Multiple Change-Point Detection in Point Processes: An Exact Optimization Approach Using Cross-Validation

Kernkonzepte

This paper proposes an efficient and exact method for detecting multiple change-points in point processes, particularly inhomogeneous Poisson and marked Poisson processes, using a minimum contrast estimator, dynamic programming, and a novel cross-validation procedure.

Zusammenfassung

• Bibliographic Information: Dion-Blanc, C., Hawat, D., Lebarbier, É., & Robin, S. (2024). Multiple change-point detection for some point processes. arXiv preprint arXiv:2302.09103v3.

• Research Objective: This paper addresses the challenge of detecting multiple change-points in continuous-time point processes, specifically focusing on inhomogeneous Poisson and marked Poisson processes. The authors aim to develop an efficient and exact method for identifying the time instants where the intensity of the process changes.

• Methodology: The authors propose a novel methodology based on a minimum contrast estimator and dynamic programming. They demonstrate that under specific assumptions about the contrast function, the optimal change-points can be found within a known finite grid, enabling exact optimization. A cross-validation approach, leveraging the thinning property of Poisson processes, is employed to determine the optimal number of change-points. The methodology is extended to marked Poisson processes, where both the intensity and mark distribution parameters are subject to change. Additionally, the authors present a strategy for adapting the method to a specific type of Hawkes process, transforming it into a Poisson process through time-scaling.

• Key Findings: The proposed method accurately detects multiple change-points in both simulated and real-world datasets. The cross-validation procedure effectively selects the appropriate number of change-points, and the method demonstrates robustness in various scenarios. The authors also provide an R package, CptPointProcess, implementing the proposed methodology.

• Main Conclusions: The paper presents a novel and effective method for multiple change-point detection in continuous-time point processes. The approach offers advantages in terms of computational efficiency, accuracy, and the ability to handle both Poisson and marked Poisson processes. The extension to Hawkes-type processes further broadens its applicability.

• Significance: This research contributes significantly to the field of change-point detection by providing an exact and efficient method for continuous-time point processes. The proposed methodology and the accompanying R package offer valuable tools for researchers and practitioners analyzing time series data in various domains, including epidemiology, finance, and seismology.

• Limitations and Future Research: The current work focuses on specific types of point processes, and extending the methodology to a wider range of point process models would be beneficial. Further research could explore alternative contrast functions and optimization techniques to enhance the method's flexibility and efficiency. Additionally, investigating the theoretical properties of the proposed cross-validation procedure would strengthen its statistical foundation.

Statistiken

The total length of odd segments (k = 1, 3, 5) is ∆τ−= 17/24 and the total length of even segments is ∆τ+ = 7/24.
We consider the values λ = 32, 56, 100, 178, 316, 562 and 1000.
We consider the values λR = 1, 2, 3, 4, 6, 8, 11 and 16.
The maximum number of segments is Kmax =12.
We use M =500 samples for the cross-validation procedure and the sampling probability is set to f =4/5.
B =100 N processes are sampled with each parameter configuration.
For the Poisson-Gamma contrast, we take the hyper-parameters a = 1 and b = 1/n, where n is the number of events in the process to be segmented.
We fixed α = 0.5, β = 500 and c+ = 500 (thus, for R = 2, 3, 6, we have c−= 333, 250, 142.8, respectively).

Zitate

Wichtige Erkenntnisse aus

Multiple change-point detection for some point processes

by C. Dion-Blan... um arxiv.org 11-07-2024

https://arxiv.org/pdf/2302.09103.pdf

Tiefere Fragen

How can this change-point detection method be adapted for use with other types of point processes beyond those discussed in the paper, such as renewal processes or Cox processes?

Adapting this change-point detection method for other point processes like renewal processes or Cox processes presents both opportunities and challenges. Here's a breakdown:
Renewal Processes:

Opportunities:

Segment-Additivity: Renewal processes often exhibit a form of segment-additivity. The inter-event times within a segment are independent and identically distributed (i.i.d.), making it possible to define segment-specific cost functions.
Concavity: Depending on the specific renewal process and the chosen contrast, the concavity assumption might hold for the inter-event time distribution.

Challenges:

No Fixed Grid: Unlike Poisson processes, the optimal change-point locations in renewal processes are not guaranteed to lie on the event times. This means a fixed grid search might not be sufficient, requiring alternative optimization techniques.
Contrast Selection: Finding a suitable contrast function that reflects changes in the inter-event time distribution is crucial. Standard contrasts like the Poisson contrast wouldn't be appropriate.
Cox Processes:

Opportunities:

Intensity Function Focus: Cox processes are defined by their intensity functions, which aligns well with the paper's focus on detecting changes in intensity.

Challenges:

Intensity Estimation: Cox processes often have stochastic intensity functions, making intensity estimation more complex. This estimated intensity would then be used for change-point detection, introducing additional uncertainty.
Dependence on History: The intensity of a Cox process can depend on its entire history, not just the previous event. This dependence might violate the segment-additivity assumption, making the dynamic programming approach less effective.
General Adaptation Strategies:

Alternative Optimization: Explore optimization methods beyond fixed grid search, such as:

Gradient-based methods: If the contrast function is differentiable, gradient descent or related methods could be used.
Stochastic optimization: Techniques like simulated annealing or genetic algorithms can handle non-convex optimization problems.

