Optimal detection of changepoints with a linear computational cost

We consider the problem of detecting multiple changepoints in large data sets. Our focus is on applications where the number of changepoints will increase as we collect more data: for example in genetics as we analyse larger regions of the genome, or in finance as we observe time-series over longer periods. We consider the common approach of detecting changepoints through minimising a cost function over possible numbers and locations of changepoints. This includes several established procedures for detecting changing points, such as penalised likelihood and minimum description length. We introduce a new ∗R. Killick is Senior Research Associate, Department of Mathematics & Statistics, Lancaster University, Lancaster, UK (E-mail: [email protected]). P. Fearnhead is Professor, Department of Mathematics & Statistics, Lancaster University, Lancaster, UK (E-mail: [email protected]). I.A. Eckley is Senior Lecturer, Department of Mathematics & Statistics, Lancaster University, Lancaster, UK (E-mail: [email protected]). The authors are grateful to Richard Davis and Alice Cleynen for providing the Auto-PARM and PDPA software respectively. Part of this research was conducted whilst R. Killick was a jointly funded Engineering and Physical Sciences Research Council (EPSRC) / Shell Research Ltd graduate student at Lancaster University. Both I.A. Eckley and R. Killick also gratefully acknowledge the financial support of the EPSRC grant number EP/I016368/1. 1 ar X iv :1 10 1. 14 38 v3 [ st at .M E ] 9 O ct 2 01 2 method for finding the minimum of such cost functions and hence the optimal number and location of changepoints that has a computational cost which, under mild conditions, is linear in the number of observations. This compares favourably with existing methods for the same problem whose computational cost can be quadratic or even cubic. In simulation studies we show that our new method can be orders of magnitude faster than these alternative exact methods. We also compare with the Binary Segmentation algorithm for identifying changepoints, showing that the exactness of our approach can lead to substantial improvements in the accuracy of the inferred segmentation of the data.