Residual TPP: A Unified Lightweight Approach for Event Stream Data Analysis
Authors: Ruoxin Yuan
Guanhua Fang
Venue: ICML 2025 poster (2025)
Keywords: Temporal point process
EasyTPP
residual
event stream
weighted method
Primary area: general_machine_learning->sequential_network_and_time_series_modeling

Abstract: This work introduces Residual TPP, a novel, unified, and lightweight approach for analyzing event stream data. It leverages the strengths of both simple statistical TPPs and expressive neural TPPs to achieve superior performance. Specifically, we propose the Residual Events Decomposition (RED) technique in temporal point processes, which defines a weight function to quantify how well the intensity function captures the event characteristics. The RED serves as a flexible, plug-and-play module that can be integrated with any TPP model in a wide range of tasks. It enables the identification of events for which the intensity function provides a poor fit, referred to as residual events. By combining RED with a Hawkes process, we capture the self-exciting nature of the data and identify residual events. Then an arbitrary neural TPP is employed to take care of residual events. Extensive experimental results demonstrate that Residual TPP consistently achieves state-of-the-art goodness-of-fit and prediction performance in multiple domains and offers significant computational advantages as well.

Residual TPP: A Unified Lightweight Approach for Event Stream Data Analysis
Ruoxin Yuan Guanhua Fang'
Abstract
et al., 2016; Biloš et al., 2019; Shehur et al., 2020), detecting anomalies in event sequences (Liu & Hauskrecht, 2021a; Shchur et al., 2021), and performing causal inference (Xu et al., 2016; Zhang et al., 2020b; 2022).
This work introduces Residual TPP, a novel, uni- fied, and lightweight approach for analyzing event stream data. It leverages the strengths of both sim- ple statistical TPPs and expressive neural TPPs to achieve superior performance. Specifically, we propose the Residual Events Decomposition (RED) technique in temporal point processes, which defines a weight function to quantify how well the intensity function captures the event char- acteristics. The RED serves as a flexible, plug- and-play module that can be integrated with any TPP model in a wide range of tasks. It enables the identification of events for which the intensity function provides poor fit, referred to as resid- ual events. By combining RED with a Hawkes process, we capture the self-exciting nature of the data and identify residual events. Then an arbitrary neural TPP is employed to take care of residual events. Extensive experimental re- sults demonstrate that Residual TPP consistently achieves state-of-the ahe goodness-of fit and pre- diction performance in multiple domains and of- fers significant computational advantages as well.
Temporal Point Processes (TPPs) (Dalcy & Vere-Jones, 2003) are widely used to model the temporal structure and dependencies in event stream data. A TPP defines the prob- ability of events occurring at specific time, conditioned on the past event history, through an intensity function. Tradi- tional statistical TPPs, such as the Possion process (King- man, 1992) and the Hawkes process (Hawkes, 1971), exhibit favorable statistical properties with fixed parametric inten- sity functions. While these models are computationally sim- ple and have been extensively applied in finance (Hasbrouck, 1991), seismology (Ogata, 1988), and other domains for decades, the strong parametric assumptions constrain their ability to capture the complexity of real- weal-world event dynam- ics. To address the limitations and enhance the expressive- ness of classical TPPs, numerous studies have leveraged the power of neural networks to construct neural TPPs. Most neural TPPs parameterize the intensity function using recur- rent neural networks (Du et al., 2016; Mei & Eisner, 2017) and attention mechanisms (Zuo et al., 2020; Zhang et al., 2020a), which enable the capture of complex dependencies. Furthermore, research has explored innovative extensions of TPPs, including their integration into meta-learning frame- works (Bae et al., 2023) and modeling the intensity process as the solution to neural jump-diffusios stochastic differ- ential equation (Zhang et al., 2024).
1. Introduction
Event stream data, characterized by sequences of events occurring over time, plays a crucial role across various do- mains, including financial transactions (Bacry al., 2015), neuroscience (Williams et al., 2020), social media (Fara- jtabar et al., 2017), and natural disaster monitoring (Fox et al., 2016). In these fields, events are typically discrete, irregularly spaced, and may exhibit complex temporal de- pendencies, distinguishing them from conventional time series data. Consequently, many studies seek to understand and model the underlying dynamics of event streams in or- der to address tasks such as predicting future events (Du
Despite these efforts, several limitations remain in the analy- sis of event stream data. (i) RNN-based TPPs struggle to ef- fectively capture long-term dependencies while transformer- based TPPs suffer from high computational costs, particu- larly when processing long sequences. (ii) There is lack of focus on developing lightweight models. Existing models tend prioritize accuracy. often at the expense of compu- tational efficiency, limiting their scalability and practical generalization ability. (ii) Unlike in time series analysis, where techniques like Seasonal- -rend Decomposition (STD) have heen successfully applied to decompose data into trend, seasonal, and residual components, no comparable decom- position methods have been proposed for TPPs to better capture the temporal structure of event streams and simplify the computational process.
'School of Management, Fudan University, Shanghai, China. Correspondence to: Guanhua Fang <fanggh@ leiggh@fudanccom edu....
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Residual TPP
In this paper, we pioneer the exploration of leveraging the statistical properties of event stream data, such as periodic- ity and self-excitation, to decompose the data and construct a lightweight model. Specifically, we introduce Residual TPP, a unified approach that combines the simplicity of traditional statistical TPPs with the expressiveness of neural TPPs, achieving superior performance. Technically, we pro- pose the Residual Events Decomposition (RED) technique, which utilizes weight function to separate residuals in tem- poral point processes. In Residual TPP, we first integrate RED with Hawkes process to capture event characteristics and identify residual events. Subsequently, an arbitrary neu- ral TPP is employed to handle the residual events, thereby reducing the computational burden of the neural TPP. Resid- ual TPP is conceptually simple, computationally efficient, and demonstrates substantial improvements in goodness-of- fit and prediction accuracy across multiple domains.
and neural network-based modeling approaches for tempo- ral point processes.
2.1. Temporal Point Processes
A temporal point process (TPP) is type of stochastic pro- cess used to model the occurrence of events over time. Specifically, temporal point process is a sequence of ran- dom variables denoted as {f:) where t, represents the time of occurrence of the i-th event, with 0 < 4 < .... ti S T. A marked temporal point process (marked TPP) extends the traditional temporal point process by not only modeling the times at which events occur, but also modeling each event type with discrete variable kj E {1,... K]. Formally, a marked temporal point process is a sequence denoted as S = kt), (ty, kr)} = {t,, K=)== In this paper, we primarily focus on marked TPP, as they represent a more general and widely applicable class of data.
In summary, the contributions of this paper are as follows:
A typical way to characterize a TPP is through its intensity function, denoted as A4 (t)H2), where H. ({((t,, k,) t, < t)) is the information filtration containing the event history up to time L. For notational simplicity, we may later write A (t (H,) as (t).). The intensity (x( represents the expected rate of events of type k occurring in an infinitesi- mal time interval [t, t - dt), conditioned on the events that have already occurred by time Therefore, the probability that an event of type k occurs over [t, f + dt) is approxi- mately A (t)dt, provided that dt is sufficiently small.
Due to the distinguished nature of event stream data (i.e. discrete event types and irregular event times), the STD methods deployed in time series data can- not be applied here. Therefore, we introduce novel RED technique for TPPs, which uses proper weight function to evaluate the performance of the intensity function and identify residual events. RED is the first TPP decomposition method and serves as a flexible, plug-and-play module that can be integrated with any TPP model to enhance its performance.
Whether for classical statistical TPPs or neural TPP models, the training objective is to learn the model parameters by minimizing the loss function, Negative log-likelihood (NLL). Given the model's intensity function (x and an event sequence S, the corresponding NLL is defined as:
Building on RED, we propose the Residual TPP, a unified approach that integrates both statistical TPPs and neural TPPs, demonstrating robust generalization capabilities.
The extensive numerical results show that the proposed Residual TPP not only achieves consistent state-of-the- art performances across multiple domains but is also more computationally efficient than existing methods. Therefore, our new model is lightweight, reducing the demand for computational resources.
E/"
Σ Σ log Au, (4: + k=1
(n) dl.
2.2. Classical Statistical TPPs
Several classical statistical models have been developed to characterize TPPs, where the intensity function usually follows a fixed parametric form.
Notation. In this paper, we denote the total event dataset, the collection of event sequences, by S J and the residual event dataset by S = w | where N is the total number of sequences. We use [S] to denote the total number of events in all sequences. Model parameters for the Hawkes process and neural TPPs are represented by 0 {0x 1k and sb- = { respectively. The subscripts n, i, and io refer the event sequence index, event number index, and event type, respectively.
Poisson Processes (Daley & Vere-Jones, 2007). The sim- plest model for TPPs is the Poisson Process, where the intensity function is defined as )u((),, a function over time domain, indicating that the probability of an event occurring in a given time interval is independent of the history.
Hawkes Processes (Hawkes, 1971). The Hawkes Process captures the self-exciting nature of events in TPPs. The occurrence of an event not only depends on the baseline intensity but is also influenced by past events, making it suitable for modeling clustered or dependent events. The occurrence of an event causes temporary increase in the
2. Background
This section provides a brief introduction to the definition of temporal point processes, their statistical modeling methods
2

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Residual TPP
intensity, which decays over time:
3. Methodology
(t) - (t)(t)+ + Σ OkKKIkk. (t 4;),
(1)
In this section, we propose the Residual TPP method to en- hance the performance of existing models while also reduc- ing the computational complexity. This method combines simple statistical TPP with neural TPP trained on the residuals, forming a simple yet powerful hybrid approach.
where (t) > 0 is the baseline intensity, t2 denotes the times of past events, and s(-) > 0 is pre-specified decay- ing function that models the influence of past events on the current intensity. The exponential function Ak (4 -6,4) 4,) BK.K, e 0.,,,,,,......... (t-A)) is frequently used in this context. A ma- jor limitation of this formulation lies in its assumption that past events cannot inhibit occurrence of future events, which unrealistic in complex real-life scenarios.
3.1. Motivation
Our motivation can be summarized as follows:
(i) Event stream data is complex due to irregular timing, heterogeneity, and the presence of noise or outliers, which may result from collection errors, unusual behavior, or rare events. Identifying and handling these outliers crucial for accurate analysis but remains challenging. Classical TPPs typically rely on strong assumptions, such as the Poisson process or self-excitation and use fixed parametric forms based on these assumptions. While these models are struc- turally simple and computationally efficient, they often per- form poorly when applied to complex real-world data. As shown in Figure 1, the Hawkes process fits the data well most of the time, but it struggles to handle commission (added) and omission (removal) outliers (Liu & Hauskrecht, 2021b). Thus, we aim to propose a new approach that not only captures the self-excitation nature of data through the Hawkes process but also effectively handles outliers.
Seff-Correcting Processes (Isham & Westcott, 1979). In self-correcting processes, the intensity function is typically modified by negative feedback term that reduces the like- lihood of subsequent events, thus correcting the system's 's behavior over time:
(4) = Vuul- - EOx); oR);
where V is a non-linear function, μu is the baseline coef- ficient, and og the self-correction term that reduces the event intensity based on the accumulated history of events.
2.3. Neural TPPs
Neural TPP models extend the classical statistical models by parameterizing the intensity function through neural net- works, rather than relying on a fixed parametric form. These models autoregressively generate future events conditioned on historical data. The most widely used neural TPPs can be broadly categorized as follows.
Rradd laawint
Origiaal Events
Adkid buthers
X
Delited Dutlers
The first class is RNN-based (Du et al., 2016) or Attention- based (Zuo et al., 2020) TPPs. In this approach, the model computes the embedding of the i-th event (4,, k.) as e, R via embedding layer. The hidden state h, is updated based on the current event embedding c; and the previous hidden state 1 The next event is then generated conditioned on h;:
2.0
50
Figure I. A simulation illustrating the impact of outliers on the Hawkes process. The ground truth intensity is a Hawkes process with added noise. where black dots represent real events generated from the ground truth. Green triangles and red crosses indicate randomly added and removed outliers, respectively. The fitted Hawkes process is estimated from the modified event sequence.
1++1, R+41 (t
h.), h f ei),
where / represents the encoder function, which can take various forms, such as recurrent architectures (e.g., LSTM, GRU) or more expressive transformer layers.
Another class, referred to as ODE-based (Jia & Benson, 2019), models the evolution of the hidden state in continuous time. Between event occurrences, the hidden state is updated continuously, typically governed by an ordinary differential equation (ODE). Specifically, the evolution of the hidden state from the (i - 1) to the i-th event is given by:
(ii) Recent advancements in neural TPPs have significantly improved performance over traditional methods. However, challenges remain: RNN-based TPPs inherit the intrinsic weaknesses of RNNs, iee. their difficulty in capturing long- term dependencies. In contrast, more expressive attention- based TPPs, while offering enhanced modeling capabilities, suffer from high computational complexity and difficulties in training over long sequences. Therefore, our approach is designed to address the issue of lightweight modeling.
h, f bb e.), where h; fODE JoDee(,-1-- (h,-1, 4-11 44). more complete list of existing neural models can be found in Appendix D.3.
3

