developing STPP models with semi-parameee direction of rescarch as it retains flexibility while also improving usability through interpretable components.
components is a promising
Reinforcement bearnnggbassed approaches have been studied in the context of temporal point processes (Li et al.) 2018 Upadhyay et al. 2018 Qu et al.] 2023); ; however, only a few works have extended them to the spatiotemporal sctting (Zhu et al.] 2021b, Li et al. 2021b), which remains relatively understudied. This is a promising avenue of research, as RL formulations learn policies that capture event dynamics, while counterfactual rollouts offer a way to examine the effects of different. interventions and their potential consequences.
For fully neural STPPs, major opportunity lies in developing techniques that expose the decision process of the model without significantly sacrificing performance. Possible directions include explicit intensity decom-
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positions Panos 2024 within neural architectures, attention mechanisms capturing covariate influence and event-event relat relationships (Meng et al. 2024a Shou et al. 2023) interpretable latent representations Meng et al. 2024b,, and counterfactual generators that systematically probe the impact of spatial or temporal perturbations. bations. Addressing these challenges will not only improve transparency but also enable the use of neural STPPs in sensitive domains where explainability essential for trust and adoption.
7.5 Applicability
Despite improvements in predictive performance, neural STPPs are rarely utilized to inform policy decisions or design interventions. Their lack of interpretability limits their applicability in real-world scenarios, where generalizable, interpretable models with uncertainty quantification are essential. Policy-makers often seek to understand the impact of interventions, necessitating models that incorporate contextual factors and causal effects in event propagation. This underscores the need for research in interpretable neural networks, neuro- symbolic methods, and counterfactual analysis for retrospective policy evaluation and deeper insights into event modeling.
7.6 Causality and Uncertainty Quantification
Policy-focused applications of neural STPPs require deeper insights into event propagation and the causal fac- tors inftuencing event rates. However, causal inference and uncertainty quantification remain underexplored in this field. There has been notable progress causal inference for neural spatiotemporal methods (Oprescul et al.
2025
Wang et al.] 2024bb Deng et al. 2023) and for spatiotemporal point processes (Papadogcorgon Christiansen et al. 2022). but the extension of these ideas to neural STPPs has been limited.
et al.
2022
Most existing neural STPPs optimize for predictive likelihood or simulation fidelity, capturing statistical dependencies in space and time but without explicitly distinguishing correlation from causation. In many real-world domains, such as modeling disease spread, human mobility, or crime policy decisions rely on de- termining whether an observed influence reflects a genuine causal relationship or shared exogenous drivers. For example, a model may learn that events one region often precede events in a neighboring area, but without causal analysis, it is unclear whether this due to true propagation, common underlying conditions, artifacts of data collection. Causal representation learning for STPPs must address spatio-temporal con- founding Wang et al. 2024a), and existing advances in causal inference for temporal point processes (Gong ee al.) 2024a Gao et al.) 2021 Zhang et al., 2022 Noorbakhsh and Rodrigues 2022) could be extended to the spatiotemporal domain, potentially leveraging structured intensity decompositions, counterfactual reasoning, and invariant reprosentation Icarning (Arjovsky et al. 2019).
Uncertainty quantification is equally important for neural STPPs, particularly in policy-facing settings where overconfident predictions can have harmful consequences. Unlike traditional statistical STPPs, where un- certainty is often explicit in the paramctric form, many neural models produce point estimates of intensity functions or event parameters without calibrated confidence measures. Bayesian methods, which could in- tegrate expert knowledge through priors and yield posterior uncertainty estimates, have not been widely adopted in neural spatiotemporal event modeling @Dubey et al. (023a).. Other promising directions include variational inference for intensity distributions (Li et al., 2024) and deep ensembles (Dubey et al. 2023b; Rahaman et al. 2021) to capture epistemic uncertainty. In spatiotemporal contexts, structured uncertainty modeling could further account for heteroskodasticity arising from uneven spatial coverage, time-varying data quality, or rare-event regimes.
Addressing causality and uncertainty jointly is particularly important. Causal effect estimates without uncertainty quantification risk overconfident and mislcading recommendations, while uncertainty estimates without causal grounding cannot separale genuine effect variability from confounding, Developing neural STPP frameworks that integrate causal structure learning with calibrated uncertainty, such as Bayesian causal neural architectures or counterfactual generative models, could lead to robust, interpretable, and trustworthy models for complex spatio-temporal intervention design.
