Conjugate Bayesian Two-step Change Point Detection for Hawkes Process
Authors: Zeyue Zhang
Xiaoling LU
Feng Zhou
Venue: NeurIPS 2024 poster (2024)
Keywords: Change Point Detection
Conjugate inference method
Bayesian method
Hawkes Process
Primary area: probabilistic_methods
TL;DR: This work employs data augmentation to propose a conjugate Bayesian two-step change point detection method for the Hawkes process.

Abstract: The Bayesian two-step change point detection method is popular for the Hawkes process due to its simplicity and intuitiveness. However, the non-conjugacy between the point process likelihood and the prior requires most existing Bayesian two-step change point detection methods to rely on non-conjugate inference methods. These methods lack analytical expressions, leading to low computational efficiency and impeding timely change point detection. To address this issue, this work employs data augmentation to propose a conjugate Bayesian two-step change point detection method for the Hawkes process, which proves to be more accurate and efficient. Extensive experiments on both synthetic and real data demonstrate the superior effectiveness and efficiency of our method compared to baseline methods. Additionally, we conduct ablation studies to explore the robustness of our method concerning various hyperparameters.

Conjugate Bayesian Two-step Change Point Detection for Hawkes Process
Zeyue Zhang 12 Xiaoling Lu Feng Zhou 1:3+
'Center for Applied Statistics and School of Statistics, Renmin University of China Innovation Platform, Renmin University of China BBejjing Advanced Innovation Center for Future Blockchain and Privacy Computing (zhangzeyue, xiaolinglu, feng. zhou) chou)@ruc... eruc.edu. cn
Abstract
The Bayesian two-step change point detection method is popular for the Hawkes process due to its simplicity and intuitiveness. However, the non-conjugacy be- tween the point process likelihood and the prior requires most existing Bayesian two-step change point detection methods to rely on non-conjugate inference meth- ods. These methods lack analytical expressions, leading to low computational efficiency and impeding timely change point detection. To address this issue, this work employs data augmentation to propose conjugate Bayesian two-step change point detection method for the Hawkes process, which proves to more accurate and efficient. Extensive experiments on both synthetic and real data demonstrate the superior effectiveness and efficiency of our method compared to baseline meth- ods. Additionally, we conduct ablation studies to explore the robustness of our method concerning various hyperparameters. Our code is publicly available at attps://gt thub. www.uu...oom/tt. Coom/AAurorr2050/aaay-.000
1 Introduction
Point process data, characterized by series of discrete events occurring over time, finds extensive applications in various domains, including finance [1], neuroscience [29], and social networks [20]. Hawkes processes [8], a subclass of point processes, have gained attention due to their ability to model self-exciting and clustering behaviors. However, traditional modeling Hawkes processes relies on the assumption that the parameters of the process, i.e., the distribution of the process, remain invariant over time. practice, this assumption often fails to hold, as the underlying dynamics of point process data can change over time (23:
The violation of the time-invariant parameter assumption poses a significant challenge in accurately modeling Hawkes process data. Real-world processes are inherently dynamic and subject to various influences, leading to fluctuations in process distribution. For instance, a social media platform, the intensity of user interactions may change due to sudden surge in activity during an emergency event or a significant drop in user engagement during a service outage. To address this issue, the change point detection (CPD) has emerged as pivotal task in analyzing point process data. CPD aims to identify the locations where the parameters (distribution) of the process undergo significant change.
In this work, our focus is on CPD within Hawkes process. Previous studies have tackled this issue, offering diverse methodologies for this purpose [2: 4; 23; 27J. Among these, the two-step estimation- prediction method [4] has gained widespread application due to its simplicity and intuitiveness. This method involves using historical data estimate model parameters and then utilizing these estimated parameters to estimate the distribution of the next event point. If the observed point aligns closely
"Correspodingg author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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with the prediction, model parameters are assumed unchanged. But if there's 's a notable deviation, it suggests parameter change, indicating a current change point.
An issue with the cntimation-prdiicaii method lies in the inaccurate estimation of model parameters. This issue stems from the constraints within the CPD system: one hand, for efficient detection, cannot use large amount of historical data as it would lead to a heavy computational burden; on the other hand, once a change point is identified, the reliance on samples post-change point for estimating subsequent model parameters restricts the availability of adequate data for initial post-change point estimations. This often results in inaccurate parameter estimations due to our reliance on limited historical dataset, subsequently affecting the prediction of the next event point. As consequence, the algorithm may wrongly identify many non-change points as change points (false positives) and vice versa (false negatives).
To alleviate the aforementioned issues, many studies propose utilizing Bayesian approaches. Com- pared to frequentist methods, Bayesian methods exhibit better robustness as they not only consider samples but also incorporate prior knowledge. When data is limited, prior knowledge acts as regular- ization, effectively preventing overfitting. Specifically, the Bayesian istimation-predictti estimates the posterior distribution of model parameters based on historical data and then leverages this posterior distribution to estimate the predictive distribution of the next event point. This predictive distribution considers all possible model specifications, which differs from the frequentist method. Similarly. if the observed point closely matches the prediction, it indicates no change point, suggesting unchanged model parameters. Conversely, a significant deviation suggests change point'.
method
Past studies have investigated the Bayesian Hawkes process [10; 21J. Due to the non-conjugacy between point process likelihoods and any priors, inferring the posterior of Hawkes process pa- rameters presents significant challenges. Currently, the majority of work utilizes methods such as Markov chain Monte Carlo (MCMC) [18] variational inference [3] to infer the posterior in non-conjugate scenarios. Inference methods derived in such non-conjugate scenarios often lack analytical expressions, leading to low computational efficiency. Therefore, they are not well-suited for the CPD system, which requires timeliness.
To address this challenge, this paper employs data augmentation strategy recently proposed in the Bayesian point process field [5; 15; 26; 29; 30]. This strategy augments the Hawkes process likelihood with auxiliary latent variables, enabling the augmented Hawkes process likelihood conditionally conjugate to the prior. Leveraging the conditionally conjugate model, we can derive an analytical Gibbs sampler that enables closed-form iterative sampling. The effectiveness and efficiency of our proposed method are demonstrated through experiments using both synthetic and real data.
Specifically, we make the following contributions: (1) We propose the conjugate Bayesian two-step change point detection (CoBay-CPD) for Hawkes process, which leverages data augmentation to address the non-conjugate issue. This novel method allows for more accurate and efficient CPD in Hawkes process. (2) We develop an analytical Gibbs sampler tailored for the proposed model, enabling closed-fore iterative sampling of the model parameters. This streamlines the inference process and alleviates the computational burden associated with non-conjugate scenarios. (3) The experiments demonstrate that our method achieves accurate and timely detection of change points in Hawkes process compared to baseline models, which highlights its practical applicability across various dynamic event modeling scenarios.
2
Related Works
In this section, we delve into the existing literature concerning CPD and Bayesian inference for Hawkes processes.
2.1 Change Point Detection for Hawkes Process
CPD in Hawkes processes has remained an exceptionally challenging endeavor. Within the frequentist framework, several methods have been proposed. For instance, techniques based on second-orde statistics have been introduced [27], as well as approaches leveraging the cumulative sum (CUSUM)
'Here "Bayesian" refers refers to the Bayesian treatment of model parameters; for the change point, it remains a point estimation.