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Residual TPP
(iii) Our approach is inspired by the Seasonal-Trend Decom- position (STD) techniques in time series analysis, which decompose a series into trend, seasonal, and residual compo- nents, allowing models to better utilize periodic information and enhance prediction accuracy. Many popular models such as Autoformer (Wu et al., 2021), FEDformer (Zhou et al., 2022), and DLinear (Zeng et al., 2023), adopt the STD approach to decompose the time series and separately model seasonal and trend components. There is significant body of research on STD methods. Classical approaches typically employ moving average kernels to estimate the trend compo- nent. More recently, several neural network-based methods have been proposed to model the periodicity in time series data (Fan et al., 2022; Lin et al., 2024a;b). However, despite similarities between event stream and time series data, cur- rent research on TPPs focuses mainly on modifying neural network architectures for improved performance. To the best of our knowledge, no studies have yet proposed decom- position methods for TPPs analogous to those used for time series. This gap arises from the unique challenges of event stream data, including irregular timing, heterogeneous event types, and complex temporal dependencies. Unlike time series, which are recorded at regular intervals and the observations are mostly continuous, event streams consist of asynchronous events that vary in type and frequency and event types take discrete values, often exhibiting high dimen- sionality and sparsity. Consequently, defining the residual of a temporal point process is inherently difficult. To address this gap, we introduce an innovative modeling and decomposition approach for temporal point processes.
activity. In the view of statistical terminology, periodicity and self-excitation correspond the first-order and second- order statistics of event stream data. These are the main features that should be captured in TPP modeling. Further explanations can be found in Appendix B.
Standard statistical TPP model, egg. the Hawkes pro- cess, can be well-suited model both periodicity and self- excitation, with its intensity function taking the form de- scribed in (1). To capture periodicity, the baseline function Pa (1) can be selected to exhibit periodic behavior. To model self-excitation, 9()) is typically chosen to be non-negative function that decays over time, such as an exponential decay or a power-law decay, reflecting the fact that recent events have a stronger impact on future events but this influence diminishes with time. Therefore, in order to capture the key statistical properties of the data in simple and efficient manner, we first fit the Hawkes process intensity X (t) to the event sequences, where the overall intensity function is given by (t) = Σ (t).
Residual Events Decomposition (RED). (Step 2) Next, we provide solution to define the residual of the temporal point process data. Our objective is evaluate the perfor- mance of the intensity function ( (1) obtained from the statistical TPP model, and to identify the data points where the intensity fit is poor. Therefore, we aim to construct a weight function such that events well captured by A (L) receive larger weights, allowing us to filter out the residual events based on these weights. To achieve this, a specific weight function is given as follows.
We denote 0 = (0)) K as the complete parameter set in x (t) = Hawkes (S; 6k), where lş the parameter set for event type k. For example, in the case of Hawkes process, O2 (+), (0)))...((())))))))))))) (-)), k' = 1, 1, K. Given a set of
3.2. Residual TPP
The Residual TPP method consists of three steps: First, a classical TPP is used to fit the statistical characteristies of the original sequence, such as periodicity and self-excitation, to obtain an intensity function. Next, the proposed Residual Events Decomposition (RED) technique is applied to sep- arate the residual components of the sequence. Finally, a neural TPP is used to capture the intensity that remains unex- plained by the classical model in the residual sequence. By combining the two intensity functions, the final TPP model is obtained. The overall algorithmic flow is illustrated in Figure 2 and Algorithm 1.
parameters @ obtained in Step 1, we define the weight of the i-th event in sequence 5 as follows:
Wi(S:0)= = 0pries (1
K Σ
20 wd)aadd-) (2)
where t0 =0 = 1,... and o2 (x) is defined as
Sp (x) (((1//2)) x x20.
(3)
Periodic/ / Self-exciting patterns modeling, (Step 1) Event stream data often exhibit two key statistical properties: peri- odicity and self-excitation. Periodicity refers to the tendency of events to recur at regular intervals, commonly observed in fields like neuroscience and economics, where events follow repetitive cycles. Self-excitation refers the phe- nomenon where the occurrence of event increases the likelihood of future events. This is important in modeling phenomena such as carthquakes, financial market crashes, or social media activity, where an event can trigger subsequent
(<(x//p))
-1 -
Here, d'(x) denotes the derivative function of an influence
function ((x), which, for x 2 O. takes the following form:
0 0 x S a,
14++2772
6'(z)= = Φ((a) (b-o)z
a < x $ b,
(4)
x > b.
Additionally, 6'(x) = ('xx') for -1 5 x < 0, where x'= ==
4

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Residual TPP
Event stream data
Intensity function
(()
Hawkes
Process
1(0)
Residual Events Decomposition
Sw
(02)()
Neural
TPPs
Figure 2. Algorithm flowchart of the Residual TPP. The event stream data here consists of three event types, represented by triangles, circles, and plus signs, respectively.
{2'[(x' ((c''+1)expp)) ex-))xx((=''11 (x+1) ex--)xxx(=x1).1 2 0}. PI, P2, a and are positive tuning parameters.
process and the neural TPP for residual events:
(x = (1 - a1- (t) + x
It evident that d' (x)) attains its maximum at 0 and has compact support. The influence function Ø(z) is increasing and twice continuously differentiable. Due to these proper- ties, (S; 0) approaches when _e (u) du is close to I, while WS(S; 0) will approach when the integral de- viates significantly from 1. The expression f (u) du represents the average number of events predicted by the Hawkes process within the time interval (4).
and the overall intensity function is X(t) Σx (t)
Here, | represents the proportion of residual events in St relative to the total number of events in dataset S.
The intensity function in the proposed model is a higher- level concept, encompassing both classical statistical TPPs and neural TPPs. Specifically, when the threshold S = 0. indicating that we believe the statistical model has adequately fit the original events, and no residual data is selected for training the neural TPP model. In this case, 0 and d (t) = X x()(t), (t), making the model equivalent to a Hawkes process. On the other hand, when the threshold w = 1, we have S' = S, meaning that the neural TPP model is trained on the entire original sequences. Here, (2) (t). which corresponds to neural
O,
Furthermore, based on the time transformation theory, if the event sequence S is generated by point process with intensity function A (t), then J, X- (u) du ~ Exp(1). In other words, weight (SS:; 0) is close to I when the number of events predicted by the Hawkes process closely matches the actual observed number of events, indicating that the model accurately captures event occurrences within this interval. This also suggests that the event occurrence within this time interval is relatively predictable, as the model can accurately forecast the probability an event occurring.
a 1 and (t) TPP. When 0 < w < 1, the resulting intensity function is combination of the Hawkes process and the neural TPP on the residual data, capturing both the self-excitation in the original sequence and the unexplained intensity from the residual data, thereby yielding more powerful model.
After obtaining the weight for each event, the residual com- ponent of sequence S can be identified by applying thresh-
old (0 < < 1) follows:
In practical applications, parameter to be tuned, allowing for more flexible balance between the statistical properties modeled by the classical TPP and the residuals captured by the neural TPP. This balance can lead to the improved model performance.
also be treated as hyper-
S = (11, W.(S:)) < w).
Residual modeling, (Step 3) After identifying the residual events, we train a neural TPP model on the residual sequence dataset S/ ap> yielding the following intensity function:
Prediction. Given a prefix of the event sequence Spot 1] {(t1, k1), (t+-1, k(11), we want predict the time and type of the next event. The probability density of the next event time L is given by
22 k (t) Neural TSP(S th),, k= 1,.,,,
(-f"_ A(5)
where the Neural TPP represents arbitrarily existing model (parametrized by th 's), such as RNN-based or attention-based models, among others.
(t( P(t+ = He.,)) = x(p)
To minimize the expected L2 loss in predicting the event time, we select the expected value:
Combined Intensity. Due to the superposition property of temporal point processes, the final intensity function for event type ie is given by the combination of the Hawkes
1 (p:(t) dt.
i, Fl1, He:
(5)
t1-1
5