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Broader Impact
Point processes, inchuding their spatiotemporal extensions, are increasingly applied in ethically sensitive domains such as police patrol allocation (Mohler et al., 2015], epidemiological surveillance (Alikhademi et al., 2022), disaster response, and urban resource planning. In such settings, the stakes of predictive modeling are high, and errors or biases in predictions may cause disparate harms across demographic, socioeconomic, or geographic groups. For example, crime prediction models have been shown to risk reinforcing existing structural biases when trained on historically biased datasets, leading to potential over-policing of certain communities (Lum and Isaac, 2016 Ensign et al. 2018 Brantingham et al.) 2018). Similarly, in public health contexts, biases in surveillance data may exacerbate disparities in detection and intervention across populations.
The use of neural STPPs for spatial interventions introduces specific ethical challenges. First, data-related issues [Westreich] 2012 Price and Ball 2015) such as selection bias, missing data, under-reporting, measurement error can lead to distorted inferences. While such issues affect both traditional statistical STPPs and their ncural counterparts, the latter may be more difficult to audit due to their complexity and lack of interprelability. As a result, it is important to critically assess whether these models can be relied upon to inform policy decisions (Dong and Xie 2024), particularly in high-impact settings where predictive outputs might be misinterpreted as causal.
Second, contextual covariates used in modeling, such satellite imagery, survey-dervved socioeconomic indicators, or night-time light intensity, may act proxies for sensitive attributes like race, wealth, or ethnicity (Weidmann and Theunissen 2001 Mellander et al., 2015 Johnson et al. 2022). When combined with event data, such proxies can inadvertently enable models to leverage sensitive attributes, leading to predictions that embed or amplify societal biases. The indirect nature of proxy variables makes these risks harder detect, nocessitating careful auditing of feature relevance and its implications for fairness.
Third, there are important privacy considerations (Cao et al. 2019h. Spatiotemporal event data, especially when linked to individuals small communities, be highly sensitive. Inference fine- fine-grained tem- poral or spatial patterns may inadvertently reveal private information, even when data is aggregated or anonymized. This raises the need for privacy-preserving modeling approaches, such as differential privacy (Dwork et al.) 2014), secure multi-party computation, or federated learning (Gong et al. 2024b Wang et al. 2024c3, when deploying neural STPPs in real-world scenarios.
Finally, governance and oversight frameworks (Furopean Union 2024 Dafoe 2018) are often lacking in the deployment of predictive spatiotemporal models. Without clear accountability structures, it can be difficult to ensure that model outputs are used responsibly and that affected communities have voice in the design, validation, and deployment of these systems.
To mitigate these risks, we advocate the following strategies: (i) fairness-aware evaluation metrics (Shang et al., 2020 Mitchell et al.) 2021) to measure and correct disparate impacts across subgroups; (ii) bias detection and de- diasiig techniques (Mohler et al., 2018; 2019 Li al., 2023) in training data and model outputs; (ili) interpretability tools (subsection 7.4 for neural STPPs, such intensity decomposition and event attribution, to allow auditing of model reasoning; (iv) transparency in reporting data provenance, preprocessing choices, and model limitations; and (v) engagement with domain experts, policymakers, and stakeholders to align modeling objectives with ethical and societal goals.
Ultimately, the potential benefits of neural STPPs, such as improved predictive performance and richer modeling of complex spatiotemporal dynamics, must be weighed carefully against the ethical risks. Their deployment should be accompanied by rigorous bias analysis, privacy safeguards, and governance mechanisms to ensure that these methods contribute to equitable and responsible decision-making.