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method [23] and sequential testing strategies [2]. However, Bayesian approaches have remained underexplored. A pioneering contribution by [4] introduced Bayesian approach based on Stein variational inference [13] tailored for Hawkes processes. However, this method derived in non- conjugate scenarios lacks analytical expressions, leading to low computational efficiency. Subsequent experiments demonstrate that our CoBay-CPD significantly enhances the computational efficiency.
2.2 Bayesian Inference for Hawkes Process
The Bayesian Hawkes process has been hot topic in research, broadly categorized into two classes: parametric and -on-aammttiiic methods. Parametric approaches [12: 21] involve applying priors to parameters in Hawkes process and inferring the posterior. Non-parametric methods [24; 25; 28), offering greater flexibility than parametric ones, model the background rate and influence function (refer to Eq. (1)) as flexible functions, apply priors (commonly Gaussian processes) on these functions, and aim to infer the posterior of these functions. As the point process likelihood is not conjugate to any priors, both parametric and non-parametric approaches face challenges in posterior inference. Most studies rely on methods such as MCMC, variational inference, or Laplace approximation [14].
2.3 Data Augmentation for Hawkes Process
In recent years, the Bayesian point process field has introduced novel data augmentation technique to address non-conjugate inference challenges. This method introduces auxiliary latent variables into point process likelihood, transforming non-conjugate problems into conditionally conjugate ones, and thus enabling the derivation of fully analytical inference algorithms. Parametric studies are referenced in [29; 30J, while non-parametric research is detailed in [16; 22; 26). With analytical expressions, the inference algorithms based on data augmentation exhibit higher computational efficiency than the non-an analytical ones derived in non-conjugate scenarios. Our study adopts the data augmentation, opting for computationally more efficient parametric approach to ensure the efficiency of CPD.
3 Methodology
In this section, we present our proposed CoBay-CPD method for Hawkes process. We outline its key derivation steps while providing further details in Appendix A.
3.1 Hawkes Process with Inhibition
The mathematical foundation of Hawkes process is defined by the conditional intensity function that represents the instantaneous rate of event occurrences time / given the history up to but not including 4. It is defined as follows:
(t) -(t Σ p(t-t,).
(1)
where j is the background rate, o(-) is the influence function representing the self-excitation effect from event occurring at 4 the summation captures the influence of all past events, H4- the historical information up to but not including i, and indicates the intensity depends on the history. The self-exciting property of Hawkes processes allows events to trigger additional events, leading to clustering and bursty behavior in event sequences.
Traditional Hawkes processes only employ positive influence functions to avoid negative intensity, limiting them to capturing excitatory interactions. To incorporate both excitatory and inhibitory effects, many studies [7: 11; 17] have proposed nonlinear Hawkes process that allows the influence functions to be negative. This study adopts the nonlinear Hawkes process proposed by [30] which is defined
x²(1) = Ao(h()), M(I) = + Σ 8lt (4),,,,
where Ā > 0 is the intensity upperbound, ar.) denotes the sigmoid function, μ E R is the baseline activation and ((-) E R is the influence function. Due to the presence of the sigmoid function a--), both μ and (0-) can be negative, allowing it to capture inhibitory effects. We choose the sigmoid as the link function due to its compatibility with the subsequent data augmentation technique.
3

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For flexible influence function, we model 6f-) as a linear combination of multiple basis functions:
(()) wrbb(.),
where (s(+)) is the b-th basis function and W/ E TR is the mixing weight. Following [30], we select the scaled and shifted beta densities with support [0, Ta) basis functions. We define the basis function with bounded support [0, T3) rather than unbounded support [0, oo) to assume that events occurring too early do influence the current time. This choice ensures more efficient computation of & (t)
afterward (details are provided in complexity analysis). Consequently, the formulation of h(t) can be expressed in vector form:
B ΣΣ
B
h(t) 1 Σ (t t.) = +
wr t.) μ Σ wb Σ
Eollt t2) ww ((t),,
4=bb=1
where jr, , wD (t) [1, 42 (t)...... [t)] and (t)
Enset ob(t t.)
represents the cumulative impact of past events on through the b-th basis function. As a result, the probability density function of the proposed model is:
p(lEN1] w, X) II (h(tttt)) exp (-["Nounaoaa))
where we assume 11.w are observed on [0, 7) and the model parameters are w and X.
3.2 Non-conjugate Bayesian CPD
The above section outlines the Hawkes process without change points. In this section, we introduce the Hawkes process with change points and how the Bayesian two-step CPD is designed detect these change points. In Bayesian two-step CPD, our goal is to identify the change point where the underlying dynamics of point process shift. The two steps in this method involve an estimation step and prediction step. Let (1:m represent the sequence of timestamps that is generated by Hawkes process with parameters 0, and we consider 0 may undergo changes at certain timestamps. We define, for the timestamp fm. the nearest change point's 's index timestamps before and after the change point are mutually independent. Following this assumption, during the estimation step, the Bayesian two-step CPD necessitates estimating the posterior of the model parameters based on fm.. To infer the posterior, we express the likelihood for the timestamps (yam after the change point as:
E (1..... m) and assume that the
II Ao(h(li)) exp (- dm
(-J(- To (b(()))d trm
pl (W
According to Bayes' theorem, the posterior of model parameters is expressed as:
p(w, (Ww A((rmmm) x p(( w, A))(w)p(A),
(3)
where we choose the prior of w as Gaussian p(w) A (w]0, K) and the prior of À as an uninfor- mative improper prior P()) x 1/X. We use a Gaussian prior on w because it is equivalent to an La regularizer, which stabilizes parameter estimation when there is insufficient observed data.
Then, in the prediction step, we leverage the posterior of model parameters to compute the predictive distribution of the next timestamp
// p(1m+1lfrmmm w, A)p(w, dddd.
p(mm+al (r.m))))) )=
This formula calculates the distribution of the next timestamp f given the observed data points tramm It is worth noting that this predictive distribution takes into account all possible model specifications, which is key distinction between Bayesian and frequentist methods.
In implementation, solving Eqs. (3) and (4) is challenging. For Eq. (3), the non-conjugate nature between the point process likelihood and the prior prevents us from obtaining analytical posterior.

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For Eq. (4), evaluating the integral is also intractable. Therefore, we resort to sampling methods for approximation: (1) Use MCMC to obtain parameter samples from the posterior {w(#), ( Jk= ~
p(w, sm)) (2) Based on the sampled parameters, use the thinning algorithm [19] to sample the
next timestamp (mm+1
+1 w(*), X )) Create confidence interval based on the
samples of (1 } -= If the actual fm+1 falls within this interval, we conclude that no change point
has occurred. Conversely, we infer the presence of a change point.
3.3 Conjugate Bayesian CPD
For non-conjugate Bayesian CPD, the MCMC algorithm in step often lacks analytical expressions, significantly impacting the timeliness of CPD due to its low computational efficiency. To address this issue, our CoBay-CPD adopts the data augmentation strategy, which augments the Hawkes process likelihood with auxiliary latent variables, enabling the augmented likelihood conditionally conjugate the prior. Based on the conditionally conjugate model, we derive an analytical Gibbs sampler to effectively obtain posterior samples of parameters, Specifically, we incorporate Pólya-Gamma variables and marked Poisson processes. Similar derivations have been presented in [30]; here we restate the key formulas for clarity.
3.3.1 Augmentation of Pólya-Gammi Variables
The sigmoid function (-) can be represented in the form of a Gaussian scale mixture:
0(2) o(2)= 1o
1r,
where ((w,,) = / log and pra (w] 0) denotes the Pólya-Gamma distribution with w E I When substituting the above expression into the product term in Eq. (2), the parameter w within the model takes on a Gaussian form.
3.3.2 Augmentation of Marked Poisson Process
A marked Poisson process be introduced to linearize the exponential integral term in Eq. (2):
exp (-/" (b((t))d trym
Epp
II
f(r fww h(t))
(wite
where II = [(wr, 4+)] denotes a realization of a marked Poisson process on the interval [L... Lm). with its probability measure denoted as px and having an intensity X(I,w) Appa(w 11 0). Notably, the key difference between our proposed change point model and the prior work by [30] lies in the fact that here, we focus on the interval with change points, [t.m. tm]. rather than the entire domain.
3.3.3 Augmented Joint Distribution
After introducing two sets of latent variables into Eq. (2), obtain the augmented likelihood:
pm 3m w, I1[w, X)=
II 2=
[Alti (enn 1x (II))
pa
II (w.t)
e fw, (()))
where w the vector w/ on each 1, 4m X(4,, wi) Xppo (1 The parameter w in the augmented likelihood takes Gaussian form. making it conditionally conjugate to the Gaussian prior. Combining the augmented likelihood with priors, we obtain the augmented joint distribution:
p(t W. II, w, X) = p(t m. w, Il A)p(w))((A.
3.3.4 Gibbs Sampler
Thanks to the conditional conjugacy of the augmented joint distribution, we can derive closed-form conditional densities for all variables, naturally leading to analytical Gibbs sampler (derivation
5