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Residual TPP
Algorithm 1 The Overall Pseudocode of Residual TPP
Theorem 3.2. When pr = p2. Ve log (l) and (1 are bounded at the true O" it holds that
INPUT: Event dataset S {S,, J influence function Φ
OUTPUT: Intensity function (t)
Initialize parameters f {0k ] K 2/-
[le(()) -- 1=
(7)
Step 1-
2: Fit the Hawkes process with full data S and obtain 0
Note that E[e(O)] is a vector and (7) holds elementwisely, The theorem says that, although f As(u)du is left- skewed, the careful design of ((x) ensures that the proposed weight function carefully balances the left and right tails of the integral of the intensity function, thereby allowing them to have equal impacts the loss function. Theorem 3.2 is the generalization of Theorem 3 in Zhang et al., 2025, where they only allow Ae(t) to take some specific forms. Here (a can take any parametric form, including neural models as well.
3: (1) f- Hawkes(S; 0s))
Step 2
4: for Sn S do
5: Initialize the residual set W 00
6: for (Im,i; kn,s) in sequences Sn do
Wi(Sps 0) t -/ppp (Stm. Ex= X (u)du
7:
8:
if (S 0) < then
9:
Sh, S, U ((ln,ishnnit
10: end if
11: end for
Therefore, Step 2 of Algorithm can safely remove those events which are well captured by the Hawkes processes. The remaining events in S are viewed as unexpected out- liers that may not have statistical interpretations (i.e., peri- odic baseline events or self-exciting events).
12: Obtain the residual event dataset S = -155 )
13:
end for
Step
14: Compute + S
15: Fit a neural TPP model to the residual S to obtain eb
16: (2) + Neural i dx)) 17: (t) + (1 (t) + (1))
4. Experiments
In this section, we present the experimental results of our method on six mainstream real-world TPP benchmarks and three synthetic datasets with six baseline models. Our code is publicly available at http:::///tthbb..oo ruoxinyuan, Res dualTPP.
Given the actual next event time tv, the most likely type of the next event is given by
(b4)
c
[r[]naxee[K] A(t,)
(6)
4.1. Experimental Setup
Datasets. We evaluate our method on six real-world bench- mark datasets: MIMIC-II (Johnson et al., 2016), Retweet (Zhou et al., 2013), Earthquake (Xue et al., 2024), Stack- Overflow (Leskovec & Krevl, 2014), Amazon (Ni, 2018) and Volcano (Xue al., 2024). The MIMIC-II dataset is avail- able at the public GitHub repository' and the others be accessed via the EasyTPP library². Detailed descriptions of these datasets are provided in Appendix D.2.
3.3. Theoretical Justification of the Weight Function
As recently shown in Zhang et al., 2025, the influence func- tion defined in (4) enjoys the "unbiasedness" properties.
Proposition 3.1. When X follows the standard exponential distribution Exp(1), it holds that E[(X- -1) dx 0.
In addition to real-world benchmarks, we construct three synthetic datasets for controlled evaluation. (i) Poisson- based: We generate non-homogenous Possion process with five event types, each with different periodic trian- gular function for X (1) and set residual events to follow k
By this property, we can show that the expected gradient evaluated at the true model parameters remains asymptoti- cally unbiased after incorporating the weight function when there is no event contamination and T + co. To be more mathematically formal, for any TPP model with intensity function (t) parametrized by 0, the weighted gradient is defined as
(2) 0.1, homogeneous Poisson process. The superposi- tion of these processes yields the Poisson-based dataset. (ii) ArNHP- based: We use the AuNHP (Yang et al., 2022) to model A with the residuals again modeled by a homoge-
e()) T Wi(S: log (o(44-1) f Je- >o(()d))
neous Poisson process x 0.1. (iii) Possion + ArtHP:
h/tpeeiwiinnuuwueeeeeee hnttps://github.. wwwto///tttteeee researcchh RasyTemmpoaa
where 0) de U1tL So(u)du-1 1).
6

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Residual TPP
Table Performance of all the methods the goodness-of-fit task. The first row presents the log-likelihood the set for the multivariate Hawkes process. For each baseline model, the first row indicates the performance of the baseline neural TPP, while the second row shows the performance the corresponding Residual TPP. Higher scores indicate better performance.
GFODNESS-OF-HI (OG-LIKELIHOOD)
MODEL
MIMIC-II
RETWEET
EARTHQUAKE
STACKOVERFLOW
AMAZON
VOLCANO
MHP
-2.839
-13.71
-4.155
-2.866
-0.534
0.983
RMTPP
-2.626
-4.382
-4.643
-2.844
-2.389
-3.776
RES RMTPP
-2.045
-3.240
-3.689
-0.864
-1.419
-3.548
NHP
-2.031
-4.146
-2.389
-2.613
-2.199
0.261
RES NHP
-1.825
-3.881
-1.930
-0.712
-1.303
0,435
SAHP
-4.672
-4.564
-3.338
-3.867
-2.370
0.128
RES SAHP
-4.488
-4.493
-3.335
-2.374
-1.548
0.256
THP
-2.048
-4.548
-3.498
-2.747
-2.365
-0,068
RES THP
-2.040
-4.597
-3.415
-2.467
-1.611
0.233
ATTNHP -2.500
-4.729
-2.896
-2.685
-2.376
-0.303
RES ATTNHP
-2.197
-4.576
-3.147
-2.482
-1.383
-0.050
ODETPP
-1.855
-4,527
-2.203
-2.492
-2.468
-0.078
RES ODETPP
-1.371
-4.491
-2.340
-2.467
-1.394
0.321
We use the same periodic non-homogenous Possion process for (1) and AtNHH for (22) Summary statistics for these synthetic datasets are provided in Table 8.
Implementation details. For simplicity, we model the event stream data using a Hawkes process with a fixed base- line intensity and an exponential decay function to capture the self-excitation property:
Each dataset is split into training, validation, and test sets. The validation set is for hyperparameter tuning, and the test set is for model evaluation. Due to the wide distribution range of event times and the sparsity of events in the Volcano dataset, we additionally scale the time series by dividing it by one thousand.
x (t) pAk +
Σ OR Bm, E4C
In our experiments, we implement this using Tick (Bacry et al., 2017). more comprehensive treatment of periodicity will be explored in our future work.
Baselines. In the experiments, we integrate the multivari- ate Hawkes process (MHP) with several neural TPP models to derive the corresponding Residual TPP. We employ all intensity-based based neural TPPs from the EasyTPP library, for integration into the Residual TPP and as baselines for com- parison. The models include two RNN-based models: the Recurrent Marked Temporal Point Process (RMTPP, Du al. (2016)) and the Neural Hawkes Process (NHP, Mei & Eisner (2017)); three attention-based models: the Self- Attentive Hawkes Process (SAHP, Zhang et al. (2020a)), the Transformer Hawkes Process (THP, Zuo et al. (2020)), and the Attentive Neural Hawkes Process (AttNHP, Yang et al. (2022)); and one TPP with a hidden state governed by a neural jump SDE: the simplified version of Neural Spatio- Temporal Point Process (ODETPP, Chen al. (2021)).. More details the baseline models are provided in Ap- pendix D.3. As clarified, the original FullyNN model (Omi et al., 2019), with fully neural network-based in- tensity. does not support multi-type event sequences and exhibit worse fitness than other neural competitors across all datasets. Consequently, we omit from the main text, with specific results provided in Appendix E.2.
We follow the architecture of all neural TPP models as outlined in the original implementations of the respective papers, utilizing the code from EasyTPP (Xue et al., 2024). ensure fair comparison, the training parameters and procedures for the corresponding neural TPP models trained on the original data kept consistent with those for models trained on the residual events filtered by RED.
Evaluation. After obtaining the Residual TPPs and the corresponding baseline neural TPPs, we evaluate the models using three standard metrics on the test set:
Goodness-of-fit We evaluate the pee-robbbbiltyy signed to the test set by the models, with higher values indicating better model performance in terms of fitting the data.
Next event-time prediction: We predict the next event's time using the minimum Bayes risk principle, based solely on preceding events, as described in (5). The accuracy is assessed using the Root Mean Squared Error (RMSE).
httppa://iithin com Datta www.iaattiiiiiiiiaiii.oii

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Residual TPP
Table Performance of all methods on next-cvent time prediction and next-event type prediction across six datascts. Lower scores
indicate better performance. As noted, the Volcano dataset is not applicable for the type prediction task, it contains only one event type.
PREDICTION PERFORMANCE (TIME RMSE/ TYPE ERROR RATE)
MODEL
MIMIC-11
RETWEET
EARTHQUAKE
STACKOVERFLOW
AMAZON
VOLCANO
MHP
0.925/27.998
21.98/53.5%
1.475/60.6%
1.619/5.55%
1.042/70.4%
4.223/N.A.
RMTPP
0.998/37.88 009//7.88% 8%
20.68/44.6
1.956/52.99
1.334/57.55
0.613/68.44
4.696/N.A
RMTPP
0.915/26.7%%
19.31/44.5
1.420/52.7%
1.315/57.5 %
0.504/67.55%
3.709/N.A.
NHP
1.010/26. 7%
21.97/42.3%
1.910/53.99
1.411/56.22
0.61
3.906/N.A.
RES NHP
0.913/20.9 %
19.14/40.9 %
1.416/52.8%
1.383/56.1
0.514/66.1 %
3.854/N.A.
SAHP
0,971/23. 8%
22.13/44.8 8
1,463/54, %
1.348/55, 6.6
0.605/68.35 33%
3.998/N.A.
RES SAHP
0.935/18.0% %
19.85/42.099 %
1.455/53.3% %
1.245/55.4
0.498/67.0% %
3.940/N.A.
THP
1,129/35.8 5%
22.01/45, .4%
1.857/54.77
1.401/56.
0.622/66 8%
3,945/N.A A.
RES THP
0.930/27.999 %
19.33/41.5 %
1.403/52.8
1.376/55.7
0.515/65.7
3,910/N.A.
ATTNHP
1.030/35.4 %
21.78/43.
1.822/54.54
1.384/57. 1
0.640/66.49
4.032/N.A.
ATTNHP
0.932/36.0%
19.41/42.9%
1.413/52.8%
1.369/57.09
0.633/65.4%
3.918/N.A.
ODETPP
1,416/22. 1% 0.934/19.2%
22.48/43.22 2%
2.396/56.00 .0%
1.469/56.00
0.697/67.66 666
3.927/N.A
RES ODETPP
19.45/42.7%% %
1.412/53.0% %
1.378/55.6
0.643/67.3%
3.865/N.A.
Table 3. End-to-end training time. "MIIP RED" reports the average runtime of Steps and 2 over 10 independent trials. For each Res
TPP, the result reflects the total time cost the 3-step procedure (MHP RED neural TPP), All training was conducted on a CPU.
END AND-TO-BEE EN-END RUNTIME (SECONDS)
MODEL
MIMIC-II
RETWEET
BARTHQUAKE
STACKOVERFLON
AMAZON
VOLCANO
MHP+RED
1.42
0.27
0.11
0.84
1.49
0.02
RMTPP
9.70
195.3
26.00
71.20
95.75
34.20
RES RMTPP
9.72
188.7
20.01
70.44
95.19
26.42
NHP
34.70
492.6
25.32
206.3
256.6
77.05
RES NHP
29.52
441.9
20.57
190.8
245.1
54.47
SAHP
128.5
498.4
24.90
498.0
505.8
37.30
RES SAHP
113.4
434.9
16.96
461.9
450.3
24.42
10.45
1183
19.20
81.20
257.7
39.90
RES THP
10.67
1029
17.77
76.44
244.8
29.82
ATTNHP
68.60
9475
162.2
1924
6375
1093
RES ATTNHP
52.92
7195
143.1
1863
5646
657.5
ODETPP 9.96
106.4
42.00
243.6
196.4
51.96
RES ODETPP
9.64
102.7
37.43
224.2
192.1
40.88
Next event-type prediction: We forecast the corre- sponding event type using both its actual time and the preceding events according to (6). The prediction accuracy is measured by the error rate.
notable improvement in predictive accuracy. Table evalu- ates the computational efficiency by reporting the training time for the baseline neural models and their corresponding Residual TPPs. Across all datasets, Residual TPP consis- tently exhibits lower training times compared to the baseline models. These experimental results demonstrate that the proposed Residual TPP is a powerful yet lightweight ap- proach for modeling event streams. It can be integrated with various types of neural TPPs, enhancing both performance and computational efficiency.
4.2. Main Results
Table reports the log-likelihood of the models on each test set. reveals that Residual TPP fits the data well and significantly outperforms the corresponding neural TPPs across all datasets. This highlights our model's capability to capture complex real-world intensity dynamics from diverse domains. The results for next event-time and event-type prediction, presented in Table 2, further confirm the supe- rior performance of our method. Residual TPP achieves
4.3. Simulation Analysis
To further validate the robustness of RED, we generate synthetic datasets under various configurations of primary
8

