Analysis framework for higher-order temporal correlations with applications to human heartbeats

TL;DR

This paper addresses the challenge of quantifying higher-order temporal correlations in event sequences by introducing burst-tree decomposition, which maps a sequence to a hierarchical tree of bursts across multiple timescales. It defines and computes novel measures—burst complexity $C_{\Delta t}$ and burst memory $M_{\Delta t}$—alongside the ordinal burst tree representation and the burst-merging kernel $K(b,b')$, then applies these to heartbeat time series from NSR, CHF, and AF groups. The results reveal distinct multiscale temporal structures among groups and show that a classifier based on these higher-order features can achieve about $80\%$ accuracy in distinguishing the groups, highlighting the method’s potential for clinical time-series analysis. Overall, the framework provides a principled, multiscale approach to capture bursts and their couplings, offering a pathway to improved physiological insight and disease detection in bursty systems.

Abstract

We propose a time series analysis framework focused on higher-order temporal correlations in the event sequence beyond the interevent time distribution by employing the burst-tree decomposition method. Bursts are clustered events that rapidly occur within shorter time periods, and they are separated by relatively longer inactive periods. The burst-tree decomposition method exactly maps the event sequence onto a tree, called a burst tree, in which each internal node represents a merge of consecutive bursts at the timescale separating those bursts. The burst tree fully reveals the hierarchical structure of bursts, hence the higher-order temporal correlations for the entire range of timescales. Those correlations are quantified using novel and existing measures derived from the burst tree, such as the burst complexity, memory coefficient for bursts, and principal and secondary cross sections of the burst-merging kernel. We apply our framework to the heartbeat time series of healthy people and of those with heart disease to reveal distinct multiscale temporal properties in physiological time series.

Figures (7)

• Figure 1: Overview of the analysis framework for higher-order temporal correlations using the heartbeat time series of one healthy subject, referred to as "the NSR subject 1." (a) A sample period of the event sequence of the NSR subject 1, where the timing of the first event was set to 0 ms. (b) The burst tree derived from the sample event sequence in (a). (c) The burst-merging kernel $K(b,b')$ estimated using the entire period of the NSR subject 1's event sequence. The white color means no data in the area. (d) The interevent time distribution $P(\tau)$ in Eq. \ref{['eq:Ptau_define']}, with the estimated values of the burstiness measure $A$ in Eq. \ref{['eq:burstiness']} and the memory coefficient $M_\tau$ in Eq. \ref{['eq:memory']}. (e) The burst size distributions $Q_{\Delta t}(b)$ for several values of $\Delta t$ after rescaling by the average burst size $\langle b\rangle_{\Delta t}$ for each $\Delta t$. (f) The burst complexity $C_{\Delta t}$ in Eq. \ref{['eq:C_deltat']} and the memory coefficient for bursts $M_{\Delta t}$ in Eq. \ref{['eq:M_deltat']} for the entire range of $\tau_{\rm min}\leq \Delta t\leq \tau_{\rm max}$. The horizontal dotted line at $M_{\Delta t}=0$ is to guide the eyes. (g) The principal and secondary diagonal cross sections $K_1(b)$ and $K_2(b)$ in Eq. \ref{['eq:diagonals']}, which are taken from the burst-merging kernel $K(b,b')$ in (c). The vertical dotted line at $b\approx 39$ in the lower panel is to guide the eyes.
• Figure 2: (a) Illustration of the burst-tree decomposition method. Vertical blue lines on the time axis in the lower panel represent events. The vertical axis in the upper panel represents a timescale $\Delta t$. Initially, when $\Delta t < \tau_{\rm min}$, each event makes a burst of size one (empty red circles). As $\Delta t$ increases from $\tau_{\rm min}$ to $\tau_{\rm max}$, the bursts are sequentially merged to form bigger bursts (filled red circles). For $\Delta t$$\ge$$\tau_{\rm max}$, all events are clustered to form one giant burst. The number next to the merged burst denotes the burst size. (b) Schematic diagram of the stochastic merging process generating an ordinal burst tree with $n=6$.
• Figure 3: Empirical results of the burstiness measure $A$ in Eq. \ref{['eq:burstiness']} and the memory coefficient $M_\tau$ in Eq. \ref{['eq:memory']}. (a) Box plots of $A$ values for NSR, CHF, and AF groups. (b) Box plots of $M_{\tau}$ values for the same three groups. In panels (a) and (b), we include the p-values for the two-sample Kolmogorov-Smirnov test comparing distributions of $A$ and $M_\tau$. (c) Scatter plot of individual results in the $(M_\tau,A)$ space, highlighting different regions occupied by different groups.
• Figure 4: Visualization of burst trees derived from event sequences for three selected subjects in each of NSR, CHF, and AF groups (top to bottom). In all cases, $70$ consecutive events are sampled for visualization. Note that the range of $\Delta t$ in the vertical axis is different across panels.
• Figure 5: Empirical results of the burst complexity $C_{\Delta t}$ in Eq. \ref{['eq:C_deltat']} and the memory coefficient for bursts $M_{\Delta t}$ in Eq. \ref{['eq:M_deltat']}. (a--c) The curves of $C_{\Delta t}$ for the entire range of $\Delta t$ in each of NSR, CHF, and AF groups. (d) Box plots of the timescale interval, defined as $\Delta\equiv \Delta t_2 - \Delta t_1$ in Eq. \ref{['eq:Delta']}, where $\Delta t_1$ and $\Delta t_2$ are the timescales satisfying $C_{\Delta t_1}=-0.8$ and $C_{\Delta t_2}=0.5$. We include the p-values for the two-sample Kolmogorov-Smirnov test comparing distributions of $\Delta$. (e--g) The curves of $M_{\Delta t}$ for the entire range of $\Delta t$ in each of NSR, CHF, and AF groups. The horizontal dotted lines at $M_{\Delta t}=0$ are to guide the eyes. (h) Box plots of the peak timescale, defined as $\Delta t_{\rm peak} \equiv {\arg\max}_{\tau_{\rm min}\leq \Delta t\leq \Delta t_{\rm upper}} M_{\Delta t}$ in Eq. \ref{['eq:Deltat_peak']} with $\Delta t_{\rm upper}=700$ ms. We include the p-values for the two-sample Kolmogorov-Smirnov test comparing distributions of $\Delta t_{\rm peak}$.