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provided in Appendix A):
m2
p(w]trm w) = II
Pro pro h(()))
(5a)
==mm
(( mm w. = (-((((() pra (w]1, h());
(5b)
p ((AllTT.imm II) = pGa (2| Nm + R, Tm);
(5c)
pl [(WW/mmoms w, 11) -N(w [wwmm 2).
(5d)
In Eq. (5c), Nm m Tm + R process, and Tm tm t.... In Eq. (5d), Σ TD K-11-1 where D is diagonal matrix with {w; (-))- in the first m Tm + 1 entries and }w the following entries, and
is the number of points on the marked Poisson
Φ = [t,) the following R entries are 1/2.. Through iterative sampling using Eq. (5), we obtain a series of samples from the model parameter posterior. The pseudocode is provided in Appendix B.1.
(t,))),,11 m EΦv, where the first m Tm + entries of v are 1/2, and
3.3.5 Algorithm, Hyperparameters and Complexity
By employing the proposed Gibbs sampler for analytical posterior sampling of model parameters and subsequently using the thinning algorithm for prediction, we establish our two-step CoBay-CPP tailored for Hawkes process. The detailed procedure is outlined in Appendix B.2.
CoBay-CPDD CCD's hyperparameters, including covariance K in the Gaussian prior, confidence intervals, and basis functions, impact its performance. In experiments, we assume K = = Oversized o² weakens the prior, causing unstable parameter estimation and oversensitive change point detection, while undersized values result in sluggish detection. Similarly, narrow confidence intervals oversen- sitize, and wide intervals slow detection. Balancing accuracy and efficiency, more basis functions enhance prediction accuracy but challenge computational efficiency. In experiments, we select all hyperparameters through volssavaii
Assuming the length of the entire sequence is N, the average length of fm.mm is M, the average length of the latent marked Poisson process is R. the average number of points within the interval of To is No. and the number of Gibbs iterations L, the computation complexity of CoBay-CPP is O(N(NN) + LRN,B + LCTH + L(M + R) + 1)2 + L(B - 1)))), where CTh represents the complexity of the thinning algorithm. The detailed analysis is provided in Appendix C.
4 Experiments
We evaluate the performance of CoBay-CPD on both synthetic and real-world datasets. For the synthetic data, our aim is to validate the capability of CoBay-CPP in accurately recovering the ground-truth -ruth parameters and change points. For the real-world data, we compare CoBay-CPD against several baseline methods determine whether our approach exhibits superior CPD performance.
4.1 Baselines
We conduct comparison between CoBay-CPD and several Bayesian change point detection (BCPD) methods which are designed to address the non-conjugate challenge for Hawkes process: (1)
SMCPD [6] combines BCPD and sequential Monte Carlo (SMC). Similar to our approach, it
sampling-bass method address the non-conjugate inference in the BCPD framework. (2)
SVCPD [4] similarly combines BCPD and Stein variational inference to address the non-conjugate inference the BCPD framework. Differently, this method infers the posterior of model parameters by the variational technique. (3) SVCPD+Inhibition is an extension of SVCPD that incorporates
a nonlinear Hawkes process with inhibitory effects. This baseline is designed because the original SVCPD only considered linear Hawkes process.
4.2 Metrics
We use four metrics to assess the performance of all methods. (1) False Negative Rate (FNR) quantifies the probability of change point being incorrectly identified as not change point,