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Residual TPP
Table Performance comparison on synthetic datasets. Goodness-of offft measured by log likelihoodd (L-L; higher better). Prediction performance is evaluated using time RMSE and type error rate (lower is better).
POSSION-BASED
ATTNHP-BASED
POSSION+ATTNHP
MODEL
L-L
TIME/TYPE
L-L
TIME/TYPE
L-L
TIME/TYPE
MHР
-0.132
0.298/62.2%
-1.435
0.329/79.4%
-0.203
0.257/68.
RMTPP
-1.030
0.337/63.2%
-1,277
0.373/78.5%
-0.788
0.271/68. 2271//8.6%
RES RMTPP
-0.531
0.305/62.3% %
-0.664
0.334/77.9
-0.027
0.237/67.88
NHP
-1.023
0.343/62.1% %
-1.267
0.374/77.49
-0.776
0.277/67.8
RES NHP
-0,527
0.301/62.0
-0.661
0.334/77.1
-0.009
0.237/67.88
SAHP
-1.033
0.330/62.198
-1.267
0.376/77.5%
-0.789
0.282/67.8
RES SAHP
-0.556
0.303/62.1%
-0.660
0.335/77.339 %
-0.002
0.238/67.88
THP
-1.026
0.336/62.6%
-1.270
0.374/77.68
-0.785
0.266/67.
RES THP
-0.566
0.300/62.0% %
-0.655
0.331/77.6%
0.004
0.234/67.8%
ATTNHP -1.026
0.337/62.
-1.266
0.374/77.3
-0.789
0.269/67.
1669/67.8%
RES ATTNHP
-0.534
0.302/62.199 %
-0.658
0.336/77.2 %
0.001
0.237/67.88
ODETPP -1.028
0.339/62, %
-1.267
0.382/77.6
-0.789
0.270/67..8
RES ODETPP
-0.565
0.302/62.1 %%
-0.663
0.332/77.6%
-0.001
0.238/67.88
intensities (1) and residual intensities (2) and evaluate the effectiveness of the RED technique across three distinct settings, as illustrated in Section 4.1.
future research. First, the baseline intensity and decay func- tion in the Hawkes process can be adjusted to better capture specific data characteristics like periodicity. Additionally, decomposition methods for TPPs remain an underexplored area. As the first decomposition technique, RED provides valuable insights for developing further methods, similar to STD in times series analysis, to simplify computational procedure and enhance model performance.
We compare the performance of Residual TPP against base- line neural TPPs on these synthetic datasets. As shown in Ta- ble 4, Residual TPP consistently improves the performance of neural TPPs through RED, even when the underlying true signal does not follow a Hawkes process. This demon- strates that the residuals identified by RED are agnostic to the true data-gent gata-generating process. RED quantifies how well given statistical model (e.g., Hawkes process) explains the observed events through its weight function. Even if the true process deviates significantly from the assumed model, RED can isolate the unexplained residuals for refinement by neural TPP. This is conceptually similar to residual learning in deep neural networks, where residuals represent deviations from a simpler base function, regardless of exact form.
Overall, the flexibility of RED as a plug-and-pll module makes Residual TPP adaptable to various TPP models and tasks. Residual TPP offers lightweight but powerful solu- tion for event stream analysis. This work not only advances event stream analysis capabilities but also opens new direc- tions for developing hybrid TPP frameworks through inno- vative decomposition strategies. Moreover, beyond event streams, the RED technique could be similarly extended to other complex, discrete, and irregular data types.
A Acknowledgements
5. Conclusion
The authors thank the anonymous reviewers for their con- structive comments on improving the quality of our paper. Guanhua Fang is partly supported by the National Natural Science Foundation of China (nos. 12301376) and Shanghai Educational Development Foundation (23CGA02).
In this study. we introduce the Residual TPP, a novel frame- work that bridges the gap between simple statistical TPPs and expressive neural TPPs through Residual Events Decom- position (RED), providing a unified solution that combines the strengths of both. Extensive experiments demonstrate that Residual TPP significantly outperforms existing mod- els, offering a robust and scalable approach for event stream data analysis. We also validate the effectiveness of RED as a novel decomposition method compatible with various existing TPP architectures.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning related to temporal point processes and continuous time event streaming analyses. There are many potential societal consequences of our work, none of which feel must be specifically highlighted here.
This framework establishes an extensible foundation for

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Residual TPP
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12

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Residual TPP
A. Technical Proof
Proof of Theorem 3.2
Suppose 0 is n-ddimensional parameter vector, i... (f1,..., 0,). By recalling the definition of e(0), the gradient can be written
e(0)
Elz o' (Ji so(u)du-1) (
Sen No, (z)dd) - -Ø' (Jo' ro(w)du- 1) So No, (Ne (olti-1)
IT,
Elsr o' (Jein so(u)du-1)
(Hanltims) (a(ti-1)
Sem (o)d()) o (Jo do(w)du So (x)
where No, (t) is the partial derivative of (t()) with respect 0% for kt = L,..., n.
Without loss of generality, only need to consider the first entry of e(0). When 0 takes the true value 0" its gradient be written
e(0)1 = Σ Φ (1re so. (oo(w)duu -1)
No- (t-_(
da (li-1) - S(" (z)
(%% Ao (u)(u) 1/" Nor (o-,(o)dd ) /T.
We define A At) = S6¹ Xx- (f)dt. When S [t,,..., tr} is counting process with intensity function Ag- (f), we know that (A"(t),, (t(()))) is standard Poisson process on [0, J'" Ag (t)dt).
Then we deploy the time transformation technique, i.e., using A (t) replace é discuss. Following the notation in Zhang et al., 2025, we define Ag (t)) oo (t) and (t))(t)) = X (t). We have
Σd 0 (1"h Xx- (u)du (u))du--)
No. (or,1(i-11)
-Ith No., (o)d 08 (Lo Xo (u- -1) 1/- X (x)
E
Ao (t+-1)
M
(M)))))
(t) dA (t) }
E
ø (1,)),) (-(t)))-((4-1))) A-(4-1)-11 1))
(NT(EL))) Xo (A+
(A- (t--1))
termi
(1())) ((A((t) dAr(1)
E
O (AA(()))
oo (A^ (t))
For simplicity, write R(AA(tt) (t)) as (t). Vt E 0,T] and CG. (A(t)) We define (() = N(AA-1())) (f), then we know that P(N + di) - 7 = 1) = ((N(A--((++d)) N(AA-1 ( =
1) = (A- = dt. other words, ........ . becomes standard homogeneous Poisson process after time transformation.
13

---

Residual TPP
By caleulations, we get that terml be simplified
(-tt-1)- 1) (X(A ( Σ ø (A))
1
SA (() ]
E
e (1 11--4-1-17 11) (x(i-i)- f
E
x(3)as)
(variable substitution)
(t. - -1- 1). (x(Ex.)
f.x x(3)dš
]
E
E
Hi.-
(take the conditional expectation respectively)
i
( [Ssmavco)
(7(TT
Σ f(tr11) =E
f(t)df,
where [(;;-1) B [. -1- 1xxx(())-)- F. x(3)dz) Hi._.
is function of f Because F, - 1 - follows the standard exponential distribution conditioning the history Hi write f() as EX~EXP()) 0 (X-1) (x(+) Jitx x())ds)
swap the order of integration of and & and get
e
(T)
o' -1 (M(0) fe T+X x(3)d5
(t1 (11-1) S/² x(3)5)) IT
Ex EXPII
df E
(f/ (A(C)) x(0) lor Jorr
] I
-EX~EXP()
o' (X -1
x(++3)dtds
1) Ex-EXDD o' 6'(X- 1) //¹ x(3)d5 /T
(swap the order integration of and
(T)
x())t -XxxXPOO ((x-1)((1-)
dī )
(extract the part that is irrelevant the expectation)
16 (x1x-1) [(xcppau dz
(X 1/2 x(7)ds IT
EX-EXPO
-Ex=XXP))
-0+ + Cg/TT,
1x-11 EX-EXRD [ (X 1) 10 (5) Here use the prop- 0 1x (Jo x(u)du) di
where C. : ==ExxXXP))
erty that (A((T) xt) fo ( x for xt) and & follow the standard exponential distribution. By the assumption that Vo log do (t), Xo( are bounded at the true 0", hence Cg is bounded as well.
As result, we get that [e( is of order O(1TT) when = 0" In other words, the gradient goes to 0 elementwisely as + 00 when P1 = P2.
14

---

Residual TPP
B. Statistical Property
In the context of temporal point processes, the terms Pl, P2, and μa refer to the mean, variance, and covariance of the event counts within time intervals, respectively. These quantities provide crucial insights into the distribution of events and their temporal dependencies. Their precise definitions are as follows.
We partition the total time span [0, T] into time intervals, each of length The number of events occurring within the j-th interval, denoted Mj. corresponds to the interval [(j )));
14 represents the empirical mean of the number of events within the time interval T:
M1
Mj.
N2 denotes the empirical variance of the number of events within the same time interval:
Σ M₁ m1)²
112
Na represents the empirical covariance of the number of events between two intervals separated by lag A:
Ms
1 Th j=1
Σ
(M, x M2+a)
Σ j=1
1
M₁+
M,
where Mj denotes the number of events in the j-th interval, and Mi+A represents the number of events in the j + A-th interval. Here,
A, with A representing the lag between the two intervals.
N3 plays crucial role in capturing the self-excitation property of temporal point processes. Self-excitation refers to the phenomenon where past events increase the likelihood of future events, creating temporal correlations between events over time. A positive value of μa suggests a correlation between the event counts in the j-th and j + Å-th intervals, indicating that events in one interval influence the occurrence of events in subsequent interval. μ4 exhibits significant variation as changes, it indicates the presence of self excitaxcitatonn. A higher value of As for smaller lags suggests stronger temporal dependence (i.e., more pronounced self-excitation), while constant small μáz across varying A values may suggest weaker or absent self-excitation.
Table 5. The Aa values for different datasets at various lag values. For each dataset, the total time span is divided into = 30 equal- length time intervals, with each interval's length given T/n. The covariance of event counts within each time interval and the corresponding lag A is calculated for each sequence in the datasct. The average of these covariance values across all sequences is then used obtain the pa for the dataset.
DATASET
=
A = 2
A = 3
=
= 10
MIMIC-II
0.024
0.012
0.007
0.002
-0.003
RETWEET
7.220
4.554
3.216
1.753
0.361
EARTHQUAKE
0.644
0.433
0.306
0.152
0.023
STACKÖVERFLOW
1.042
0.871
0.718
0.451
-0.091
AMAZOM
3.674
2.911
2.279
1.259
-0.010
VOLCANO
3.272
1.285
0.413
0.022
-0.138
Table presents the values of μa across different datasets for various lag values. The results indicate clear evidence of self-excitation in most cases, as the values of μ3 consistently show a positive correlation between event counts in different time intervals. This correlation is particularly pronounced at smaller lag values, which strongly suggests the presence of self-excitation behavior. As the lag A increases, the values of μа generally decrease, which aligns with the expected behavior of self-excitation, where the influence of past events on future events diminishes over time.
is worth noting that the MIMIC-II dataset consists of relatively short sequences with fewer events, which limits the ability to effectively capture the covariance between event counts in different time intervals. In contrast, the other datasets, with a larger number of events, yield more consistent results in identifying self-excitation patterns.
15

