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. Recent work in the penalised optimisation setting has focussed on developing an exact pruning-based approach which, 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. To this end we present a method that enables us to find the solution path for all choice of penalty values across a continuous range. This permits an evaluation of the various segmentations to identify a suitable 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.

In this paper we build on an approach proposed by for nonparametric changepoint detection. This approach defines the best segmentation for a data set as the one which minimises a penalised cost function, with the cost function defined in term of minus a non-parametric loglikelihood for data within each segment. Minimising this cost function is possible using dynamic programming, but their algorithm had a computational cost that is cubic in the length of the data set. To speed up computation, resorted to a screening procedure which means that the estimated segmentation is no longer guaranteed to be the global minimum of the cost function. We show that the screening procedure adversely affects the accuracy of the changepoint detection method, and show how a faster dynamic programming algorithm, pruned exact linear time (PELT) (Killick et al. 2012), can be used to find the optimal segmentation with a computational cost that can be close to linear in the amount of data. PELT requires a penalty to avoid under/over-fitting the model which can have a detrimental effect on the quality of the detected changepoints. To overcome this issue we use a relatively new method, changepoints over a range of penalties (Haynes et al. 2016), which finds all of the optimal segmentations for multiple penalty values over a continuous range. We apply our method to detect changes in heart-rate during physical activity. Electronic supplementary materialThe online version of this article

In recent years, various means of efficiently detecting changepoints in the univariate setting have been proposed, with one popular approach involving minimising a penalised cost function using dynamic programming. In some situations, these algorithms can have an expected computational cost that is linear in the number of data points; however, the worst case cost remains quadratic. We introduce two means of improving the computational performance of these methods, both based on parallelising the dynamic programming approach. We establish that parallelisation can give substantial computational improvements: in some situations the computational cost decreases roughly quadratically in the number of cores used. These parallel implementations are no longer guaranteed to find the true minimum of the penalised cost; however, we show that they retain the same asymptotic guarantees in terms of their accuracy in estimating the number and location of the changes.

Kaylea Haynes, a PhD student in statistics at Lancaster University, applies changepoint analysis to Fitbit data to optimise her exercise regime

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