[Paper Review] Off-line detection of multiple change points with the Filtered Derivative with p-Value method

Pierre R. Bertrand, Mehdi Fhima+1 more|arXiv (Cornell University)|Mar 22, 2010

Complex Systems and Time Series Analysis23 references4 citations

TL;DR

This paper proposes the Filtered Derivative with p-Value (FDp-V) method for off-line detection of multiple change points in time series with unknown change point count. It combines the fast, linear-complexity Filtered Derivative with a p-value-based false alarm removal step, achieving O(n) time and memory complexity while maintaining high detection accuracy on simulated and real data, including ECG and financial time series.

ABSTRACT

This paper deals with off-line detection of change points for time series of independent observations, when the number of change points is unknown. We propose a sequential analysis like method with linear time and memory complexity. Our method is based at first step, on Filtered Derivative method which detects the right change points but also false ones. We improve Filtered Derivative method by adding a second step in which we compute the p-values associated to each potential change points. Then we eliminate as false alarms the points which have p-value smaller than a given critical level. Next, our method is compared with the Penalized Least Square Criterion procedure on simulated data sets. Eventually, we apply Filtered Derivative with p-Value method to segmentation of heartbeat time series, and detection of change points in the average daily volume of financial time series.

Motivation & Objective

• Address the challenge of detecting multiple change points in large-scale time series with unknown change point count.
• Overcome the high O(n²) computational complexity of traditional methods like Penalized Least Squares Criterion (PLSC) and dynamic programming.
• Develop a method with linear time and memory complexity suitable for modern high-frequency, large-volume data.
• Reduce false positives in change point detection by integrating statistical significance testing via p-values.
• Validate the method on simulated data and real-world applications such as heartbeat and financial time series segmentation.

Proposed method

• Apply the Filtered Derivative method to detect potential change points with O(n) time and memory complexity.
• Use a second step to compute p-values for each detected potential change point under the null hypothesis of no change.
• Eliminate false alarms by removing points with p-values below a predefined significance threshold.
• Leverage strong approximation theory and extreme value limit theorems to derive asymptotic distributional properties of the test statistic.
• Utilize the standardization of the Filtered Derivative to ensure stability and comparability across different data segments.
• Apply the method to both simulated data and real-world time series, comparing results with the Penalized Least Squares Criterion (PLSC).

Experimental results

Research questions

• RQ1 Can a change point detection method achieve O(n) time and memory complexity while maintaining high detection accuracy on large-scale time series?
• RQ2 How effective is the integration of p-value thresholding in reducing false positives in the Filtered Derivative method?
• RQ3 Does the FDp-V method outperform or match the performance of PLSC on simulated and real data under varying signal-to-noise conditions?
• RQ4 What is the asymptotic distribution of the Filtered Derivative under the null hypothesis of no change point?
• RQ5 Can the FDp-V method reliably detect structural changes in high-frequency physiological and financial time series?

Key findings

• The FDp-V method achieves O(n) time and memory complexity, making it scalable to large datasets with n ≥ 100,000 observations.
• The method successfully reduces false positives by eliminating change points with p-values below the significance level, improving detection specificity.
• On simulated data, FDp-V detects change points with high accuracy comparable to PLSC, but with significantly lower computational cost.
• The asymptotic distribution of the standardized Filtered Derivative converges to the Gumbel extreme value distribution, supporting the validity of the p-value computation.
• In real-world applications, FDp-V successfully segments 24-hour ECG data and detects structural changes in daily financial volume time series.
• Theoretical results confirm that the p-value-based filtering step is asymptotically valid under the null hypothesis, ensuring control of Type I error rates.