Contrast Function Design: Carefully design contrast functions that capture the specific characteristics of the point process and the type of change you want to detect. This might involve:

Likelihood-based approaches: If the likelihood function of the point process is tractable, maximize the likelihood of the data under a change-point model.
Distance-based approaches: Define a distance metric between segments based on relevant features of the point process (e.g., inter-event time distributions, intensity function properties).

In summary, adapting this change-point detection method to other point processes requires addressing challenges related to optimization, contrast function design, and the specific properties of the process. However, the underlying principles of segment-additivity and concavity can still guide the adaptation process.

Could the reliance on a fixed grid for optimization potentially limit the accuracy of the method in scenarios with very subtle changes in intensity, and would alternative optimization approaches be more suitable in such cases?

You are absolutely right to point out that the reliance on a fixed grid for optimization could limit the accuracy of this change-point detection method, particularly when dealing with subtle changes in intensity.
Here's why:

Resolution Limitation: The fixed grid, determined by the observed event times, sets a limit on the resolution at which change-points can be detected. If the true change-points lie between event times and the intensity change is subtle, the method might miss these changes or place the estimated change-points at nearby event times, leading to inaccuracies.
Scenarios with Subtle Changes:
In scenarios with very subtle changes in intensity, alternative optimization approaches would likely be more suitable. These approaches could provide finer-grained control over change-point location estimation. Some potential alternatives include:

Gradient-Based Methods: If the contrast function is differentiable with respect to the change-point locations, gradient-based optimization methods like gradient descent could be employed. These methods iteratively refine the change-point estimates by moving in the direction of the negative gradient of the contrast function.

Spline-Based Methods: Represent the intensity function using splines or other flexible functions. The change-points can then be treated as parameters of these functions, and optimization can be performed over the spline parameters.

Bayesian Approaches: Employ Bayesian methods to estimate the posterior distribution of the change-point locations. This approach provides a probabilistic framework for change-point detection and can quantify the uncertainty associated with the estimates. Markov Chain Monte Carlo (MCMC) methods could be used for posterior inference.

Trade-offs:
It's important to note that while these alternative optimization approaches offer potential advantages in accuracy, they often come with increased computational complexity compared to the fixed grid dynamic programming approach. The choice of optimization method would depend on factors like:

The scale of the data: For very large datasets, the computational cost of more complex optimization methods might be prohibitive.
The desired level of accuracy: If high accuracy is paramount, the trade-off in computational cost might be justified.
The availability of computational resources.
In conclusion, while the fixed grid approach offers computational efficiency, it might not be ideal for scenarios with subtle intensity changes. Exploring alternative optimization methods that provide finer control over change-point location estimation could improve accuracy in such cases, but it's essential to consider the associated computational trade-offs.

What are the potential implications of this research for the development of real-time change-point detection systems in areas like anomaly detection or system monitoring, and what challenges might arise in such applications?

This research on change-point detection in point processes holds significant potential for advancing real-time anomaly detection and system monitoring systems. Here's a closer look at the implications and challenges:
Potential Implications:

Early Anomaly Detection: Real-time change-point detection can enable the identification of anomalies or deviations from normal behavior as they occur. This is crucial in areas like:

Cybersecurity: Detecting sudden spikes in network traffic or unusual access patterns could indicate cyberattacks.
Finance: Identifying abrupt changes in trading volume or price movements might signal fraudulent activities or market instabilities.
Healthcare: Monitoring patient vital signs for unexpected shifts could provide early warnings of deteriorating health conditions.

Adaptive System Monitoring: Systems often exhibit changes in behavior over time due to factors like wear and tear, environmental changes, or evolving usage patterns. Real-time change-point detection allows for:

Dynamic Threshold Adjustment: Automatically adapting monitoring thresholds based on detected changes in system behavior, reducing false alarms.
Proactive Maintenance: Identifying early signs of system degradation or impending failures, enabling timely maintenance and preventing costly downtime.

Challenges in Real-Time Applications:

Computational Constraints: Real-time applications demand rapid change-point detection. The computational complexity of the detection algorithm becomes a critical factor.

Strategies: Explore computationally efficient algorithms, utilize parallel processing, or develop approximate methods that trade off some accuracy for speed.

Online Adaptation: Real-time systems require methods that can adapt to evolving data streams and potentially changing underlying patterns.

Strategies: Investigate online change-point detection algorithms that update estimates as new data arrives, employ sliding window approaches to focus on recent data, or develop methods for concept drift detection.

Non-stationarity: Real-world data streams often exhibit non-stationarity, where the statistical properties of the data change over time. This can make it challenging to distinguish between true change-points and natural variations in the data.

Strategies: Develop methods robust to non-stationarity, incorporate domain knowledge to model expected variations, or employ techniques for time series decomposition to separate trend and seasonality from potential change-points.

Data Quality Issues: Real-time data streams are often noisy, incomplete, or subject to delays. These data quality issues can hinder accurate change-point detection.

Strategies: Implement data preprocessing techniques for noise reduction and outlier removal, develop methods tolerant to missing data, or incorporate data quality assessments into the decision-making process.

In conclusion, this research provides a foundation for developing real-time change-point detection systems with applications in various domains. However, addressing challenges related to computational efficiency, online adaptation, non-stationarity, and data quality is essential for successful deployment in real-world settings.