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A grid-based methodology for fast online changepoint detection

Per August Jarval Moen

Abstract

We propose a grid-based methodology for online changepoint detection that allows offline changepoint tests to be applied to sequentially observed data. The methodology achieves low update and storage costs by testing for changepoints over a dynamically updating grid of candidate changepoint locations. For a broad class of test statistics, including those based on empirical averages and certain likelihood ratios, we show that the resulting online procedure has update and storage costs that grow at most logarithmically with the sample size. We further show that finite-sample power guarantees for the offline test translate directly into non-asymptotic upper bounds on the detection delay, under a mild robustness assumption. Building upon the methodology, we construct methods for detecting changes in the mean and in the covariance matrix of multivariate data, and prove near-optimal non-asymptotic upper bounds on their detection delays. The effectiveness of the methodology is supported by a simulation study, where we compare its performance for detecting mean changes with that of state-of-the-art online methods. To illustrate its practical applicability, we use the methodology to detect structural changes in currency exchange rates in real time.

A grid-based methodology for fast online changepoint detection

Abstract

We propose a grid-based methodology for online changepoint detection that allows offline changepoint tests to be applied to sequentially observed data. The methodology achieves low update and storage costs by testing for changepoints over a dynamically updating grid of candidate changepoint locations. For a broad class of test statistics, including those based on empirical averages and certain likelihood ratios, we show that the resulting online procedure has update and storage costs that grow at most logarithmically with the sample size. We further show that finite-sample power guarantees for the offline test translate directly into non-asymptotic upper bounds on the detection delay, under a mild robustness assumption. Building upon the methodology, we construct methods for detecting changes in the mean and in the covariance matrix of multivariate data, and prove near-optimal non-asymptotic upper bounds on their detection delays. The effectiveness of the methodology is supported by a simulation study, where we compare its performance for detecting mean changes with that of state-of-the-art online methods. To illustrate its practical applicability, we use the methodology to detect structural changes in currency exchange rates in real time.

Paper Structure

This paper contains 53 sections, 29 theorems, 278 equations, 19 figures, 1 table.

Table of Contents

  1. Online changepoint detection using the static grid
  2. Computational properties
  3. Statistical properties
  4. Some more online detectors detecting changes in covariance matrices
  5. Sparse changes in covariance
  6. A likelihood-ratio detector for a change in mean or covariance in Gaussian data
  7. Some online detectors for changes in regression coefficients
  8. A direct approach
  9. Discussion: High-dimensional data, non-Gaussianity and temporal dependence
  10. Comparison of rates for the multivariate mean-change problem
  11. Comparison with ocd
  12. Case 1:
  13. Case 2:
  14. Case 3:
  15. Comparison with MdFOCuS
  16. ...and 38 more sections

Key Result

Lemma 1

For all $t \geq 2$, the grid $G^{(t)}$ in thegrid satisfies

Figures (19)

  • Figure 1: Plot of the elements of the reversed static geometric grid $t - G^{(t)}_{\mathrm{stat}}$ for $t = 9, \ldots, 12$.
  • Figure 2: Evolution of the grid $G^{(t)}$ in \ref{['thegrid']} for $t = 17, \ldots, 20$.
  • Figure 3: Plot of the elements of the reversed dynamic geometric grid $t - G^{(t)}$ for $t = 9, \ldots, 12$.
  • Figure 4: Average detection delay of the detectors (on log scale) for varying change magnitudes ($\phi$) and changepoint sparsities $k = 1,5,10,100$.
  • Figure 5: Relative update time (top) and memory consumption (bottom) of the detectors, shown as a function of $t$ (left) and $p$ (right).
  • ...and 14 more figures

Theorems & Definitions (53)

  • Lemma 1
  • Theorem 1
  • Proposition 1
  • Example 1
  • Proposition 2
  • Example 2
  • Proposition 3
  • Example 1: continued
  • Proposition 4
  • Theorem 2
  • ...and 43 more