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Residual TPP
C. Detailed Analysis of RED
C.1. Influence Function
1.0
a-0.1, b-1
a-0.5, b-1
0.8
a= bm3
0.4
0.2
-0.5
0.0
0.5
2.0
2.5
Figure 3. The influence function ,o (1:) under different parameter settings (a, b), with p1 P2 1. The parameters and b determine the transition points and the degree of truncation applied to the function.
Figure visualizes the behavior of the influence function %/ (a)) defined in (3), with the parameters set as P1 = P2 The function exhibits piecewise behavior, where the varying parameters and define the transition points and the extent of the truncation. Specifically, 222 oz (1) remains relatively high for E [0, aj and then decays quadratically in the interval [a, b). Beyond b, 0% d22,,22 (z) becomes zero. The function attains its maximum at 0, and maintains certain symmetry for negative and positive arguments due to the definition: o' o'(x)
ø' d'(x') for -1 < x < 0, where x' is defined as - 1) = (xr + exp(pxx - 1), o 2 0}. The interplay between the parameters a and b alters the transition regions of the function. Smaller values of a and b impose stricter constraints on x, which, in turn, affects the sensitivity of the weight function to the underlying integration values.
x' = {x'((x' + exp((=x'
This visualization demonstrates the flexibility of the influence function, the parameters a. b, P1 and p2 allow for fine-tuning of the sensitivity and truncation behaviors, which essential for robust statistical modeling. Specifically, the parameters b can be adjusted based on the characteristics of the data without affecting the order of the function values d' 2p (I)). as shown by the following proposition.
Proposition C.1. Let P1 P2 1. For any 21, 22 [-1, let 6' pp (x) be denoted o' (z) when the parameters 01 and b = by and as (b)x)) when the parameters are 02 and h = by where 01 02 bj b2 >> 0. It holds that:
(x11) 2 (x2) + (5(Z1) (x22)
Proof. When 02 E [-11 0), both of (x) and d (x)
are monotonically decreasing, Therefore, we have:
(21) 2 of (x2)
+ x5(x1) 2 (x(2))
When 32,32 E [0, oo), both of (x) and & (x) are monotonically increasing. Hence, we get:
d( of (x2)
22
(2(x1) (2(2))
Without loss of generality, consider when म E |-1, and x2 oo). from the definition of d' d' (I), we know that of (21) = o'(n) (z,) = o' (z1) of (z1), where x satisfies the equation (x¹ + 1) exp(-z' 1) = (x) + 1) exp(p -I1 1),
with 2 2 0. Similarly, we have dz(x1) = (z)). Therefore, the inequality becomes:
(x1) 2 (22) * d(41) 2 (x2) + z z x2 + d5(x1) 2 (2(22) + ds(21) 2 (x(22).
16

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Residual TPP
C.2. Weight Function
Based the definition of the influence function, we calculate the weight value for each event using Equation (2). When the integral _ ( (u) du is close to 1, (5; 0) approaches 1. Conversely, (S(;; 0) approaches the integral deviates from 1. This allows us to isolate the residual events.
In all experiments presented in this paper, we set P1 = P2 = 1. The parameters a and are adjusted based on the characteristics of the data from different datasets to ensure that the proportion of residual events remains within an appropriate range. The variation of parameters and b influences the values of the weights and the proportion of non-zero weights, but does not affect the ranking of the weight values, as demonstrated in Proposition C.I. This further justifies the use of the weight function define to separate residual events.
MIMIC
Retweet
Earthquake
0.6
probabiity
0.3
0.4
probabilty
0.4
probabiltty
0.5
0.4
0.3
0.2
0.1
0.1
0.2
0.4
0.0
1.0
0.4
Weight Value
Weight Value
Weight Vallue
0.7
StackOverflew
Amazon
Volcano
0.6
0.5
probabilty
probability
0.4
0.3
0.2
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.6 Weight Value
0.4
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Weight Value
Weight Vallue
Figure 4. Weight distribution across different datasets: (a) MIMIC-II, (b) Retweet, (c) Earthquake, (d) StackOverflow, (e) Amazon, (I) Volcano. The weight computation is performed as follows: First, Hawkes process with fixed baseline intensity and exponential decay function is fitted to the data to obtain the intensity function. The parameters of the influence function are set to a = 1, 2, P1 P2 Based on this, weight values all events across all sequences in dataset are computed. histograms show distribution of these weight values for each dataset.
Figure 4 illustrates the distribution of weight values computed across different datasets under identical settings. Each distribution exhibits a truncation near a weight value of 0.8, with a significant portion of the probability mass concentrated A higher weight value indicates that the number events predicted by the Hawkes process closely matches the actual number of observed events, suggesting that the model accurately captures event occurrences within this interval. Conversely, lower weight value indicates poorer performance of the intensity function.
Given this observation, it is natural to consider filtering the events based on their weights, which would enable us to isolate the residual component of the events. These residual events represent unexpected events that may not have clear statistical interpretations. Consequently, we can further utilize neural TPPs to model the intensity of this residual component. This filtering process could therefore particularly useful for capturing statistical properties such as periodicity and self-excitation,, while the residual component can be further analyzed using additional modeling techniques.
17

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Residual TPP
C.3. Combined Intensity
In our approach, the final intensity function for event type is defined combination of the intensity functions from the Hawkes process and the neural TPP for residual events. Specifically, the combined intensity function is given by:
(t) (1 - 2 (t) (t),
where a =
[S] represents the proportion of residual events in S relative to the total number of events in the original dataset S. This formulation leverages the superposition property of point processes, enabling the combination of contributions from both the Hawkes process and the neural TPP in a way that aligns with intuitive observations from the data. The weight parameter a is directly driven by the residual events present in the dataset, making the model's behavior data-driven and interpretable.
One might wonder the advantages of our proposed method in comparison the approach of fitting the data directly using the intensity function of the form
(t) (1 ) T (1) + (1),
(8)
where 01 and 02 are treated as learnable parameters. Here we give two explanations.
i. The latter approach (8) does not explicitly account for the proportion of residual events in the data; instead, the parameters on and 02 are optimized during training to minimize the loss function and determine the best-fit values. In contrast, our method explicitly identifies the residual events and non-residual events.
ii The latter approach is much more computationally heavy. Fitting the model in form of (8) requires more computational time than that of fitting (2) (t) only. On the other hand, our method is plug-and-play module- Hence, Step 2 of the proposed algorithm effectively reduces the sample size and hence makes Step more efficient. As a result, our method is more lightweight than the original approach, which fits the full data with X (t).
C.4. An Alternative Choice of Influence Function
The modular structure of the RED framework allows the substitution of d' (x(x) with any valid influence function that satisfies the "unbiasedness" property. To further enrich this work, we investigate an alternative influence function given by
(1+ a)) (a + 1) + eexp(x) -1
('xx)
This function continuously differentiable over its domain and preserves the "unbiasedness" property established in Proposition 3.1.
2.2
2.0
0-3.0
Figure 5. The smooth influence function ('(x)
(a+a)a+1) (0
-1 under different parameter settings o.
As reported in Table 6, the Residual TPP method implemented with both the original and alternative influence functions (denoted as "old" and "new". respectively) consistently outperforms the baseline models, demonstrating the robustness of the RED framework the choice of influence function.
18

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Residual TPP
Table Performance Comparison on MIMIC-II Earthquake Datasets. Goodness of Lig-Likelihood, higher better); Prediction (Time RMSE/ / Type Error Rate, lower is better).
MIMIC-II
EARTHQUAKE
MODEL
LOG-LIKELIHOOD
TIME/TYPE
LOG-LIKELIHOOD
TIME/TYPE
MНР
-2.839
0.925/27.9%
-4.155
1.475/60.6%
RMTPP
-2.626
0.998/37.8 8%
.4.643
1.956/52. .9%
RES RMTPP(OLD)
-2.045
0.915/26.7 77%
-3.689
1.420/52.7 %
RES RMTPP(NEW)
-2.150
0.923/24.499 4%
-3.667
1.415/52.899
NHP
-2.031
1.010/26.7 7%
-2.389
1.910/53 9%
RES NHP(OLD)
-1.825
0.913/20.55
-1.930
1.416/52.89 %
RES NHP(NEW)
2.040
0.900/1 8.0
-1.916
1.419/52.95%
SAHP
-4,672
0.971/23 8%
-3.338
1.463/54. %
RES SAHP(OLD)
.4.488
0.935/18 8.0 %
-3.335
1.455/53.39 %
RES SAHP(NEW)
-4.571
0.925/22. 1%
-3.312
1.458/53.0 %
THP
-2.048
1.129/35.55
-3.498
1.857/54.
RES THP(OLD)
-2.040
0.930/27.9 99
-3.415
1.403/52.8 %
RES THP(NEW)
-2.879
0.911/26.79 %
-3.533
1.417/52.8%
ATTNHP
-2.500
1.030/35.4
-2.896
1.822/54.5%
RES ATTNHP(OLD)
-2.197
0.932/36.00
-3.147
1.413/52.8 %
RES ATTNHP(NEW)
-2.918
0.914/32.09 %
-1.969
1.419/52.89
ODETPP
-1.855
1.416/22.15 1%
-2.203
2,396/56. .00%
RES ODETPP(OLD)
-1.371
0.934/19.2%
-2.340
1.412/53.09 %
RES ODETPP(NEW)
-2.365
0.921/19.3%
-1.889
1.411/53.18
19

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Residual TPP
D. Experiment implementation details
D.I. Experimental Environment
All experiments this study were implemented using PyTorch (Paszke et al., 2019), trained using the Adam (Kingma, 2014) optimizer, and executed on a CPU.
D.2. Descriptions of Datastes
MIMIC-II (Johnson et al., 2016). This electronic medical dataset contains records 650 patients' clinical visits to Intensive Care Units (ICUs) over seven-year period. Each sequence represents the medical history of single patient, where each event is defined by its timestamp and an associated disease diagnosis code, serving as the event type.
Retweet (Zhou et al., 2013). The Retweet dataset includes 5,200 sequences of tweets, each consisting of an original tweet followed by a series retweets. Each event in a sequence characterized by timestamp and the user tag of the individual retweeting the post. Furthermore, users are grouped into three categories based on their number of followers: "small", "medium", and "large," providing information as event type.
Earthquake (Xue et al., 2024). This dataset comprises 4,296 sequences of timestamped earthquake events recorded across the Conterminous United States from 1996 to 2023, Each earthquake event is classified into one of seven categories based its magnitude.
StackOverflow (Leskovec & Krevl, 2014). This dataset includes two years award collections from 2,200 users on StackOverflow, question-and-annwwe website, where each user receives a sequence badges. We treat each user's reward history as a sequence, with each event the sequence representing the receipt of one of 22 distinct types of badges.
Amazon (Ni, 2018). This dataset is derived from the product review behaviors of the 5,200 most active users on Amazon, spanning the period from January 2008 October 2018. Each sequence represents a user's review history, where each event is defined by its timestamp and the category of the reviewed product. The dataset particularly useful for studying consumer behavior and temporal dynamics in e-commerce.
Volcano (Xue al., 2024). This dataset consists 431 sequences timestamped volcanic eruption events recorded over several centuries across the globe. Since the dataset contains only single event type, it represents a simpler scenario compared to other datasets. To address the extensive original time span and enhance model usability, scale the time series by dividing the timestamps by thousand, bringing the events into manageable time range for analysis.
Table Statistics the real-world datasets. The table columns, from left to right, represent the dataset, number of event types, number events, sequence length, and number of sequenees.
DATASET
# TYPES
EVENTS
SEQUENCE LENGTH
# SEQUENCES
MIN
MEAN
MAX
TRAIN
VALID
TEST
MIMIC-II
2.419
33
527
65
RETWEET
493,708
10
40
97
9,000
1,535
1.520
EARTHQUAKE
70,723
11
16
18
3,000
400
896
STACKOVERFLOW
22
142,777
41
65
101
1,400
400
400
AMAZON
16
330,000
14
45
94
6,454
922
1,851
VOLCANO
9.127
244
400
181
D.3. Descriptions of Baseline Neural TPPs We provide detailed overview of the neural TPP models utilized in this study:
RMTPP (Du et al., 2016): RMTPP utilizes Recurrent Neural Networks (RNNs) to capture the temporal dependencies in event sequences and learn latent representation of event history. The core idea is use RNN to model the
20