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Reviews:
[{"review_id": "rzRNI7qGjU", "reviewer": "Reviewer_PUWP", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors provide, to the best of my knowledge, the first survey on neural spatiotemporal point processes (STPPs). Neural STPPs have been a research area of great interest in the ML community in recent years, and having a comprehensive survey of recent developments would be a benefit to the community.\n\n## Strengths\n- The survey is quite broad, covering most of the key literature in the area that I am aware of.\n- It is well organized, divided into sections on modeling, parameter estimation, evaluation, and applications.\n\n## Weaknesses\nSeveral of these sections are lacking in depth, with the discussion looking more like what is found in the related work section for a methods paper, rather than a survey. See my requested changes for more details.", "claims_and_evidence_yn": "No", "claims_explanation": "The claims are partially supported--the authors claim to provide comprehensive coverage on neural STPPs, and they succeed in terms of breadth, but several sections are lacking depth.", "audience_yn": "Yes", "audience_explanation": "Yes, as there is a large body of ML research on neural STPPs.", "requested_changes": "- Section 3.2.2 on continuous-time normalizing flow (CNF)-based methods is extremely short and lacking in detail. Given the long and detailed description on kernel-based methods in Section 3.2.1, I would suggest adding a few paragraphs and perhaps a figure or equations illustrating the difference between these CNF and kernel-based methods.\n- Sections 3.3.1 and 3.3.2 are also lacking in depth. The advantages of multi-event prediction models are described. What are the disadvantages?\n- Last sentence of Section 5: provide some references or justification as to why recent work favors WD (What is WD? Wasserstein distance?) and MMD for assessing generative quality.\n- Section 6 on applications is also lacking in depth. Beyond just providing a citation to the application paper, it would be useful to provide some type of information (either in the body text or in the form of a table) on how the different studies relate in the methods they used--what model, what estimation approach, and what evaluation metric.", "broader_impact_concerns": "There is a brief ethical statement that should probably be expanded. The authors discuss data-related issues such as selection bias, missing data, etc. in Section 7.2. There is no discussion on how neural STPPs compare to traditional statistical STPPs in the presence of such issues, and whether they can be trusted for drawing conclusions, e.g., to inform policy decisions.", "additional_comments": "- In Section 2.1, the last sentence refers the reader to Figure 1, which illustrates the $t_i$'s and $\\\\mathbf{s}_i$'s, but not the history $\\\\mathcal{H}_t$, which should be added.\n- Needs a thorough proofread. Lots of minor typos throughout, including the following in just the first few pages:\n  - Section 2.3, second sentence: add space between \"the\" and $\\\\mathcal{H}_t$.\n  - Section 2.3, last sentence is missing a period.\n  - Section 3, second paragraph: Capitalize first word in first sentence.\n  - Vaswani et al. (2017) reference for \"Attention is all you need\" is missing the other authors.\n  - All equations have the word \"equation\" duplicated.\n- Table 1 should probably be moved to a later page, closer to Section 5.", "extra_scores": {}, "extra_text": {}}, {"review_id": "3eI6YfDVqM", "reviewer": "Reviewer_M8P1", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "### Summary.\nThis work reviews modern spatio-temporal point processes (STPPs) that leverage deep neural networks to model the intensity function and discusses the open challenges that remain for advancing the development of STPPs. Specifically, Section 3 introduces methods for modeling the intensity function, including history event-based encoders, kernel functions, and continuous-time normalizing flows for single-event prediction, as well as transformer-based normalizing flows or diffusion models for multi-event prediction. Section 4 then describes parameter estimation methods, including maximum likelihood estimation (MLE), amortized inference, score matching, and reinforcement learning. Section 5 presents the evaluation metrics for the trained models, followed by applications of STPPs in Section 6. Finally, Section 7 discusses open challenges in this area.\n\n### Contribution.\n\nThe main contribution of this work is (1) to categorize deep neural network-based spatio-temporal point processes (STPPs) from a modeling perspective, and (2) to present future research directions necessary for advancing the development of STPPs.", "claims_and_evidence_yn": "No", "claims_explanation": "This work is written as a review of recent trends in spatio-temporal point processes (STPPs), but it lacks detailed descriptions of methods and empirical results to support the authors' claims. As a result, my comments are based solely on the authors’ categorization and high-level assertions.\n\n> Regarding the categorization presented in Section 3, the current description primarily enumerates existing methods based on modeling techniques. However, it is unclear why certain transitions from older to newer approaches are considered. For instance, it is not evident why the kernel method is introduced instead of using a history-based encoder, or what specific advantages the kernel method offers. Clarifying the motivations and benefits behind each modeling choice would help readers better understand not only the modeling techniques themselves but also the goals and advantages that guide their development.\n\n> In addition, the criteria for categorization seem unclear. Section 3.1 appears to be organized by architectural choices, whereas Sections 3.2 and 3.3 are categorized by task type. This inconsistency may confuse readers about the overall structure and intention behind the categorization.\n\n> As for the open challenges discussed in Section 7, the rationale for each point tends to be presented from a general machine learning perspective rather than being grounded in the specific context of STPPs. As a result, the suggested future research directions come across as overly broad.\n\n> In particular, Section 7.4 on Applicability states that neural STPPs are rarely used to inform policy decisions or design interventions. The authors attribute this mainly to the lack of interpretability and therefore suggest studying \"interpretable neural networks, neurosymbolic methods, and counterfactual analysis for retrospective policy evaluation.