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Table 1: The FNR, FPR, MSE and RT of CoBay-CPD and other baselines on the synthetic dataset.
Model
FNR(L)
FPR(% 4
MSF(L)
RT(minute 4)
SMCPD
0.38 + 0.41
0.76 + 0.26
0.07 = 0.01
5.50 = 0.31
SVCPD
0.50 + 0.35
0.76 +
0.06 = 0.00
7.78 = 0.01
SVCPD+Inhi
0.33 1 0.24
0.60 0.00
0.16 0.01
23.09 1 0.60
CoBay-CPD
0.13 + 0.22
0.46 1 0.26
0.05 0.00
4.62 = 0.10
calculated as True Fositives False Negatives False negative cases are critical in many applications, as they indicate that certain crucial changes fail receive attention. (2) False Positíve Rate (FPR) quantifies the probability of a stable point being incorrectly identified as a change point, True Negatives
True Positives
calculated as False Fositives Trme Negatives Minimizing false positive cases is also important because frequent false alarms will waste resources. (3) Mean Square Error (MSE) measures the distance between the predicted next timestamp (the average of {(*) and the actual timestamp, calculated as 1 -E (c!h) 44)². This metric evaluates how accurately the model predicts the next data point. (4)
Running Time (RT) measures the efficiency of the method by its runtime.
4.3 Synthetic Data
We validate the efficacy of CoBay-CPD using synthetic dataset. Our goal is to verify whether CoBay-CPD can accurately recover the ground-truth parameters and change points.
Datasets The synthetic data is created by concatenating three segments of Hawkes process data, each characterized by different parameters. Specifically, three segments of Hawkes processes adhere to the model configuration outlined in Section 3.1, with their influence functions assumed to be a mixture multiple beta densities. Three segments employ identical basis functions and mixing weights. However, they have different intensity upperbounds, namely d 5, λ2 10 and À3 3. We utilize the thinning algorithm to simulate data for three Hawkes processes, and concatenate them to form the synthetic data. The two change points are located at the 43-rd and 136-th points, indicated by grey lines in Fig. 1a in Appendix D.2. Further details can be found in Appendix D.
Results We evaluate the performance of change point detection using CoBay- and other baseline models on the synthetic dataset. For CoBay-CPD, we adopt a prior distribution p(w) = N(w K). where K = 0.5I. The detection outcomes are presented Fig. la in Appendix D.2. The change points identified by CoBay- DBDay-CDDDaa are 44, 136, while SMCPD detects 96, 136, SVCPD detects 44, 96 and SVCPD+Iniibtiin detects 51, 136. This discrepancy suggests that CoBay-CPD achieves more accurate change point detection. Furthermore, the estimated parameter À from CoBay-CPD for the synthetic data is depicted in Fig. in Appendix D.2. The estimated parameter N closely aligns with the ground truth, demonstrating the accuracy of parameter estimation by CoBay-CPD. Notably, there are prominent changes around the change points in Fig. 1b. This phenomenon arises due the initiation of new Hawkes process with distinct parameters at the occurrence of change point. The challenge of accurately estimating parameters with limited data in such scenarios alleviated by the Bayesian framework. In similar situations, frequentist methods tend to perform poorly.
We also compare CoBay-CPl CPD against baseline methods terms FNR, FPR, MSE and RT. The results are presented in Table 1. As anticipated, CoBay- CPD outperforms the alternatives. This superiority can be attributed to CoBay- CCD's utilization of nonlinear Hawkes process model, which encompasses both excitation and inhibition effects. In contrast, SMCPD and SVCPD employ simpler linear Hawkes process model, constraining their expressive power. Additionally, CoBay-CPD employs Gibbs sampler to accurately characterize the parameter posterior, whereas both SVCPD and SVCPD+Inhibitio utilize variational-bar methods to approximate the parameter posterior. As result, their change point detection accuracy is compromised.
4.4 Real-world Data
In this section, we conduct comparison between CoBay-CPD and baselines on two real datasets.
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Table 2: The FNR, FPR, MSE and RT of CoBay-CPD and other baselines real-world datasets.
WannaCry
NYC Vehicle Collisions
FNR())
[PR(J)
MSE[(x ))
RT(minntte I)
FNR())
FPR($ ()
MSF())
[T(minute L)
SMCPD
0.38 0.06
0.02 = 0.01
3,59 0.08
11.65 0.07
0.56 = 0.16
2 1466000 0.55
0.02 = 0.00
24.67 = 0.26
SVCPD
0..44+0.11
0.01 + 0.01
3,47 +-0.6
9.72 + 0.06
0.58 + 0.36
1.00
0.43
0.02 + 0.00
19.30 0.09
SVCPD+Inhi
0.54 0.09
3.54
2.54
0.36
CoBay-CPD
0.21 + 0.04
0.05 + 0.02
3.42 +42-+:00 0.00
6.24 + 0.49
0.13 + 0.16
0.89 0.16
0.01 + 0.00
8.70 + 0.26
Table 3: Ablation study. The FNR, FPR, MSE and RT of CoBay-CPD with different hyperparameters.
Metric
Number Basis Functions
Conlidence Interval
Prioe Covariance
2
95%
90%
85%
d² 0.01
o² 0.5
5² 10
FNRCJ
0.38
0.22
0.13
0.22
FPRI%
1.07 1 0.50
0.91
4: 0.30
0.61
1: 0.00
0.46 :: 0.26
0.46
0.26
1.83 1: 0.43
0.76
0.26
0.46 4: 0.26
0.91 0.30
MSE()
0.05 0.00
0.05 0.00
0.05
0.00
0.04 0.00
0.05
0 0.00
0.04
+ 0.00
0.04
0.00
0.05
+ 0.00
0.05
5 0.01
RT(minute
1,57 0.03
2.61
0.08
3.62
0.10
5.03
0.02
4.62
0.10
4,50
0.111
4.74
0.02
4.62 0.30
4.41 0.100
Datasets We analyze two datasets from the domains of network security and transportation, with specific details provided below. More comprehensive information regarding data preprocessing can be found in Appendix E. (1) WannaCry Cyber Attack [4]: The WannaCry virus infected more than 200,000 computers around the world 2017 and received much attention. The WannaCry Cyber Attack data contains 208 traffic logs information observations. Each observation contains the relevant timestamp. (2) NYC Vehicle Collisions' [27]: The New York City vehicle collision dataset comprises approximately 1.05 million vehicle collision records, each containing information about the time and location of the collision. For our experiments, we select the records from Oct.14th 2017.
The real-works data, unlike synthetic data, does not have ground-truie change points. Therefore, use the points where timestamps surge as the ground-truth change points in the WannaCry dataset, and utilize the reported change points from [27] the ground-truth change points in the NYC dataset.
Results Figures 3a to 3d in Appendix E.2 display the change point detection outcomes of different methods applied WannaCry data. It clear that CoBay-CPD exhibits the most favorable detection performance. The change points identified by our method are consistent with the actual change points. The SMCPD, SVCPD and SVCPD+Inhibition detect a relatively limited number of change points, resulting in missed change points. Table 2 presents various metrics of four methods for change point detection in the WannaCry data. Clearly, because SMCPD, SVCPD, and SVCPD+Inhibitinn detect too few change points, their FNR is high and FPR is low. In contrast, CoBay-CPP exhibits the lowest FNR, a reasonably balanced FPR, the smallest MSE, and requires the least runtime.
Figures 4a to 4d in Appendix E.2 show the change point detection outcomes of four methods for the NYC data. Notably, SVCPD detects fewer change points, while SMCPD identify an excessive number. The change points detected by CoBay-CPD are 43, 110, 160, 194, 284, 338, 398, corresponding to the times 2:30, 9:00, 12:00, 13:10, 16:00, 17:55, 20:00. These timestamps coincide with peak traffic hours on workdays. Table 2 presents various metrics of four methods for change point detection in the NYC data. Consistently, SMCPD exhibits high FNR and FPR. The high FPR due an excessive number of change points detected by SMCPD. while the high FNR is due to the inaccurate detection of numerous change points by SMCPD. Whereas SVCPD shows a high FNR and low FPR due to detecting too few change points. SVCPD+Innibition achieves relatively balanced FNR and FPR, indicating the beneficial impact of employing a nonlinear Hawkes process. CoBay-CPD demonstrates superior accuracy and efficiency, with the lowest FNR, FPR, MSE, and RT in change point detection compared to all baseline models.
4.5
Ablation Study
In this section, we conduct hyperparameter analysis and stress tests of CoBay-CPD on synthetic data.
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Table 4: The results of stress tests. We conduct experiments to verify the performance change of CoBay-CPD w.r.t the number of change points, the difference between adjacent X's, and the closeness between two change points.
Number of Change Points
AA
A
Metric
0.1
5
10
15
FNR
0.00
0.00
0.13 0.22
0.11
= 0.14
1.00 0.00
0.25
0.43
0.00
0.00
0.33 4: 0.24
0.00 1: 0.00
0.00 4: 0.00
FPR L) 0.43
0.60
0,46
0.26
0.31 0.50
1.61 0.57
0.35
0.60
0.43 1: 0.60
1.00
4: 0.70
0.93
4: 0.65
0.31 1: 0.54
MSE ()
0.04
0.00
0.05 0.00
0.07 0.01
0.02 0.00
0.03 + 0.00
0.04 + 0.00
0 + 0.00
0.02 + 0.00
0.02 + 0.00
Number of Basis Functions The number of basis functions impacts the expressiveness of the Hawkes process, influencing the model's detection performance. We assess the model's detection performance across varying numbers of basis functions, from 3, as shown the Table 3. In experiment, we set the other hyperparameters as: 90% confidence interval and K = 0.5I. Observably, as the number of basis functions increases. FNR and FPR decrease, indicating enhanced detection accuracy, while RT increases, indicating increased computational burden.
Confidence Interval We try 3 different confidence intervals: 95%, 90%, and 85% for the next timestamp, as shown in Table 3. In the experiment, we choose 4 basis functions and K 0.5I. A wider confidence interval results in fewer detected change points, leading to larger FNR and smaller FPR. Conversely, a narrower confidence interval leads to the detection of more change points, resulting in numerous incorrect change points. Consequently, the FPR increases significantly, while the FNR also shows a slight rise. So the compromise, 90% confidence intervals, is the best.
Prior Covariance The Gaussian prior covariance K 2 also has large impact on the detection results. In this experiment, we choose 4 basis functions and 90% confidence interval. If o² is too large, FNR and FPR will increase. This is because the prior too loose, causing the posterior samples of model parameters to spread excessively and fail to concentrate around the true values. On the contrary, when 6² is too small, the posterior samples of model parameters are too concentrated in certain position that may be wrong value, resulting in a larger FPR, as shown in Table 3.
Stress Tests The stress tests assess how well a model performs under difficult or extreme conditions. We conduct three stress tests experiments: one involving the number of change points (more indicating greater difficulty), another focusing on the difference between adjacent X AA (smaller indicating greater difficulty), and the third examining the closeness between adjacent change points, A/ (smaller indicating greater difficulty). The results are shown in Table 4. (1) Experiments with different numbers of change points reveal consistent performance across varying numbers. (2) Regarding AA, model effectively detects change points even when AA is small (c.g., AA 1). However, excessively small Aλ values lead to decreased performance, as the parameters on both sides of the change point become too similar to distinguish. (3) Regarding At, the model maintains good even when two change points are close, although performance slightly declines. More experimental details can be found in Appendix F.
Limitations and Broader Impacts
Although our proposed CoBay-CPD method offers an efficient and accurate solution to the change point detection problem in Hawkes process, it still has some limitations. For instance, extending CoBay-CPD to multivariate Hawkes processes remains challenging because the current method requires change points to occur at specific event locations. However, in multivariate Hawkes processes, a change point in one dimension (an event location) does not necessarily correspond to event locations in other dimensions. This challenge needs to be addressed further in future research.
The introduction of CoBay- CPD for Hawkes process holds promise for both positive and negative social impacts. This method effectively addresses the non-conjugate inference challenge, improving the efficiency and accuracy of change point detection across various fields. However, as automated change point detection methods become more accurate and efficient, there is risk of overreliance on these systems without proper validation or human oversight. This could lead to erroneous decisions or missed opportunities for critical interventions.

---

Conclusions
In summary, this work introduces a novel conjugate Bayesian two-step change point detection method for Hawkes process, which effectively addresses the non-conjugate inference challenge. Leveraging data augmentation, we transform the non-conjugate inference problem to conditionally conjugate one, enabling the development of analytical Gibbs sampler for efficient parameter posterior sampling. Our proposed upproach surpasses existing methods, showcasing superior accuracy and efficiency in detecting change points. The contributions of this research hold great potential for advancing event-driven time series analysis and change point detection across various applications.
Acknowledgments and Disclosure of Funding
This work was supported by NSFC Projects (Nos. 62106121, 72171229), the MOE Project of Key Research Institute of Humanities and Social Sciences (22JJD1 10001), the Big Data and Responsible Artificial Intelligence for National Governance, Renmin University of China, the fundamental research funds for the central universities, and the research funds of Renmin University of China (24XNKJ13).
References
[1] Bacry, E., Mastromatteo, I,, and Muzy, J.-F. Hawkes processes in finance, Market Microstructure and Liquidity, 1(01):1550003 2015.
[2] Bhaduri, M., Rangan, D., and Balaji, A. Change detection in non-stationary hawkes processes through sequential testing. In ITM Web of Conferences, volume 36, pp. 01005. EDP Sciences, 2021.
[3] Blei, D. M., Kucukelbir, A., and McAuliffe, J. D. Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518):859-87) 2017.
[4] Detommaso, G., Hoitzing, H., Cui, T. and Alamir, A. Stein variational online change- point detection with applications to hawkes processes and neural networks. arXiv preprint arXiv:1901.07987 2019.
[5] Donner, C. and Opper, M. Efficient Bayesian inference of sigmoidal Gaussian Cox processes. Journal of Machine Learning Research, 19(1):2100-274, 2018.
[6] Doucet, A., Johansen, A. M., et al. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704):3, 2009.
[7] Gerhard, F., Deger, M., and Truccolo, W. On the stability and dynamics of stochastic spiking neuron models: Nonlinear hawkes process and point process glms. PLoS computational biology, 130))9e 2017.
[8] Hawkes, G. Spectra of some self-exciting and mutually exciting point processes. Biometrika. 58(1):83-90, 1971.
[9] Kingman, J. F. Poisson processes, volume Clarendon Press, 1992.
[10] Linderman, S., Johnson, M., Miller, A., Adams, R., Blei, D., and Paninski, L. Bayesian learning and inference recurrent switching linear dynamical systems. In Artificial Intelligence and Statistics, pp. 914-922, 2017.
[11] Linderman, S. W. Bayesian Methods for Discovering Structure Neural Spike Trains. PhD thesis, Harvard University, 2016.
[12] Linderman, S. W. and Adams, P. Scalable bayesian inference for excitatory point process networks. arXiv preprint 100.:v:557.032222 2015.
[13] Liu, Q. and Wang, D. Stein variational gradient descent: general purpose bayesian inference algorithm. Advances in neural information processing systems, 29, 2016.
10