---

Residual TPP
Table Statistics of the synthetic datasets. Possion-based dataset represented sequences follow periodic non homomooogeuu Possion process with homogeneous Poisson anomalies. AttNHP--aae baaed dataset represented TPP sequences are generated by AttHH model and than randomly add homogeneous Poisson anomalies. Possion+AttNHH dataset represented sequences follow periodic non homogenous Possion process with commission events generated by ArNHH. The table columns, from left to right, represent the dataset, number of event types, totoal number of events, number residual events generated by (2), sequence length, and number of sequences.
DATASET
# TYPES
EVENTS
RESIDUALS
SEQUENCE LENGTH
# SEQUENCES
MIN
MEAN
MAX
TRAIN
VALID
TEST
POSSION- TOSSION-BASED
5
33,243
4,253
20
29
100
600
200
200
ATTNHP-BASED
100,052
12,807
9
44
79
1,000
500
500
POSSION+ATTNHI
54,737
16,625
24
126
600
200
200
sequence of past events, which then produces a dynamic, evolving feature vector that encodes the event history. This latent representation is then passed through an exponential transformation to define the intensity function.
NHP (Mei & Eisner, 2017): NHP extends the traditional Hawkes process by introducing a continuous-time LSTM network to encode event sequences. Unlike standard Hawkes models, NHP allows the intensity function to decay over time automatically
result of the learned parameters within the LSTM, without requiring inter-event times as inputs. This enables the model to learn complex temporal patterns without the need to manually specify the relationship between event timings.
SAHP (Zhang et al., 2020a): SAHP introduces self- attention mechanism to aggregate historical events for modeling the intensity function. This method enhances the expressiveness of the intensity function by allowing it to adapt to different time scales and dependencies. The self-attention mechanism enables the model to efficiently capture long-range dependencies in the event sequence.
THP (Zuo et al., 2020): THP incorporates a Transformer architecture to model the intensity function of temporal point process. The Transformer is known for its ability capture long-range dependencies through self-attention mechanisms, which is particularly useful for event sequences with complex, non-linear temporal dependencies.
AtNNH (Yang et al., 2022): Att extends the Transformer framework model event sequences, generating rich, context-sensitive embeddings for both observed and potential future events. These embeddings are based on the event's temporal context, enabling the model to generate more accurate predictions by considering the surrounding events and their relative importance.
ODETPP (Xue et al., 20e4):The hidden state evolution in the ODETPP is governed by a neural ODE. It is simplified version of the Neural Spatio-Temporal Point Processes (NSTPP) proposed by Chen et al. (2021) by removing the spatial component. NSTPP leverages the Neural ODE framework to model high-fidelit spatio-temporal distributions, combining concepts from Neural Jump SDEs (Jia & Benson, 2019) and continuous-time normalizing flows (Chen et al., 2018).
FullyNN (Omi et al., 2019): FullyNN is model with fully neural network-based intensity function. It uses feedforward neural network to model the integral of the intensity function, deriving the intensity function as its derivative. This approach enables exact evaluation of the log-likelihood function, which involves the integral, without relying on numerical approximations.
D.4. Likelihood computation
We can estimate the model parameters by locally maximizing the Negative log-likelihood function with any stochastic gradient method:
K Σ / (u(t) dt.
NLL(As) Σ log Av. (t.)
(9)
km
Note that computing the NLL can be challenging due to the presence of the integral in the second term of Equation (9). In this study, we utilize a Monte Carlo method for integral estimation to evaluate the intensity function and its gradient,
21

---

Residual TPP
technique first applied to the NLL computation for TPPs by Mei and Eisner (2017), and subsequently adopted by most models in the field (Omi et al., 2019; Xue et al., 2024).
We randomly sample time points ~ Unif(O, Т) from the interval [0, T and compute the corresponding intensity (t) for each event type k. The expected value of the integral is approximated by TA(t), and its gradient is computed via backpropagation, enabling efficient estimation of VA. To enhance the accuracy of the integral and its gradient, we repeat this process over samples, thereby reducing the variance of the noisy estimator.
E. More Experimental Results
E.1. Extending RED with Different Base Models
To further demonstrate the capability of the RED technique, conduct additional experiments using two configurations: (i) simple TPP + RED + simple TPP, and (ii) neural TPP RED + neural TPP. These configurations allow us to compare the performance of Residual TPP models that apply the RED technique with different base models for residual filtering, against their respective base models without RED.
As shown Table 9, the combination MHP RED MHP may yield worse results, as the model complexity twice that of single MHP. Apparently, the residuals do not follow MHP, leading to overfitting.
We then the NHP representative neural base model for residual filtering. For each baseline neural TPP, compare its performance with two RED variants: the original version using Hawkes process the base model and the new vairant using NHP. The results in Table 9 demonstrate that Residual TPPs with the RED technique consistently outperform the baselines, regardless whether Hawkes or NHP is used as the base model. This highlights that the RED technique, as a plug-and-play module, can effectively enhance the performance of TPPs.
Table Performance comparison of Residual with RED method using different base models on example benchmark datasets: MIMIC-II and Earthquake. For each baseline neural TPP, the first row reports its original performance, the second row (e.g. Res RMTPP) shows Residual TPP with RED using Hawkes as base model and third row (c-g. NHP+RMTPP) presents the new Residual TPP with RED using NHP as base model. We evaluate the model's goodness-of Lig-Liiklihood, higher is better) and prediction performonce (Time RMSE/ / Type Error Rate, lower better).
MIMIC-II
EARTHQUAKE
MODEL
LOG- LIKELIHOOD
TIME/TYPE
LOG-LIKELOOOOD
TIME/TYPE
MHP МHP+МHP
-2.839
0.925/27.9%
-4.155
1,475/60.65
-3.883
0.919/43.0%
-3,746
1,452/71.88
RMTPP
-2.626
0.998/37.88
-4.643
1.956/52.99
RES RMTPP
-2.045
0.915/26.77 %
-3.689
1.420/52.7 %
NHP+RMTPP
-2.059
0.929/37.89
-4.098
1.797/52.898
NHP
-2,011
1.010/26.7
-2.389
1.910/53.9
RES NHP
-1825
0.913/20.9 %
-1.930
1.416/52.8 %
NHP+NHP
-1.792
1.061/22. %
-2.089
1.711/52.80
SAHP
-4.672
0.971/23. 8%
-3.338
1,463/54. %
RES SAHP
-4.488
0.935/18.0 %
-3.355
1.455/53.30
NHP+SAHI
-4.126
0.935/20.C 9%
-3.152
1.452/52.8 %
THP
-2,048
1.129/35. 5.
-3,498
1.857/54.
RES THP
-2.040
0.930/27.9 %
-3,415
1.403/52.8 %
NHP+THP
-2744
1.067/35.5%
-4.112
1.755/53.0%
ATTNHP
-2.500
1.030/35.499
-2,896
1.822/54.5%%
RES ATTNHP
-2.197
0.932/36.00
-3.147
1.413/52.8
AHT+AT
-2.649
1.094/23.8 %
-2.133
1.757/52.8%
ODETPP
-1.855
1.416/22.1
-2,203
2.396/56.05
RES ODETPP
-1.371
0.934/19.2 %
-2.340
1.412/53.0 %
NHP+ODETPP -2.480
1.263/22.7%
-2.089
2.099/54.85
Nevertheless, we omit these extended results from the main text, as one of the key advantages our method its lightweight
22

---

Residual TPP
nature. Our primary motivation is to capture statistical properties using simple TPP and refine residuals using neural TPP. thereby accelerating neural TPP computation with fewer events. While neural TPP RED neural TPP configuration may further improve accuracy, it comes at the cost of substantially increased computational complexity.
E.2. FullyNN
To be self-complete, compare the proposed method with FullyNN model. It can be clearly from Table 10 Table 12 that there is significant improvement in all performance metrics by using our RED technique.
Table 10. Performance of FullyNN and Residual FullyNN on the goodness-of of-fit task. Higher scores indicate better performance.
GOODNESS-OF-FI (LoG-LIKELIHOOD)
MODEL
MIMIC-II
RETWEET
EARTHQUAKE
STACKOVERHLOW
AMAZON
VOLCANO
FULLYNN RES FULLYNN
-10.01 -2.971
-11.45 -4.810
-10.00
-12.72
-9.737
-9.337 -1.344
-4.132
-3.448
-1.963
Table Performance of FullyNN Residual FullyNN on the next event time prediction task. Lower scores indicate better performance.
PREDICTION PERFORMANCE (TIME RMSE)
MODEL
MIMIC-II
RETWEET
EARTHQUAKE
STACKÖVERFLOW
AMAZON
VOLCANO
FULLYNN RES FULLYNN
4.580 1.247
21.92 19.76
4.313
4.316
3.862
6.267
1.523
2.379
1.196
5.629
Table 12. Training time per epoch for FullyNN and Residual FullyNN. All training was conducted on CPU.
TRAINING TIME PER EPOCH (SECONDS)
MODEL
MIMIC-1I
RETWEET
EARTHQUAKE
STACKÖVERFLOW
AMAZON
VOLCANO
FULLYNN
0.672 0.570
2.360 2.124
0.341
1.695
4.980
0.150
RES FULLYNN
0.317
1.686
4.740
0.116
23


