[Paper Review] Efficient penalty search for multiple changepoint problems

Kaylea Haynes, Idris A. Eckley+1 more|arXiv (Cornell University)|Dec 11, 2014

Optimal Experimental Design Methods14 references22 citations

TL;DR

This paper proposes CROPS, a computationally efficient method to find optimal changepoint segmentations across a continuous range of penalty values in multiple changepoint problems. By leveraging dynamic programming and a novel penalty search strategy, CROPS achieves linear computational complexity in data size and the difference between optimal changepoint counts, enabling rapid comparison of segmentations for robust penalty selection without relying on model-specific criteria.

ABSTRACT

In the multiple changepoint setting, various search methods have been proposed which involve optimising either a constrained or penalised cost function over possible numbers and locations of changepoints using dynamic programming. Such methods are typically computationally intensive. Recent work in the penalised optimisation setting has focussed on developing a pruning-based approach which gives an improved computational cost that, under certain conditions, is linear in the number of data points. Such an approach naturally requires the specification of a penalty to avoid under/over-fitting. Work has been undertaken to identify the appropriate penalty choice for data generating processes with known distributional form, but in many applications the model assumed for the data is not correct and these penalty choices are not always appropriate. Consequently it is desirable to have an approach that enables us to compare segmentations for different choices of penalty. To this end we present a method to obtain optimal changepoint segmentations of data sequences for all penalty values across a continuous range. This permits an evaluation of the various segmentations to identify a suitably parsimonious penalty choice. The computational complexity of this approach can be linear in the number of data points and linear in the difference between the number of changepoints in the optimal segmentations for the smallest and largest penalty values. This can be orders of magnitude faster than alternative approaches that find optimal segmentations for a range of the number of changepoints.

Motivation & Objective

• To address the challenge of selecting an appropriate penalty in penalized changepoint detection when the assumed data model is misspecified.
• To enable comparison of segmentations across a continuous range of penalty values, rather than relying on single-default penalties like AIC or BIC.
• To develop a computationally efficient method that avoids the high cost of repeatedly solving constrained optimization problems for each penalty value.
• To provide a practical alternative to existing methods like Segment Neighbourhood that are computationally intensive.
• To improve robustness in changepoint detection by allowing data-driven evaluation of multiple penalty choices.

Proposed method

• CROPS uses the PELT algorithm to efficiently minimize a penalized cost function over a continuous range of penalty values β.
• It exploits the duality between penalized and constrained optimization: for each β, it finds the corresponding number of changepoints m* that yields the same segmentation.
• The method computes optimal segmentations for all β in a continuous interval by tracking how the optimal number of changepoints changes with β.
• It leverages dynamic programming with pruning to maintain linear time complexity in data size n and in the difference between the maximum and minimum optimal changepoint counts.
• The algorithm efficiently traverses the penalty range by identifying breakpoints where the optimal segmentation changes, avoiding redundant computation.
• It supports any cost function and can be combined with any changepoint detection algorithm, though PELT is used for its efficiency.

Experimental results

Research questions

• RQ1 How can we efficiently compute optimal segmentations for a continuous range of penalty values in multiple changepoint problems?
• RQ2 What is the computational cost of comparing segmentations across multiple penalty values, and can it be reduced below existing methods?
• RQ3 How does model mis-specification affect standard penalty choices like AIC, BIC, or Hannan-Quinn, and can a range-based approach mitigate this?
• RQ4 Can a single algorithm efficiently output segmentations for all penalty values in a continuous interval, rather than for discrete numbers of changepoints?
• RQ5 How does the proposed method compare in speed and accuracy to existing approaches like Segment Neighbourhood or fixed-penalty methods?

Key findings

• CROPS achieves linear computational complexity in the number of data points n and in the difference between the number of changepoints in the optimal segmentations for the smallest and largest penalty values.
• The method is up to two orders of magnitude faster than the Segment Neighbourhood approach in simulations, significantly improving computational efficiency.
• Using the SIC penalty (BIC) on well-log data led to overfitting with 218 segments, demonstrating the fragility of default penalty choices under model mis-specification.
• The elbow method on unpenalized cost vs. number of segments suggested 10–12 segments as plausible, and CROPS enabled direct comparison of such segmentations across penalties.
• CROPS recovers segmentations that are optimal under penalized criteria for any β in the range, avoiding the need to solve constrained problems for each m.
• The approach enables robust, data-driven penalty selection by allowing visual and quantitative evaluation of segmentations across a continuous penalty spectrum.