---

[14] MacKay. J. Information theory, inference and learning algorithms. Cambridge university press, 2003.
[15] Malem-Shine Salemm-hinitski, N., Ojeda, C., and Opper, M. Flexible temporal point processes modeling with nonlinear hawkes processes with gaussian processes excitations and inhibitions. In Proceedings of the 38th International Conference on Machine Learning, pp. 1-31, 2021.
[16] Salem-Shinitski, N., Ojeda, C., and Opper, M. Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects. Entropy, 24(3):356, 2022.
[17] Mei. H. and Eisner, The neural Hawkes process: neurally self melf-modulating multivariate point process. In Guyon, I., von Luxburg, U., Bengio, S., Wallach, H. M.. Fergus, R., Vishwanathan, V. N., and Garnett, (eds.), Advances Neural Information Processing Systems 30; Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA. USA. pp. 0754-67660 6764, 2017.
[18] Neal, M. Probabilistic inference using markov chain monte carlo methods. 1993.
[19] Ogata, Y. Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50(2):779-40 1998.
[20] Pinto, J. C. L., Chahed, T., and Altman, E. Trend detection in social networks using Hawkes processes. In Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2015, pp. 1441-1448. ACM, 2015.
[21] Rasmussen, G. Bayesian inference for hawkes processes. Methodology and Computing in Applied Probability, 15:623-641 2013.
[22] Sulem, D., Rivoirard, V., and Rousseau, J. Scalable variational bayes methods for hawkes processes, arXiv preprint arXiv:2212.00293, 2022.
[23] Wang. H., Xie, L., Xie, Y., Cuozzo, A., and Mak, S. Sequential change-point detection for mutually exciting point processes. Technometrics, 65(1):44 2023.
[24] Zhang, R., Walder, C., and Rizoiu, M.-A. Variational inference for sparse Gaussian process modulated Hawkes process. arXiv preprint 1r9ivv/905.1/ 10496v2, 2019.
[25] Zhang, R., Walder, J., Rizoiu, M., and Xie, L. Efficient non -arameeeet Bayesian Hawkes processes, In International Joint Conference on Artificial Intelligence, pp. 4299 4305, 2019.
[26] Zhou, F., Li, Z., Fan, X., Wang, Y., Sowmya, A., and Chen, Efficient inference for non- parametric Hawkes processes using auxiliary latent variables. Journal of Machine Learning Research, 21(241):1-31, 2020.
[27] Zhou, F., Li, Z., Fan, X., Wang, Y., Sowmya, A., and Chen, F. Fast multi-resolution segmentation for nonstationary Hawkes process using cumulants. International Journal of Data Science and Analytics, 10:321-330, 2020.
[28) Zhou, F,, Luo, S., Li, Z., Fan, X., Wang, Y., Sowmya, A., and Chen, F. Efficient em-variational inference for nonparametric hawkes process. Statistics and Computing. 31(4):46, 2021.
[29] Zhou, F., Zhang, Y., and Zhu, Efficient inference of flexible interaction in spiking- neuron networks. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net, 2021.
[30] Zhou, F., Kong, Q. Deng, Z., Kan, J., Zhang, Y., Feng, C., and Zhu, Efficient inference for dynamic flexible interactions of neural populations. Journal of Machine Learning Research, 23 (211):1-49, 2022.
11

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A Derivation of CoBay-CPD
In this section we provide proof of data augmentation and Gibbs sampler, respectively.
A.1 Data Augmentation
Focus the interval with change points [ the probability density (likelihood) of CoBay-CPD can be presented as:
Xo(h(L)) exp (-1 Frre
p14) w, = II
{o(h(1))dl)
trm
T
Substitute the augmentation of Pólya-Gamma Variables o(2) Jo ef( (w] O) and the sigmoid symmetry property a(2) = - o((=2) to the above equation, we can obtain:
(-/ fm (o(())) = exp (-f(mm) f.
(1 (-
mm 10
edw,, -1- ((())) Appa O)dwdt) )
exp
According to Campbell's 's theorem [9], the exponential integral term can be rewritten as
fm exp (-/ frmm
(a(h(t))dt (-(f(oohh(t)dtt
Eps.
II (w.))n
(w. h(8))
where = (W,, f- denotes a realization a marked Poisson process on the interval [l.m. 4m]. with probability measure denoted as ps and having an intensity X(w, (,,1(
Therefore, the likelihood of CoBay-CPD can be rewritten as :
do (h(t:)) exp (-11mm (k()t))dt
p( 1m
w. 1) = II
i=
(L (er ppa (wssl1,
II
Eyp
II (w,t)en
((w))(())))
(e (p (IIX)
II (w.t)el
h(),
where w is the vector of wi. It is straightforward to see the integrand is the augmented likelihood:
7
p((()) w. 11/,, X)
II
[A ( (P (TTA)
II (w,A)EII
(w,))()))
Fine
A.2 Gibbs Sampler
Based on the augmented joint distribution
p((4) w, II, w, Å/T..
II A, Tm ((())((()))))))))))))))((()))(
we can derive the conditional densities of all variables in closed form. By sampling from these conditional densities iteratively, we construct an analytical Gibbs sampler.
A.2.1 Derivation for
(t,)((t))) m w) II
II (w.t)eI
((())
1=mm
12

---

where II ef(w))h(t) is constant as II is given. In addition, combined f(w, 2) = - 23 w/22 (w,t)eI1
log 2 and P (b, 0) e
Pro(wlb, c) with the above derivation, we can deduce that,
p(wk(tit w) x /ood x
II p 1=
A.2.2 Derivation for 1I
The posterior of IT is dependent on (1:) i=mm" w and X,
(x(I)) II ed (w)))
p(ll w, A)
Px Px [I ef( h(t))
where Campbell's theorem can be applied to convert the denominator, the equation above can be transformed as
Px (I X) [[..een (w)))(())
p(T] W, X)
exp (- S.m -o² (1 - h(())) ((ow-- ()dd)wt
(-/(mm Cm 1o00
f(u. (f( h(t)) Xpra(w[1, O)dwvd() )
= II (c((w) (wpa(w|1, (w.t)EII
[PPl:- 0)) exp
Crm
The above posterior is in the likelihood form of a marked Poisson process with intensity m A) = (ww,))(t)))
(-((h(())ppa(ul1,,
A.2.3 Derivation for Ā
p p(iltt) (m II)
II
[Att, we (wr. (X(() 1/X
(D (oralw/1,,lte ((t4)) 1/2,
II Sppo(w|1,0)) Jim Jo Apeatal 1.ddddt (w.1)e1
where PPG (w(1, transformed as
So the above equation are
II) & Amm--=+14 +1+ 11 ftm di- (ti
Let Nm = m Tm + 1, [II is the number of points on the marked Poisson process, and
Tm fm f... then we can get
p(A III Paa (N|N R, T.))
A.2.4 Derivation for
For w, we utilize the Gaussian prior p(w) (WwO0, where K is the prior covariance matrix. Then the derivation for w is follows
B+1
p(wlft: 1
II) x
II
2
II
II (ww)))(())
2
b
(w.t)el1
where hi(1) EE Enso da(l -1--1- 1;) and (w, = =122 =/2-2220 w// log 2 and prG (w]b, 0)-
PPG (w (w)lbcc.. Therefore, obtain
pl p(w|(4)mm WI N(wm, E),
where Σ = $D4 T + KK where D is diagonal matrix with (wi mm in the first m + entries and (wr R in the following R entries, and & (( )m (t,))) where the first m Tm. + 1 entries of are 1/2, and the following R entries of v are -1/2.
ΣΦv.
13