\" However, this direction seems misaligned. To me, it would be more appropriate to emphasize the need for improving the interpretability of STPPs themselves, rather than broadly targeting applicability as the main future direction.\n\n> Similarly, in Section 7.5, the authors argue that causality and uncertainty in STPPs have been underexplored and should therefore be studied. While this is a reasonable claim, the justification feels too generic and insufficiently tied to the unique characteristics of STPPs. Strengthening this section by explicitly connecting these research directions to the structural and functional properties of STPPs would make the argument more compelling.", "audience_yn": "Yes", "audience_explanation": "To the best of my knowledge, review papers on deep neural network-based spatio-temporal point processes (STPPs) are still limited, with the exception of [1]. While this prior work overlaps in its discussion of kernel methods, the present paper offers a broader perspective. In particular, if this work were to more explicitly introduce continuous time-based methods, such as continuous normalizing flows or diffusion models for STPPs, it could be of significant interest to readers seeking to understand recent trends in generative approaches to STPP modeling.\n\n[1] Deep Spatio-temporal Point Processes: Advances and New Directions – arXiv", "requested_changes": "* Regarding Section 3, could the authors clarify the reasoning behind the proposed categorization? Additionally, would it be possible to include an “umbrella-style” illustration that visually summarizes the categorization of STPPs? Such a figure could help readers better understand the overall structure and relationships between the different modeling approaches.\n\n* In Section 3, could the authors elaborate on the advantages and objectives of each modeling method? It would also be helpful to explain the motivations for the transitions between modeling paradigms—e.g., why certain methods (e.g., kernel-based approaches) are introduced as alternatives to previous techniques (e.g., history-based encoders), and what specific benefits they offer.\n\n*  For Section 7, could the authors provide more concrete justifications for the importance of addressing each open challenge? Connecting these challenges more explicitly to the unique characteristics and limitations of STPPs would help readers appreciate why they are pressing research directions, rather than general challenges in machine learning.", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}, {"review_id": "jR6BDfsI3E", "reviewer": "Reviewer_dApF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "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": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The submission provides a comprehensive review of parameter estimation and inference methods for spatiotemporal point processes (STPPs), focusing on neural network-based approaches. Maximum likelihood, score matching, and reinforcement learning techniques for parameter estimation. Task-specific evaluation metrics for assessing STPP performance, including distributional fit, prediction accuracy, and uncertainty quantification. Real-world applications in natural disasters, crime, traffic, and epidemiology. Open challenges including reproducibility, benchmarking datasets, computational costs, and interpretability.\n\nStrengths:\n- Comprehensive coverage of both classical and modern inference techniques, including cutting-edge neural and non-parametric approaches.\n- Balanced discussion on evaluation metrics, emphasizing practical trade-offs between likelihood-based and distributional measures.\n- Real-world relevance is emphasized via application sections grounded in diverse domains.\n- Identifies reproducibility and benchmarking as key gaps, showing awareness of broader research needs.\n\nLimitations:\n- While it mentions challenges like interpretability, the review does not deeply engage with specific methods to address these issues.\n- Heavy reliance on recent preprints and workshop papers (e.g., Lüdke et al., 2025) suggests some cited methods may not yet be widely validated.\n- The text is dense and assumes the reader already has foundational knowledge of STPPs and neural methods, which may limit accessibility.\n- There is less focus on causal inference or fairness aspects, both of which are increasingly relevant in real-world STPP applications.", "claims_and_evidence_yn": "Yes", "claims_explanation": "- The submission systematically references recent literature and explicitly notes trade-offs between methods (e.g., computational complexity vs. flexibility).\n- Metrics and applications are tied back to specific studies, providing concrete examples.\n- Some claims would benefit from clearer quantitative comparisons or reproducible benchmarks, which the authors themselves acknowledge as a limitation.", "audience_yn": "Yes", "audience_explanation": "TMLR readers interested in time series, structured prediction, or generative models would find it relevant. The paper discusses neural point processes, estimation methods, evaluation challenges, and reproducibility, all key concerns in machine learning research and practice.", "requested_changes": "- Clarify and expand on interpretability strategies for neural STPPs beyond just identifying it as a challenge (e.g., explainability techniques, causal discovery).\n- Provide a table or structured list summarizing the strengths and weaknesses of each inference method discussed (MLE, score matching, RL).\n- Include a more concrete proposal or roadmap for unified benchmarking datasets, what domains, what characteristics, potential public data sources?\n- Recheck consistency in citations, ensure all mentioned works are properly referenced, especially newer or unpublished studies like Lüdke et al., 2025.\n- Lightly simplify technical explanations where possible to make the review accessible to a broader audience.", "broader_impact_concerns": "- Crime prediction models and epidemiological surveillance carry clear ethical risks: privacy violations, reinforcing social biases, or misinterpretation of predictive outputs as causal.\n- The review does not explicitly address data privacy, algorithmic bias, or governance frameworks, which is increasingly expected for works in this area.\n- Suggested broader impact section additions: discuss risk mitigation strategies, such as privacy-preserving learning or fairness-aware evaluation metrics.", "additional_comments": "boderline reject", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)