Ordinal Patterns Based Change Points Detection

TL;DR

This work develops a rigorous probabilistic framework for ordinal patterns in time series by proving functional central limit theorems for relative frequencies of ordinal patterns under linear increment models, covering both short-range and long-range dependence. By representing ordinal patterns as linear inequalities of the increment vector, the authors obtain explicit CLTs for pattern probabilities and establish a reduction principle linking SRD and LRD regimes. They apply these results to turning-rate based change-point detection, deriving self-normalized CUSUM statistics and variance estimators, and demonstrate an EEG application that identifies transitions between sleep stages. The theoretical contributions enable statistically principled detection of distributional changes in non-Gaussian, potentially long-range dependent time series, with practical guidance for variance estimation and implementation. Overall, the paper provides a solid bridge between ordinal-pattern theory, empirical processes, and change-point analysis with real-world applicability to biomedical signals.

Abstract

The ordinal patterns of a fixed number of consecutive values in a time series is the spatial ordering of these values. Counting how often a specific ordinal pattern occurs in a time series provides important insights into the properties of the time series. In this work, we prove the asymptotic normality of the relative frequency of ordinal patterns for time series with linear increments. Moreover, we apply ordinal patterns to detect changes in the distribution of a time series.

Figures (5)

• Figure 1: The six ordinal patterns of order $r+1=3$.
• Figure 2: Top: EEG recordings of a healthy individual (4th patient from the CAP Sleep Database terzano2001atlas), originally sampled at 512 Hz, resulting in a time series of 19,553,792 data points. Due to the high resolution and length, detailed features are difficult to discern. Bottom: Corresponding turning rate series, where each data point represents an 8-second segment of the EEG recording. The sleep cycles are visible.
• Figure 3: Histogram of $SC_{n_b}$ for $n=5000$ and 1000 simulations of MA(1) without (left) and with (right) change of $\rho(1)$. In the right plot, the autoregressive parameter changes from $\theta=0.4$ to $\theta=0.7$ after 50% of the observations. Under the null hypothesis the estimated 0.95 quantile is 6.335.
• Figure 4: Power of the test for different values of $h=\phi_2-\phi_1$ with $n=500$ and $n=1000$ data points and Laplace distributed innovations. The curves correspond to different values of $\tau$, where changes occur at $1/10$ for the blue curve $(\tau=0.1)$, at $1/4$ for the green curve $(\tau=0.25)$, and at $1/2$ for the red curve $(\tau=0.5)$. $\phi_1$ is set to $0.4$. For $h=0$ (no change), the power corresponds to the significance level $5\%$.
• Figure 5: Extract of EEG recordings for the $5$-th patient of the dataset terzano2001atlas. The recordings cover approximately 30 minutes of observations extracted from the C4-P4 channel, in the temporal window going from 01:08:2 (REM) to 01:47:33 (S2). The time series contains $1.2\times 10^6$ data points.

Theorems & Definitions (47)

• Lemma 1
• proof
• Theorem 1: Short-Range Dependence
• Theorem 2: Long-Range Dependence
• Definition 1
• Lemma 2
• proof
• Theorem 3: Short-Range Dependence
• proof : Proof of Theorem \ref{['theorem:convergence_OP_SRD']}
• Theorem 4: Long-Range Dependence