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B Algorithm
B.1 Gibbs Sampler
Algorithm 1 Gibbs Sampler
Input: fr basis functions (o)(-)) covariance K;
Output: Parameter posterior samples [w-(k), (()) 1==1;
1: for iteration do
2: Sample w by Eq. (5a);
3: Sample IT through thinning by Eq. (5b);
4: Sample by Eq. (5c);
5: Sample w by Eq. (5d).
6: end for
B.2 CoBay-CPD
Algorithm 2 (m + 1)-th round of CoBay-CPD
Input: mm, basis functions (-)) covariance K;
Output: Change point at the current position or not;
1: Sample {w/kk) {(k) K by Appendix B.1;
2: Sample (t(0) p(tm+1l(r...m w(k), { and create confidence interval of (fm+ e.g., the 5% and 95% quantiles denoted as +1 and 1 the actual tm+1 falls within the interval [1 th +1]. classify not change point; otherwise, a change point.
C Analysis of Complexity
Assuming the length of the entire sequence is N, the average length of fr..mm is M, the average length of the latent marked Poisson process is R. the average number of points within the interval of To is Ng. and the number of Gibbs iterations is L, the computation complexity of CoBay-CPP is ONN(N B + LRNNB- + LCTH + L(M + (B + 1)² + L(B 1)²)), where Cth represents the complexity of the thinning algorithm marked poisson process, not in the prediction step. This is because thinning algorithm in prediction step only samples one point at a time, which is very fast, so its computational complexity can ignored. We ignore the complexities of other sampling operations since they are fast. The first term corresponds to the precomputation of ( on f...sm the second term to the computation of (() on II, the third term to the sampling of the marked Poisson process, the fourth and fifth terms the computation of mean and covariance. By limiting the maximum length of 6m,, and To. we can reduce the values M, No and R, thereby accelerating the computation of (l). Moreover, as the length of LT..... om decreases, CTh will decrease as well.
D Synthetic Data Experiment
D.1 Data Processing
We generate a synthetic data concatenated by three segments Hawkes process data. these three
segments Hawkes process data, we assume 4 scaled beta densities: $1,2.3.4 = Betaíā = 50, 3 50, scale = 6, shift = {-2, 0, the baseline activation. However, they have different intensity upperbounds, namely A 5, X = 10 and Xą We use the thinning algorithm to generate a sequence according the intensity function specified above.
0,01 1}) the basis functions with support [0, T. 6] and μ = 0 as
14

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D.2 Result Presentation
In experiment, we assume the basis functions are same the ground truth, and set the other hyperparameters as 90% confidence interval and K 0.51. The estimation of À and the estimated that, the estimated
= [44 Wi,..., wa]" from CoBay-CPD are shown in Fig. 1. We parameter Ā and w [n, wi...., wa] of synthetic data from CoBay-CPD closely oscillates around true value, indicating the accuracy of parameter estimation by CoBay-CPD.
12.5
10.0
number cvangs
5.0
1000
6000
Unie
50
1000
150
SMCPP
Galay-530
cred cia Innava
5VCPD
CuBay-EIO
True
Change Point Detection
(b) Estimated
Ally
15
(c) Estimated μ
(d) Estimated w
(e) Estimated w22
Caboe
(f) Estimated Wa
(g) Estimated Wa
Figure Synthetic data: (a) the change point detection results of CoBay-CPD and alternatives, illustrating the change point detection performance; (b) the estimated À from CoBay-CPD, indicating the accuracy of parameter estimation of (oBay-CPD((C)-gg) the estimated parameter (a) μ. (b) w1. (c) w2. (d) Wy and (c) wy of synthetic data from CoBay-CPD.
E Real-world Data Experiment
E.1 Data Processing
The preprocessing details two real- world datasets are shown below.
WannaCry Cyber Attack In May 2017, the WannaCry virus infected more than 200,000 computers worldwide, causing least hundreds of millions of dollars in damage, and received much attention. The WannaCry Cyber Attack data contains 208 traffic logs information observations. Fach observation contains the relevant timestamp. In this paper, the points where timestamps surge taken the ground truth change points, shown in Fig. 2a.
NYC Vehicle Collisions The New York City vehicle collision dataset comprises approximately 1.05 million vehicle collision records, each containing information about time and location the collision. For our experiments, we select the records from October 14th, 2017, which contains 477 vehiele collision records. We utilize the change points detected in [27] as the ground truth, shown in Fig. 2b.
15

---

100 1.6
206
38c
266
383
493
coanting srmees
counting pxcens
time
(a) WannaCry Data
(b) NYC Vehicle Collisions Data
Figure 2: (a) The WannaCry data with ground-truth change points (grey lines). (b) The NYC Vehicle Collisions data with ground-truth change points (grey lines). The upper plot illustrates the increasing of timestamps as events accumulate. The lower plot reverses the axes, representing a counting process.
730
5000
100
200
100
200
200
200
1.00
1:00
lime
lims
Change
Change
(a) CPB
(b) SMCPD
4.00
100
202
100
202
number OF EVONTS
number Of EverEs
160
300
100
300
400
data
cedile réeral
change point
data
crecible iéeeval
change 2Ott
prodicedd moan
clange Po Loe
prodiced moan
change 50 Inu
(c) SVCPD
(d) SVCPD+Inhibition
Figure 3: The WannaCry data. The upper plot illustrates the increasing of timestamps as events accumulate. The lower plot reverses the axes, representing counting process. The change point detection result of (a) CoBay-CPD, (b) SMCPD, (c) SVCPD, (d) SVCPD+Inhibiton.
E.2 Results Presentation
For WannaCry, we adopt a prior distribution p( w) [Ww)0, K) for CoBay-CPD, where K is
diagonal matrix with diagonal entries of 0.5. Moreover, we choose 90% confidence interval and
scaled shifted beta densities: 21,3,344 Beta( 50, 3 50, scale 6, shift {-2. 1, 1}) as basis functions. The complete graph of experimental results of four methods on WannaCry Cyber Attack Dataset shown in Fig. 3. Figures to 3d display the change point detection outcomes of different methods applied to WannaCry data. The blue line is the real data, the red solid line is the mean of the predicted points, green dotted line the confidence interval, and the orange, purple, yellow and turquoise line are the detected change point location.
16

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10
20
120
200
350
400
100
200
300
400
400
400
1ime
1911
data
ciedible interaal
Change postt
data
inberual
Change
predicted mean
chanpe paint true
predicted mean
change
(a) CoBay-CPD
(b) SMCPD
20
number of evens
number of ovent
2200
2200
10
15
20
10
15
20
25
4ite
cedile nteevl
change
dile
cistible interva sort.tue
Change
prodicted mozn
prodicted mein
(c) SVCPD
(d) SVCPD+Inhibiton
Figure 4: The NYC Vehicle Collisions data. The upper plot illustrates the increasing of timestamps as events accumulate, The lower plot reverses the axes, representing a counting process. The change point detection result of (a) CoBay-CPD, (b) SMCPD, (c) SVCPD, (d) SVCPD+Inhibito.
For NYC Vehicle Collisions, choose scaled shifted beta densities; 01,2,3,4 Beta(ā 10, 8 30, scale = 6, shift {-2, -1, O, 1}) as basis functions, 90% confidence interval and K = 0.51, which is same that in WannaCry. The change points in the NYC Vehicle Collisions obvious as those in the WannaCry, making change point detection a more challenging task for this dataset. The complete graph of experimental results of four methods NYC Vehicle Collisions Dataset is shown in Fig. 4. The blue line is the real data, the red solid line is the mean of the predicted points, the green dotted line is the confidence interval, and the orange, purple, yellow and turquoise line are the detected change point location. Figures 4a to 4d show the change point detection outcomes of four methods for the NYC data. Notably, SVCPD detects fewer change points, while SMCPP identify an excessive number.
not as
F
Stress Tests
F.1
Test Number Change Points
We conduct a stress test with the number of change points: 1,2,3. We generate three sets of synthetic data concatenated by some segments Hawkes process data. In all these segments of Hawkes process data, we assume scaled beta densities: 01.2,3,4 Beta(a 50, 50, scale 6, shift 2, - 0, 1}) as the basis functions with support [0, T = 6) and = 0 as the baseline activation. However, they have different intensity upperbounds. For # change points = 1, we
let λι 5, λι = 10; For # of change points 2, we let X21 = 5, X22 10. and λ22 = 3: For
# change points 3. we let λg1 5, Asa 10, Aga = 3, and Ar4 8. We use thinning algorithm to generate these sequences according to the intensity specified above.
17