Reviews:
[{"review_id": "P7rVNY6u3v", "reviewer": "Reviewer_H11P", "summary": "The paper proposes Residual TPP, a hybrid framework combining classical statistical TPPs (e.g., Hawkes processes) and neural TPPs through Residual Events Decomposition (RED). This computationally efficient approach leverages Hawkes processes for self-excitation/periodicity and neural TPPs for residuals, reducing training costs while improving performance. Empirical validation shows state-of-the-art results on six real-world datasets (e.g., MIMIC-II, Retweet, Volcano) for goodness-of-fit, event time/type prediction.", "questions": "", "limitations": "", "rating": 3, "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": "Thank you for the detailed suggstions.\n\n>Q1：Why is $\\phi'(x)$ defined with piecewise quadratic decay instead of smoother alternatives?\n\nWe acknowledge this concern and have incorporated new experiments using function $\\phi'(x)=\\frac{(1+\\alpha)(x+1)}{(x+1)+\\alpha\\exp(x)}$，which is smooth across its domain and preserves the “unbiasedness” in Prop 3.1. As shown in Tab1 (https://anonymous.4open.science/r/ResidualTPP-3695/Re_H11P.pdf), ResTPP with two different influence functions both achieve better performance than the baselines. However，the core novelty of our work lies in introducing RED as the first general decomposition framework for TPPs. The modularity of RED allows $\\phi'(x)$ to be replaced with any valid influence function with “unbiasedness” property. We also encourage future research to explore enhanced variants.\n\n>Q2：How is the residual threshold $w$ selected?\n\nFig 4 in Appendix C.2 shows the weight values distribution on different datasets. As observed，each distribution exhibits a truncation near a weight value of 0.8，with a substantial portion of the weights concentrated at 0. Given this observation，it suggests that $w$ can naturally be chosen as any value within $(0,0.8)$.\n\n>Q3：Does RED’s preprocessing introduce overhead that negates training time savings for small datasets？Will it lead to overfitting？\n\nIn response to Reviewer **QaBc**'s Q2，we give a detailed explanation and compare the end-to-end runtimes between ResTPP and baseline models in Tab 2 (https://anonymous.4open.science/r/ResidualTPP-3695/Re_H11P.pdf). MIMIC-II and Volcano are small datasets with a few hundred short sequences. As shown，even with these small datasets，RED’s preprocessing time is negligible compared to the training time of neural TPPs，highlighting the efficiency of our method.\n\nRegarding overfitting，Hawkes process (HP) is a statistical model with few parameters，making it less prone to overfitting. To further address your concern，we conduct an additional experiment on small HP datasets. We simulat a 1D HP with intensity function $\\lambda(t)=0.2+0.6\\int_0^te^{-1.2s}dN(s)$. See Fig 2 from the same link for details. The proportion of residual events filtered by RED is only 13%，indicating that most self-exciting patterns have already been captured. Fitting neural TPP to this small fraction of events will not lead to overfitting.\n\n>Theoretical Claims：\n\nThank you for raising this theoretical point. We find that the cumulative probability functions of the integral $\\int_{t_{i-1}}^{t_i}\\sum_{k=1}^K\\lambda_k^{(1)}(u)du$ for residual and non-residual events are overlapped with each other. Therefore it seems that we cannot separate them perfectly. Regarding unbiased estimation，if there are no residual events，then RED can guarantee the unbiasedness property by choosing $w=0$. However，if there exist residual events following an arbitrary TPP，then it would be hard to establish the unbiasedness result. We leave it as future work.\n\n>Experimental Designs Or Analyses & Essential References Not Discussed：\n\nWhile Meta TPP is novel and intersting，it's not a hybrid framework like ours. Moreover，it's not an intensity-based model，meaning RED technique cannot be directly applied to Meta TPP，as our RED method relies on the intensity function.\n\nAs mentioned in Section 3.1，many popular models like Autoformer，FEDformer and DLinear adopt STD approach to decompose time series. However，they cannot be easily adapted for TPP for comparison due to TPPs' complexity (i.e. discrete event types and irregular spaced event times). Our proposed RED technique makes the first attempt to develop decomposition-based TPP variants. While RED is conceptually similar to residual learning and ensembling modeling，its design and operation mechanics are specifically tailored to the unique challenges of TPPs.\n\nWe will cite these papers and include a detailed discussion in the Camera-ready version.\n\n>Other Comments Or Suggestions：\n\nHP's baseline intensity $\\mu_k(t)$ can be modified to periodic functions to capture periodicities. However，in the original RED，we did not do so because：First，unlike time series，periodicity in event stream data typically appears in specific fields like neuroscience. Most commonly used TPP benchmarks do not exhibit periodicities but instead show self-excitation，as discussed in Appendix B. Hence，the standard HP already performs well on these benchmarks. Second，fitting more complex statistical TPPs increases computational complexity，whereas we aim to keep our method simple and efficient. For complex dependencies that cannot be captured，neural TPP can be used for refinement. Our additional experiments in response to Reviewer **QaBc**'s suggestion and Tab2&3 in https://anonymous.4open.science/r/ResidualTPP-3695/Re_QaBc.pdf may also help clarify. Due to space limitations，we will include other statistical TPPs in the Camera-ready version.\n\nWe sincerely appreciate your time. We hope our revisions have addressed your concerns and improved the paper.", "claims_and_evidence": "1. RED is the first decomposition technique for TPPs, inspired by time series decomposition. This addresses a gap in TPP analysis.\n2. RED is model-agnostic and integrates with various neural TPP architectures (RNN-, attention-, ODE-based).", "methods_and_evaluation_criteria": "The residual threshold $\\(w\\)$ is treated as a hyperparameter without systematic guidelines for tuning. The impact of $\\(w\\)$ on performance/complexity trade-offs is underexplored.", "theoretical_claims": "The theoretical justification for RED’s weight function (Section 3.3) relies on Proposition 3.1 and Theorem 3.2 but seems to lack a formal proof of how RED ensures unbiased estimation or optimal residual separation.", "experimental_designs_or_analyses": "Missing comparisons with recent hybrid TPP frameworks (e.g., Meta TPP [Bae et al., 2023]) and decomposition-inspired methods (e.g., Autoformer [Wu et al., 2021] adaptations for TPPs). Add some possible benchmarks against hybrid models and decomposition-based TPP variants.", "relation_to_broader_scientific_literature": "RED is the first decomposition technique adopted into TPPs, which bridges the gap between simple statistical TPPs and expressive neural TPPs and provides insights for developing further methods.", "essential_references_not_discussed": "While RED is novel for TPPs, its conceptual similarity to residual learning in deep networks (e.g., ResNet) and ensembling modeling (e.g., boosting) is underemphasized. Clarify how RED differs from generic residual learning frameworks.", "other_strengths_and_weaknesses": "The choice of $\\(\\phi^{\\prime}(x)\\)$ in Equation (4) is heuristic. No ablation studies validate its superiority over alternative influence functions.", "other_comments_or_suggestions": "The baseline Hawkes process assumes non-inhibitory effects and fixed parametric forms (e.g., exponential decay). This may limit its ability to capture complex periodicities. The authors may also compare with other statistical TPPs as the base model for RED.", "questions_for_authors": "1. Why is $\\(\\phi^{\\prime}(x)\\)$ defined with piecewise quadratic decay (Eq. 4) instead of smoother alternatives (e.g., sigmoidal transitions)?  \n2. How is $w$ selected in practice? Is cross-validation used?  \n3. Does RED’s preprocessing (Hawkes fitting + RED decomposition) introduce overhead that negates training time savings for small datasets?  Will it lead to overfitting?", "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": "", "extra_scores": {}}, {"review_id": "GgS7Zo1H9T", "reviewer": "Reviewer_Jjzj", "summary": "The paper proposes decomposing a TPP into two models - one is a traditional model like Hawkes and the other is a neural model. First, Hawkes model is fit to the sequence. Then the residual events are found using an influence function. Neural model is fit to the residual points and the overall model is the sum of these two intensities. The experimental section shows that this works better than using a simple neural TPP model on different dataset-model combinations.", "questions": "", "limitations": "", "rating": 3, "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": "Thank you for your insightful and positive feedback! \n\n>Q1：Is there any alternative to using an influence function?\n\nPlease refer to our response to Reviewer **H11P**'s Q1 and Table1 in https://anonymous.4open.science/r/ResidualTPP-3695/Re_H11P.pdf.\n\n>Q2：This also seems like a novel way of finding anomalies in a TPP sequence.\n\nWe agree with you and have supplemented our work with a simulation experiment demonstrating that RED can successfully identify anomalies in TPP sequences. We simulate a 1D Hawkes process with intensity function $\\lambda(t)=0.5+0.8\\int_0^t e^{-s}dN(s),0<t\\leq 20$, generating a dataset with 300 sequences. We then divide $(0,20]$ into 20 subintervals of length 1 and randomly select one subinterval in each sequence to insert anomaly events. The times of these anomalies are chosen randomly and uniformly within the select interval, and anomalies account for 21% of the total events. We apply the RED technique to calculate the weight values of all events, perform a moving average, and predict the anomaly interval with the smallest weights. RED achieves an accuracy of 89.0%. This demonstrates RED’s ability to successfully detect anomalous intervals in TPPs. Due to space limitations, we will include additional experiments on applying RED to anomaly detection in the Camera-ready version.\n\nHowever, we chose to omit this part in our original paper, as the primary contribution lies in introducing RED as the first general decomposition framework for TPPs, rather than its application to specific tasks such as anomaly detection. While this topic falls outside the scope of the current paper, future research may explore enhanced RED variants for anomaly detection in TPPs.\n\n\n>Q3：I don't see an experiment of starting with a neural TPP and fitting a second neural TPP as a residual model. Or alternatively, taking a simple TPP and fitting a simple TPP residual. Additionally, one could take it a step further and find another residual. Do you expect this would work or not?\n\nThanks for your valuable suggestions. We have added experiments using simple TPP + RED + simple TPP and neural TPP + RED + neural TPP to further refine our work. As shown in Table 1 (https://anonymous.4open.science/r/ResidualTPP-3695/Re_Jjzj.pdf), the combination of MHP + RED + MHP may yield worse results, as the model complexity of “MHP + MHP” is twice that of a single MHP. Apparently, the residuals do not follow MHP, leading to overfitting.\nThen we select NHP as the example base neural model for residual filtering. For each baseline neural TPP, we compare its performance with the original RED using Hawkes and the RED using NHP. The results demonstrate that Residual TPPs with the RED technique consistently outperform the baseline, whether using Hawkes or NHP as the base model. This highlights that the RED technique, as a plug-and-play module, can effectively enhance the performance of TPPs.\n\nWe also would like to clarify that one of the key advantages of our method is its lightweight nature. Our goal is to capture statistical properties with a simple TPP and refine the residual part using a neural TPP, thereby accelerating neural TPP computation with fewer events. While combining neural TPP + RED + neural TPP may yield better performance, it would also introduce significantly higher computational complexity.\n\nAdditionally, this work is the first decomposition method for TPPs. We use a self-defined weight value to filter and obtain residuals. We believe future work can explore more advanced decomposition methods to derive alternative residuals, offering significant potential for further development.\n\n>Q4：What does section 3.3 say about the connection between the way points are discarded from the Hawkes model and the fact additive intensity implies uniformly rejecting some points?\n\nWe leverage the superposition property of the TPPs. The weight function is used to decide whether an event comes from the Hawkes model or residual model. It shares the similar spirit of using rejection sampling to determine the event type.\n\n>Essential References Not Discussed：It could use a bit more overview of the existing works, and their connections to this paper. No specific papers in mind. I think Lüdke et al. \"Add and Thin\" (2023) is also relevant as it has some similar ideas, but this paper is distinctly different.\n\nThanks for the suggestion! Lüdke et al.(2023) introduce ADD-THIN, a diffusion-inspired TPP model that allows sampling entire event sequences at once and excels in forecasting. This inspired us to explore how future research could develop new decomposition techniques to enhance the forecasting ability of standard autoregressive TPP models. More literature review on Meta TPP, UNHaP and related works can be found in our response to Reviewer **H11P** and **QaBc**. We will include more discussions in the Camera-ready version.