Multilayer Horizontal Visibility Graphs for Multivariate Time Series Analysis

TL;DR

This work introduces Multilayer Horizontal Visibility Graphs (MHVG) as a parameter-free, topological mapping of multivariate time series into multilayer networks, leveraging Cross-Horizontal Visibility to capture cross-variable and lagged dependencies. It defines inter-layer edges, a comprehensive set of MHVG topological features (including a novel Ratio Degree), and provides a theoretical and empirical evaluation on synthetic and real datasets. The results show that inter-layer and all-layer features enhance clustering and classification tasks, offering interpretable descriptors that complement traditional univariate analyses. The approach is general, scalable with potential parallelization, and paves the way for more robust multivariate time series mining with topology-based features.

Abstract

Multivariate time series analysis is a vital but challenging task, with multidisciplinary applicability, tackling the characterization of multiple interconnected variables over time and their dependencies. Traditional methodologies often adapt univariate approaches or rely on assumptions specific to certain domains or problems, presenting limitations. A recent promising alternative is to map multivariate time series into high-level network structures such as multiplex networks, with past work relying on connecting successive time series components with interconnections between contemporary timestamps. In this work, we first define a novel cross-horizontal visibility mapping between lagged timestamps of different time series and then introduce the concept of multilayer horizontal visibility graphs. This allows describing cross-dimension dependencies via inter-layer edges, leveraging the entire structure of multilayer networks. To this end, a novel parameter-free topological measure is proposed and common measures are extended for the multilayer setting. Our approach is general and applicable to any kind of multivariate time series data. We provide an extensive experimental evaluation with both synthetic and real-world datasets. We first explore the proposed methodology and the data properties highlighted by each measure, showing that inter-layer edges based on cross-horizontal visibility preserve more information than previous mappings, while also complementing the information captured by commonly used intra-layer edges. We then illustrate the applicability and validity of our approach in multivariate time series mining tasks, showcasing its potential for enhanced data analysis and insights.

Figures (16)

• Figure 1: Schematic diagram of the network-based features approach to time series mining tasks. The first column displays a set of synthetic multivariate time series, the second column the corresponding multiplex horizontal visibility graphs (based on lacasa2015network) and the multilayer horizontal visibility graphs proposed in this work, the third column shows the set of topological features extracted from the two types of multiple-layer networks, and the last column a principal component analysis derived from the extracted topological feature sets. The description of the notation corresponding to the topological features presented in the third column is given in Section \ref{['sec4']}.
• Figure 2: Illustrative example of two toy multilayer networks with five entities, $V = \{1,2,3,4,5\}$, and two layers $\boldsymbol{L} = \{L_1, L_2\}$. (a) represents a toy multilayer network and (b) a toy multiplex network. Solid lines represent the intra-layer edges and dashed lines represent the inter-layer edges. Source: Modified from vanessa2020.
• Figure 3: Illustrative example featuring two supra-adjacency matrices: (a) depicting a supra-adjacency matrix of a toy multilayer network, and (b) showcasing a supra-adjacency matrix of a toy multiplex network. Colored blocks represent intra-layer graphs, while gray blocks represent inter-layer graphs.
• Figure 4: Illustrative example of the Horizontal Visibility Graph (HVG) algorithm. (a) represents a toy time series and corresponding visibility between data bars (observations), where solid blue lines represent the horizontal visibility lines according to the HVG method. (b) represents a network generated by the corresponding mapping. Source: Adapted from vanessa2020.
• Figure 5: Illustrative example of the Multiplex Horizontal Visibility Graph (HVG) algorithm: (a) represents a toy multivariate time series with three components, $\boldsymbol{Y} = \{\boldsymbol{Y}^{1}, \boldsymbol{Y}^{2}, \boldsymbol{Y}^{3}\}$, and (b) the resulting multiplex network with three layers generated by the Multiplex HVG mapping. Source: Adapted from vanessa2020.