---

F.2
Test 2: Difference between Adjacent Parameters
We conduct stress test with Δλ 0.1, 1.5. We generate three sets of synthetic data by concatenating two segments of Hawkes process data. Within each segment, we assume four scaled beta densities:
$1,2,3,4 = Beta 50, 8 50, scale = 6, shift = -2, - 0, 1) as the basis functions with support the baseline activation. However, they possess different intensity upper bounds. Specifically. for Δλ 0.1, we set mn = 10 and A12 10.1; for AA we set λ21 10 and X22 9; for AA 5, we set tsy 10 and A32 = 5. We use the thinning algorithm to generate these sequences according the intensity specified above.
[0, To 6). and μ
F.3 Test Closeness between Adjacent Change Points
We conduct a stress test with A, 5, 10, 15. Three sets of synthetic data are generated by concate- nating three segments of Hawkes process data. Within each segment, assume four scaled beta densities: 01,2.3 2.3,2.311 Beta
50, 3 50, scale 6, shift -2, -1, 0, 1) the basis functions with support [0, & = 6). and μ = 0 as the baseline activation. They have different intensity upper- bounds X1 = 10, X2 = 5, and A = 15. We use the thinning algorithm to generate these sequences according to the intensity specified above. We adjust the data length of the second segment from 5 to 10 to 15, thereby controlling the interval A, between two adjacent change points from 5 to 10 to 15.
F.4 Stree Tests Supplements
Due to the page limit, we only presented the stress test results for method in the main paper. However, based on a suggestion from an anonymous reviewer to include the stress test results for the baselines, have provided them in Tables 5 to 7.
Table 5: The FNR, FPR and MSE of CoBay-CPD and other baselines on synthetic dataset with different number of change points.
Model
2
3
FNR()
EPR(%
MSE())
FNR())
FPR(S
MSR())
FNR())
FPR(%
MSE())
SMCPD
0.33 4 0.47
0.63 0.63
0.88 0.03
0.38 = 0.41
0.76 :: 0.26
0.07 4: 0.01
0.67 = 0.19
1.23 4:
0.08 0.01
SVCPD
0.67 + 0.47
0.6 = 0.95
0,06 + 0.01
0.50 = 0.35
0.76 + 0.26
0.06
0.00
0.50 = 0.32
1.64
0.15 = 0.05
SVCPD+Iahi
1.88
0.01
0.23
CoBay-CPD
0.00 + 0.00
0.43 = 0.60
0.04 + 0.00
0.13 = 0.22
0.46 :: 0.26
0.05
=: 0.00
0.11 0.14
0.31 0.50
0.07 0.01
Table 6: The FNR, FPR and MSE of CoBay-CPD and other baselines synthetic dataset with different difference between adjacent 's (AX).
Model
FNR())
FPR(% !)
MSF())
FNR())
FPR(%
MSR(J)
FNR())
FPR(% ()
MSF())
1,20
0.70
SVCPD
1.00 0.00
2.41
= 0.98
0.05 : 0.01
0.8 = 0.37
5.29 + 1.05
0.06 0.01
0.67 = 0.47
0.63 2: 0.95
0.06 0.01
SVCPD+Inhi
0.67 1: 0.47
1.41 0.83
0,06 1 0.00
0.33 2 0.47
1.17
1: 0.52
0.06 3: 0.00
0.33 0.47
1.88 1 0.63
0.08 0.01
CoBay-CPD
1,00 2 0.00
1.61 0.57
0.02 : 0.00
0.25 = 0.43
0.35 + 0.60
0,00
0.00
0.43 : 0.60
0.04 0.00
Table 7: The FNR, FPR and MSE of CoBay-CPD and other baselines synthetic dataset with different closeness between two change points (At).
Model
15
ENR())
FPR(% J)
MSE())
FNR())
FPR(%
MSB(J)
FNR())
FPR(% 1)
MSB())
SMCPD
0.75
0.24
SVCPD
0.42 0.19
1.24
= 0.56
0.03 0.01
0.75 = 0.25
0.46 2: 0.65
0.05 0.01
0.08 = 0.19
3.01
1.15
0.06
0.01
SVCPD+Inhi
0.58 0.19
1.24 à 1.33
0.05 1 0.01
0.25 - 0.38
0.23 1 0.52
0.05 1 0.00
0.17 0.24
2.01
1.33
0.05
0.00
CoBay-CPD
0.33 1: 0.24
1.00 = 0.70
0.01 1 0.00
0.00 = 0.00
0.93 1 0.65
0.02 0.00
0.08 0.19
0.80
0.57
0.03
0.00
18