\n\nThank you again for your time and valuable suggestions. We would be happy to clarify any further concerns.", "claims_and_evidence": "All claims seem to be supported by evidence.", "methods_and_evaluation_criteria": "The benchmarking was done on traditional datasets and using well established baseline models.", "theoretical_claims": "I checked all the theoretical claims but Theorem 3.2.", "experimental_designs_or_analyses": "The experimental design is valid.", "relation_to_broader_scientific_literature": "To the best of my knowledge this is a novel approach for finding \"anomalies\" in a TPP, applied to fitting a residual model.", "essential_references_not_discussed": "It could use a bit more discussion on the influence functions and overview of the existing works, and their connections to this paper. No specific papers in mind. I think Lüdke et al. \"Add and Thin\" (2023) is also relevant as it has some similar ideas, but this paper is distinctly different.", "other_strengths_and_weaknesses": "Strengths: original approach, clear motivation and implementation, theoretical justification, good empirical results.\n\nWeaknesses: not enough discussion on the influence function choice, either showing why this is the chosen function compared to alternatives or better positioning in the influence function literature. Some other issues in the question section.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Is there any alternative to using an influence function?\n\n- This also seems like a novel way of finding anomalies in a TPP sequence. Would you agree? If yes, why not add an experiment on that?\n\n- I don't see an experiment of starting with a neural TPP and fitting a second neural TPP as a residual model. Or alternatively, taking a simple TPP and fitting a simple TPP residual. Additionally, one could take it a step further and find another residual. Do you expect this would work or not?\n\n- What does section 3.3 say about the connection between the way points are discarded from the Hawkes model and the fact additive intensity implies uniformly rejecting some points?", "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": "", "extra_scores": {}}, {"review_id": "26VTHLa1jk", "reviewer": "Reviewer_QaBc", "summary": "Residual Temporal Point Process (TPP) is introduced as a novel method for event stream data analysis, unifying statistical and neural TPP approaches through Residual Events Decomposition (RED). RED uses a weight function to quantify how well the intensity function captures event characteristics and identify residual events. The method integrates RED with a Hawkes process to capture the self-exciting nature of event stream data, and then uses a neural TPP to model the residual events. Experiments show that Residual TPP achieves state-of-the-art goodness-of-fit and prediction performance across multiple domains and is computationally efficient. The RED technique is the first TPP decomposition method and can be integrated with any TPP model to enhance its performance.", "questions": "", "limitations": "", "rating": 3, "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": "Thank you for your detailed and thoughtful feedback!\n\n>Q1：Is the threshold $w$ arbitrarily chosen? If not how do you choose a good value?\n\nPlease refer to our response to Reviewer **H11P**'s Q2.\n\n>Q2：Why the proposed stepwise approach is lightweight?\n\nResidual TPP follows a 3-step procedure：(1) fitting Hawkes process (HP)；(2) applying RED；(3) training a neural TPP on residuals. We originally evaluate efficiency by comparing the runtime/epoch for Step 3, as the neural TPP training is computationally intensive. Hence, the majority of ResTPP's computational complexity comes from this step. In contrast, HP fitting in Step 1, performed using Tick library, is efficient and fast, while Step 2, which computes weights based on HP, is even faster due to the fixed parametric intensity form.\n\nWe acknowledge your point and have included end-to-end runtime comparisons (HP fitting + RED + neural TPP training) against baselines, as shown in Tab1 (https://anonymous.4open.science/r/ResidualTPP-3695/Re_QaBc.pdf). These results further highlight ResTPP’s overall efficiency, demonstrating the computational advantage despite its stepwise nature.\n\n>Experimental Designs Or Analyses:\n\nWhile neural TPPs can theoretically capture both $\\lambda^{(1)}+\\lambda^{(2)}$ as their width or depth approaches infinity, they face practical limitations that deep neural networks require large datasets and long training times to learn complex patterns. \n\nThe superior performance of ResTPP comes from two reasons：  \n(1) Theoretically, the model space of a neural TPP is contained within the model space of RED+neural TPP. In other words, optimizing over a larger model space can reduce estimation error.  \n(2) Practically, RED+neural TPP helps aviod overfitting. The first- and second- order features of TPPs are well captured by MHP, leading to better generalization.  \nOverall, with the same model architecture, RED technique can easily enhance performance compared to using a neural TPP alone. \n\n>Strengths And Weaknesses：The RED technique has limited scope where the true signal is from HP.  \n>Suggestion 1：I think the authors should motivate more the concept of periodicity. In the paper it is limited to hawkes data.  \n>Suggestion 3：can the author think of creating a synthetic example where residuals are known and conduct experiments?\n\nThank you for the comment, but we respectfully disagree. While the paper uses HP as a representative example of a statistical TPP for clarity, RED is intentionally designed as a plug-and-play module compatible with any base TPP model. The use of HP is explained in Appendix B and does not limit RED's scope.\n\nTo validate RED's robustness, we further simulate datasets using diffferent true signals $\\lambda^{(1)}$ (not Hawkes) and residual intensities $\\lambda^{(2)}$.  \n(1) Poisson-based：We generate a non-homo Possion process with 5 event types, each with a different periodic triangular function for $\\lambda^{(1)}$, and set residuals to follow $\\lambda^{(2)}=0.1$, a homo Poisson process. The combination of these two generates a Poisson-based dataset.  \n(2) AttNHP-based：We use the AttNHP model for $\\lambda^{(1)}$ and homo Poisson process $\\lambda^{(2)}=0.1$ for residuals.  \n(3) Possion+AttNHP：We use the same periodic non-homo Possion process for $\\lambda^{(1)}$ and AttNHP for $\\lambda^{(2)}$.  \nDescriptive statistics for the simulated dataset are provided in Tab2 (https://anonymous.4open.science/r/ResidualTPP-3695/Re_QaBc.pdf).\n\nWe compare the performance of ResTPP and baseline neural TPPs on these simulated datasets. As shown in Tab 3, ResTPP consistently enhances the performance of neural TPPs through RED, even when the true signal does not follow HP or exhibits periodicity.\n\n>Suggestion 2：I think the authors can explain eqn 2-4 better.\n\nAppendix C.1 explains $\\phi'(x)$, with Fig 3 visualizing its behavior under different parameter settings. Fig 4 in Appendix C.2. shows the distribution of weights.\nWe have cited Zhang et al. (2025, arXiv) *Learning...Outliers* in Section 3.3, as our weight function is inspired by their work. Still, we appreciate your suggestion and will make it clearer in Camera-ready version.\n\n>Relation To Broader Scientific Literature：\n\nThanks for pointing out valuable related work. Loison et al.(2024) introduce UNHaP, a framework that differentiates structured physiological events, modeled through MHP, from spurious detections, modeled as Poisson noise. UNHaP assumes that true signal follows HP with specific Poisson noise, whereas our method offers greater flexibility in handling arbitray noise.\nZhang et al.(2021) propose a more computationally demanding method to select exogenous events through the best subset selection framework, whereas our method is more lightweight and efficient.\nWe will include more discussions on the literature review in the Camera-ready version.\n\nWe have carefully addressed the main concerns and hope our revisions meet your expectations. Thanks again for your time and expert feedback.", "claims_and_evidence": "Regarding efficiency: the authors proposed Residual TPP as an efficient approach compared to other neural TPPs.  Residual TPP however is a 3 step procedure. The author evaluation efficiency by run time / epoch for step 2 of their model with baseline models where learning is end to end. In my opinion this is unfair comparison and somewhat problematic. \n\nRegarding season trend decomposition – the authors mentioned the residual TPPs are inspired by works in time series where such decomposition is common (they do cite many  TS works) however, they do not include any reference to support their claim of Periodic patterns in modeling event streams (see line 208 – 219).  I find it less convincing to decompose here and/or it can be motivated more. Here are two examples I have read before tha have similar taste:\nLoison et al. UNHaP: Unmixing Noise from Hawkes Process to Model Physiological Events, AISTATS 25\nZhang et al. Learning to Select Exogenous Events for Marked Temporal Point Process NeurIPS 21.", "methods_and_evaluation_criteria": "The key equations in the proposed methods 2-4 are less well explained. (later on i found similar eqns are from Zhang et al arxiv paper  “Learning under Commission and Omission Event Outliers”. (posted jan 23, 2025); but the author did not cite or mention in their paper. )\n\nI understand the authors are trying to decide on a point whether it should be considered as in the normal pattern in the hawkes process or the residual by the weight W_i(S;theta) . It is hard to understand the each term mean when they put together and why they should work, other than the justification of asymptotics in 3.3.  \n\nThe benchamarks and evaluations are typical.  My concern on Evaluation on computational efficiency are explained above.", "theoretical_claims": "I did not check the correctness of proposition 3.1 and theorem 3.2 due to the fact I dont quite grasp eqn 2-4.\n\nI later note proposition 3.1 is lemma 1 in Zhang et al arxiv paper  “Learning under Commission and Omission Event Outliers”. (posted jan 23, 2025).", "experimental_designs_or_analyses": "I am not sure about the experimental validity for a fair comparison of proposed model and baselines.  The authors did make some effort - “To ensure a fair comparison, the training parameters and procedures for the corresponding neural TPP models trained on the original data are kept consistent with those for models trained on the residual events filtered by RED.” \n\nRegarding the results of Table 2 &3 I am less convinced the Res version outperforms its neural TPP counterparts, since neural TPP should be able to capture both the λ(1)(t) + λ(2)(t)  compared to the case the proposed approach which is using hawkes to model λ(1)(t) and neural tpp to model λ(2)(t)  and then combined for inference. May be the proposed Res has more hyperparameters to tune to get better results?", "relation_to_broader_scientific_literature": "The key contributions of the paper are okay  but the authors should argue why theirs are different from :\nLoison et al. UNHaP: Unmixing Noise from Hawkes Process to Model Physiological Events, AISTATS 25\nZhang et al. Learning to Select Exogenous Events for Marked Temporal Point Process NeurIPS 21. \nZhang et al arxiv paper  “Learning under Commission and Omission Event Outliers”. (posted jan 23, 2025).", "essential_references_not_discussed": "Maybe the authors should mention\nLoison et al. UNHaP: Unmixing Noise from Hawkes Process to Model Physiological Events, AISTATS 25\nZhang et al. Learning to Select Exogenous Events for Marked Temporal Point Process NeurIPS 21.", "other_strengths_and_weaknesses": "Other Strength:\nFlexibility: RED is a plug-and-play module that can be integrated with any TPP model to enhance its performance\n\nOther Weakness:  \nLimitation:The RED technique has limited scope where the true signal is from hawkes process.", "other_comments_or_suggestions": "1.\tI think the authors should motivate more the concept of periodicity in event streams. In the paper it is limited to hawkes data. \n2.\tI also think the authors can explain eqn 2-4 better - maybe with an example (or just refer to the Zhang et al paper) \n3.\tLast suggestion is can the author think of creating a synthetic example where the residuals are clearly known and conduct experiments?", "questions_for_authors": "1. Is the threshold w arbitrarily chosen?  If not how do you choose a good value?\n2. Why the proposed stepwise approach is lightweight? residual tpp is. but the approach may not be.", "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": "", "extra_scores": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: This works proposes Residual TPP, a 2-stage framework where a Hawkes process first fits the data and then a neural model takes care of the "residual events", i.e., those for which the Hawkes process provides a poor fit. Experiments demonstrate that Residual TPP achieves strong results in multiple domains and offers computational advantages.

All the reviewers give positive ratings to this paper. Initially, there were concerns about run-time analysis and reference issues, which were mostly addressed over the rebuttal period.

I recommend acceptance for this paper.
Decision: Accept (poster)