Online Neural Networks for Change-Point Detection

TL;DR

These algorithms demonstrate linear computational complexity and are suitable for change-point detection in large time series and compare them with the best known algorithms on various synthetic and real world data sets.

Abstract

Moments when a time series changes its behavior are called change points. Occurrence of change point implies that the state of the system is altered and its timely detection might help to prevent unwanted consequences. In this paper, we present two change-point detection approaches based on neural networks and online learning. These algorithms demonstrate linear computational complexity and are suitable for change-point detection in large time series. We compare them with the best known algorithms on various synthetic and real world data sets. Experiments show that the proposed methods outperform known approaches. We also prove the convergence of the algorithms to the optimal solutions and describe conditions rendering current approach more powerful than offline one.

Figures (9)

• Figure 1: Example of a time series with two change-points at moments $t_{1}=400$ and $t_{2}=800$. Observations between these points have different probability distributions: $P_{1}(x(t))$ for $0 < t < t_{1}$, $P_{2}(x(t))$ for $t_{1} < t < t_{2}$ and $P_{3}(x(t))$ for $t_{2} < t < 1200$.
• Figure 2: Example of change-point detection using the proposed algorithms. (Top) A time series with two change-points at moments $t_{1}=400$ and $t_{2}=800$. (Bottom) Change-point detection score $\bar{d}(t)$ estimated by the algorithms ONNC and ONNR.
• Figure 3: Example of change-point detection using the online and offline algorithms. (Top) A time series with one change-point at moments $t=0$. (Bottom) Change-point detection score $\bar{d}(t)$ estimated by the algorithms.
• Figure 4: Change-point detection score estimated by the algorithms ONNC and ONNR after the time shift: $\bar{d}'(t) = \bar{d}(t+l+n)$, where score $\bar{d}(t)$ is shown in Figure \ref{['fig:cpd_example_shift']}. Positions of the score peaks are considered as positions of the detected change-points.
• Figure 5: Example of change-point detection score $\bar{d}'(t)$ estimated by ONNC and ONNR algorithms (bottom) for a time series in mean jumps data set (top).

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Corollary 2.1
• Corollary 2.2
• Corollary 2.3
• Theorem 3
• Theorem 4
• Corollary 4.1
• Corollary 4.2
• proof : The proof of Theorem \ref{['lem:1']}