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
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Reviews:
[{"review_id": "9xFTe2OA9R", "reviewer": "Reviewer_E9Cj", "summary": "This paper considers the Bayesian two-step change point detection model for Hawkes process. Through data augmentation techniques such as the use of Polya-Gamma random variables and the marked Poisson process, conditional conjugacy is achieved and an efficient Gibbs sampler can be designed for posterior computation in the first step. Extensive numerical experiments are provided on both synthetic and real data.", "questions": "1. What are the benefits of using beta density as the basis functions for \\phi, compared to using say functions based on differently scaled RBF or Matern kernels?\n2. Just like the sigmoid function can be polya-gamma augmented, the probit function also has its own data augmentation techniques. Can some similar Gibbs sampler be obtained if we use the probit function as \\sigma?\n3. How well is the mixing of the data augmented Gibbs sampler?", "limitations": "See weaknesses and questions.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and organized.\n2. Sufficient background is provided, and clear motivation is discussed.\n3. The targeted problem, i.e. posterior computation for Bayesian change point detection problem, is an interesting, important, and challenging problem.\n4. The proposed method is concise, and very applicable in real problems.\n3. Extensive numerical experiments and analyses are conducted.", "weaknesses": "The method is only Bayesian in the first step, i.e. the step of obtaining the posterior predictive distribution of the next event occurrence t_{m + 1}. The determination of change point is purely based on whether the next event occurrence lies with in the posterior predictive credible interval, which no longer takes into account of the uncertainty at this step. It would be better and more intuitive if a jointly Bayesian model could be used - of course, this is also more computationally challenging.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q: The method is only Bayesian in the first step, ...... more computationally challenging. \n\nA: Thank you for your valuable suggestion. As you mentioned, we are indeed only using Bayesian in the first step at the moment. In future work, we will consider using a complete Bayesian approach.\n\n> Q: What are the benefits of using beta density as the basis functions for $\\phi$, compared to using say functions based on differently scaled RBF or Matern kernels? \n\nA: This is a good question. We experimented with beta density as the basis functions for $\\phi$ because we followed the convention in the previous work [30]. As stated in [30], ``Although basis functions can be in any form, to make the weights indicative of functional connection strength, basis functions are chosen to be probability densities with support $[0, \\infty]$\". Furthermore, they restricted the support to $[0, T_\\phi]$ to accelerate the computation of ${\\Phi}(t)$. Therefore, beta density is a natural choice. \n\n> Q: Just like the sigmoid function can be polya-gamma augmented, the probit function also has its own data augmentation techniques. Can some similar Gibbs sampler be obtained if we use the probit function as $\\sigma$? \n\nA: This is an interesting question. The probit method also has its own data augmentation techniques. Theoretically, a similar Gibbs sampler can also be obtained if we use the probit function to replace $\\sigma$. However, we have not tried this method. \n\n> Q: How well is the mixing of the data augmented Gibbs sampler? \n\nA: Thank you for your question. In our experiments, we set the burn-in to 90, considering the samples starting from the 91st as samples from the stationary distribution. We found that the mixing of the data augmented Gibbs sampler performed very well.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "exaLaJaly5", "reviewer": "Reviewer_xKxH", "summary": "This paper proposes a conjugate Bayesian two-step change point detection method for the Hawkes process using data augmentation. It addresses the computational inefficiency of existing methods by providing analytical expressions. The new method proves to be more accurate and efficient, as demonstrated by extensive experiments on both synthetic and real data.", "questions": "Gibbs sampling is still used; is that computationally expensive? Could you please discuss the computational complexity of your algorithm?", "limitations": "Yes, discussed in a separate section.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This paper proposes a novel method that ensures the posterior distribution of the bounded historical period Hawkes process is conjugate. Innovatively, it employs Pólya-Gamma variables and marked Poisson processes. This innovative approach is highly useful, extending beyond change point detection to potentially a wide range of applications for the Hawkes model.\n\nEmpirical results show the proposed method outperforms counterparts.", "weaknesses": "The implementation details have not been adequately discussed and presented. While some figures and details are provided in the appendix, I suggest moving the important ones to the main content and discussing them in detail. Additionally, I recommend including more background information on Pólya-Gamma variables and marked Poisson processes, as these concepts may not be familiar to readers outside this domain.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q: The implementation details have not been adequately discussed ...... as these concepts may not be familiar to readers outside this domain. \n\nA: Thank you for your suggestion. Due to page limit, some content had to be placed in the appendix. However, we appreciate your feedback and will consider making a more reasonable adjustment to the content placement in the camera-ready. Additionally, we will include more background information on Polya-Gamma variables and marked Poisson processes to aid readers who may not be familiar with these concepts.\n\n> Q: Gibbs sampling is still used; is that computationally expensive? \n\nA: Thanks to the data augmentation technique, the data augmented Gibbs sampler has completely analytical expressions, making its use not expensive.\n\n> Q: Could you please discuss the computational complexity of your algorithm? \n\nA: Indeed, we have already done this. The discussion on computational complexity is provided in detail in Section 3.3.5, ‘Algorithm, Hyperparameters, and Complexity,’ on page 6, line 215 of the main content.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FVDZveaAKU", "reviewer": "Reviewer_5yaf", "summary": "The paper aims to detect change points (in terms of model parameters) in point processes and proposes a conjugate Bayesian two-step change point detection method for Hawkes processes. This is achieved by applying data augmentation and a novel Gibbs sampler for closed-form updates for model parameters. For both synthetic and real-world data, the proposed method demonstrates superior performance in accuracy and efficiency compared to existing baselines.", "questions": "Could you help explain the statistical significance of results for synthetic data, e.g., how many runs to average the results for the std.? It seems to me that 1 std. is relatively large compared to the average, and there are large overlaps of 1 std. between different models.", "limitations": "Limitations are discussed in the paper.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "* The paper is very well-written in its structure, notation, explanation, and discussion. This work is put in the context of literature and is self-contained. All symbols are defined before using them.\n* The main paper includes all important pieces with great clarity and necessary details while and concise at the same time.\n* Experimental results demonstrate significant improvements from baselines (especially on real-world datasets). The settings are clearly described for readers to interpret the results.", "weaknesses": "The results for synthetic data are not as strong as those for real-world data, please see my question below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q: Could you help explain the statistical significance of results for synthetic data, e.g., how many runs to average the results for the std.? It seems to me that 1 std. is relatively large compared to the average, and there are large overlaps of 1 std. between different models. \n\n\nA: Thanks for your question; this is a very good point. We averaged the results over 5 runs to obtain the final results. The randomness in our experiments comes from the sampling of model parameters from the corresponding parameter posterior and the sampling of the next point given the sampled parameters. The interval of our model takes all this randomness into account, which is why the experimental variance is a bit large.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VxR7aAJxlM", "reviewer": "Reviewer_Wgzk", "summary": "The paper proposes a new Bayesian inference method for change point detection in Hawkes Processes using conditionally conjugate priors and Gibbs Sampling. In their experiments, their new method turns out to perform better than competing methods both in terms of accuracy and speed.", "questions": "- In the synthetic datasets experiments, I wonder whether the data simulation benefits your method over the other methods? Asked differently are your and the other methods only differing in the inference algorithm applied or also in the specific model they assume? In the latter case, simulating data from the same model that is assumed by your method (but not by others?) could potentially bias the results. I would like to understand this point better.\n- In terms of innovations of the conditional conjugate approach, what are the concrete innovations over the approach proposed in [30] as cited in your paper?\n- You only investigate the use of 1 to 3 basis functions. As someone experienced with splines, this seems relatively little. Can you elaborate why you only need few basis functions in most real world scenarios?\n- Is your method implemented somewhere in a user friendly manner such that people can readily apply it to their own data?", "limitations": "As a non expert for these models, I wonder if your new method limitations also for univariate Hawkes Processes? You only mention limitations for potential multivariate extentions.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors do a good job in motivating the relevance of the model and the need for improved (Bayesian) inference methods. The benchmarks appear to be sound and show clear benefits of the new method compared to alternatives. Obtaining conditionally conjugate priors for such a model is not an easy task and I think the authors do a good job in communicating their theoretical results both in the main text and in the appendix (where they provide the proofs).", "weaknesses": "- The synthetic data experiments seem to be relatively limited, showing only the results of a single process with 2 change points. I think that a bigger simulation study with more variation in different aspects (beyond what the stress test study did) would have strengthend the paper. \n- The stress test in the ablation study seems to vary the difficulty of the problem, which makes sense. However, the authors only show the results of their own method there, although I see no reason to not also show the results of the other methods for these cases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q: The stress test in the ablation study seems to vary the difficulty of the problem, which makes sense. However, the authors only show the results of their own method there, although I see no reason to not also show the results of the other methods for these cases.\n\nA: Thank you for your valuable suggestion. Due to page limit, we only presented the results of the stress tests for our own method in the paper. In the rebuttal PDF, we have included results for other baseline methods as well. It is evident that our method still shows significant advantages. Please see Table 1, 2 and 3 in the rebuttal PDF. Thank you once again for your suggestion and we will add these new results to the camera ready. \n\n> Q: In the synthetic datasets experiments, I wonder whether ...... could potentially bias the results. I would like to understand this point better.\n\nA: Thank you for your asking. The data simulation method could indeed favor our model over others because our model assumes a nonlinear Hawkes process, while the SMCPD and SVCPD models assume a linear Hawkes process. To address this potential bias, we modified the SVCPD model to also assume a nonlinear Hawkes process, resulting in the SVCPD+Inhi model. Our model still outperformed all these models, demonstrating the effectiveness and robustness of our approach. Thank you for highlighting this important point. \n\n> Q: In terms of innovations of the conditional conjugate approach, what are the concrete innovations over the approach proposed in [30] as cited in your paper?\n\nA: [30] used conditional conjugate methods for parameter estimation without change points, but we made an extension to this method, using conditional conjugate methods for change point detection. This is the main difference between our work and [30]. \n\n\n> Q: You only investigate the use of 1 to 3 basis functions. As someone experienced with splines, this seems relatively little. Can you elaborate why you only need few basis functions in most real world scenarios? \n\nA: We verified this through experiments. In fact, we tried 1-5 basis functions. However, due to page limits, we only showed the results for 1-3. We found that having too many basis functions might lead to overfitting, so using fewer basis functions might be more effective. \n\n\n> Q: Is your method implemented somewhere in a user friendly manner such that people can readily apply it to their own data? \n\nA: The code can be run with a single Python file, and anyone can directly call this interface. Once everything is finalized, our code will be made publicly available on GitHub, making it easy for everyone to use on their own data. Thank you for your interest. \n\n\n> Q: As a non expert for these models, I wonder if your new method limitations also for univariate Hawkes Processes? You only mention limitations for potential multivariate extentions. \n\nA: For univaraite Hawkes process, these limitations do not exist.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their efforts in providing insightful comments and constructive feedback. 
We are pleased that the reviewers have recognized the significance of our paper in solving an interesting change point detection problem in Hawkes process [R1, R2, R3, R4], conducting comprehensive numerical experiments [R1, R2, R3, R4], and maintaining clear and concise writing [R1, R2, R3]. 
In the following, we address reviewers’ comments point by point.

Meta-review: This paper uses data augmentation to design a conjugate Bayesian two-step change point detection algorithm for the Hawkes process. This is more accurate than existing methods that rely on non-conjugate inference. The reviewers agree that this is well-written and clearly motivated, the problem considered is important, and the experiments are extensive. The reviewers have asked many questions which the authors need to address in the revised manuscript.
Decision: Accept (poster)