# Detecting changes in slope with an L0 penalty

arXiv: https://arxiv.org/abs/1701.01672 · HTML: https://mlsanity.com/paper/1701.01672/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2017.

## Built on by (most cited first, 4 of 4)

- [Automatic Change-Point Detection in Time Series via Deep Learning](https://mlsanity.com/paper/2211.03860/) (2022)
- [Predictive change point detection for heterogeneous data](https://mlsanity.com/paper/2305.06630/) (2023)
- [Geometric-Based Pruning Rules For Change Point Detection in Multiple Independent Time Series](https://mlsanity.com/paper/2306.09555/) (2023)
- [A convolutional framework for detecting event-driven dynamics in energy price series](https://mlsanity.com/paper/2609.00402/) (2026)

## Abstract (arXiv)

Whilst there are many approaches to detecting changes in mean for a univariate time-series, the problem of detecting multiple changes in slope has comparatively been ignored. Part of the reason for this is that detecting changes in slope is much more challenging. For example, simple binary segmentation procedures do not work for this problem, whilst efficient dynamic programming methods that work well for the change in mean problem cannot be directly used for detecting changes in slope. We present a novel dynamic programming approach, CPOP, for finding the "best" continuous piecewise-linear fit to data. We define best based on a criterion that measures fit to data using the residual sum of squares, but penalises complexity based on an L0 penalty on changes in slope. We show that using such a criterion is more reliable at estimating changepoint locations than approaches that penalise complexity using an L1 penalty. Empirically CPOP has good computational properties, and can analyse a time-series with over 10,000 observations and over 100 changes in a few minutes. Our method is used to analyse data on the motion of bacteria, and provides fits to the data that both have substantially smaller residual sum of squares and are more parsimonious than two competing approaches.