Source-linked AI summary
Network structure of multivariate time series
Lucas Lacasa, Vincenzo Nicosia, Vito Latora
TL;DR
The paper tackles the need for scalable analysis of high-dimensional multivariate time series. It maps each component into a multiplex visibility graph and uses network structure to quantify shared dynamics, distinguishing dynamical phases and financial crises from stable periods.
Problem
High-dimensional multivariate time series require analysis methods beyond approaches primarily developed for univariate signals.
Method
The method maps each time-series component into a layer of a multiplex visibility graph and analyzes inter-layer degree correlations through mutual information.
Results
Multiplex network descriptors capture dynamical phases and synchronization in coupled chaotic maps and distinguish financial crises from periods of stability.
Takeaways & Limitations
The framework provides an alternative for analyzing large, heterogeneous, and non-stationary multivariate time series without ad hoc phase-space partitioning.
Abstract
from arXiv · showhide
Our understanding of a variety of phenomena in physics, biology and economics crucially depends on the analysis of multivariate time series. While a wide range of tools and techniques for time series analysis already exist, the increasing availability of massive data structures calls for new approaches for multidimensional signal processing. We present here a non-parametric method to analyse multivariate time series, based on the mapping of a multidimensional time series into a multilayer network, which allows to extract information on a high dimensional dynamical system through the analysis of the structure of the associated multiplex network. The method is simple to implement, general, scalable, does not require ad hoc phase space partitioning, and is thus suitable for the analysis of large, heterogeneous and non-stationary time series. We show that simple structural descriptors of the associated multiplex networks allow to extract and quantify nontrivial properties of coupled chaotic maps, including the transition between different dynamical phases and the onset of various types of synchronization. As a concrete example we then study financial time series, showing that a multiplex network analysis can efficiently discriminate crises from periods of financial stability, where standard methods based on time-series symbolization often fail.
I. INTRODUCTION
The paper addresses the challenge of analyzing high-dimensional multivariate time series by mapping them into multilayer visibility networks. It proposes this framework for extracting dynamical information and applying it to chaotic systems and financial instability.
- I. INTRODUCTION: The multiplex visibility graph maps each component of a multidimensional signal into a separate network layer.The layers are formed using the Horizontal Visibility Graph algorithm.
- I. INTRODUCTION: Multivariate time series describe systems with many degrees of freedom, but existing visibility-network approaches are primarily framed for univariate series.This motivates a network-based method specifically designed for multidimensional signals.
- I. INTRODUCTION: The framework is intended to extract graph-theoretical information and construct feature vectors for automatic classifiers from multivariate signals.The approach is described as simple, accurate, and computationally efficient.
- I. INTRODUCTION: A projection of the multilayer network yields a single-layer network analogous to a functional network, with the paper stating that it can outperform standard construction methods.The supplied figure caption also identifies this projection as a way to represent information flow among units.
- I. INTRODUCTION: The method is validated on high-dimensional spatio-temporal chaos and applied to empirical multivariate financial series to characterize financial instability.The examples cover coupled chaotic maps and periods of financial instability.
II. RESULTS
The method constructs horizontal visibility graphs for each time-series component and combines them into a multiplex network. Inter-layer degree correlations are summarized through mutual information to quantify shared structure and information flow.
- II. RESULTS: Visibility algorithms map an ordered real-valued time series into a graph, with Horizontal Visibility linking data points through an ordering criterion.Two nodes are connected when every intermediate datum is smaller than the lower endpoint value.
- II. RESULTS: Each component of an M-dimensional time series becomes one layer of an M-layer multiplex visibility graph.The construction applies to empirically measured or deterministic and stochastic dynamical systems.
- II. RESULTS: The multiplex graph is represented by layer-specific adjacency matrices, whose entries indicate whether two time points are connected at each layer.The paper focuses on undirected horizontal visibility graphs, while other linking criteria can also be used.
- II. RESULTS: The inter-layer mutual information Iα,β measures correlations between the degree distributions of two layers.Its joint distribution P(k[α], k[β]) records nodes having specified degrees at layers α and β.
- II. RESULTS: Averaging Iα,β across layer pairs produces I, a scalar measure of typical information flow in the system.The values Iα,β can also weight the edges of a complete graph whose nodes represent layers.
A. Information flow and phase diagram in Coupled Map Lattices.
Diffusively coupled map lattices exhibit multiple dynamical phases as coupling changes, and multiplex-network measures capture transitions and synchronization regimes.
- A. Information flow and phase diagram in Coupled Map Lattices.: Diffusively coupled map lattices balance chaotic local dynamics against spatial homogenization from coupling.The model uses a ring of coupled sites, with coupling strength ϵ controlling the interaction.
- A. Information flow and phase diagram in Coupled Map Lattices.: The phase diagram includes Fully Developed Turbulence, Pattern Selection, and spatio-temporal intermittency, with distinct synchronization properties.For M = 5 logistic maps, increasing ϵ produces transitions from high-dimensional chaos through pattern selection to partially synchronized chaotic states.
- A. Information flow and phase diagram in Coupled Map Lattices.: FDT produces a chain-like backbone, PS produces noninteracting communities, and STI produces slightly overlapping dense communities.These structures correspond to qualitatively different topological values across phases.
- A. Information flow and phase diagram in Coupled Map Lattices.: Multiplex-network quantities detect the onset of complete synchronization in the corresponding coupled-map regime.The method also captures transitions among partially synchronized dynamical states.
B. Scaling up the system.
Scaling the analysis to 200 coupled logistic maps reveals additional phases and preserves qualitatively distinct network structures for different dynamics.
- B. Scaling up the system.: 200 diffusively coupled logistic maps exhibit short phases such as Brownian motion of Defects between FDT and PS.Increasing system dimension makes the phase description more cumbersome and motivates projections and coarse-grained variables.
- B. Scaling up the system.: The backbone graph of layers is unique for each dynamical phase and qualitatively different across phases.Edges are added in decreasing inter-layer mutual information until the backbone becomes connected.
- B. Scaling up the system.: The inter-layer information quantities capture relevant dynamics in high-dimensional systems.Their use supports a structural description of the enlarged coupled-map system.
C. Multiplex analysis of financial instabilities.
Multiplex visibility analysis of 35 major US-stock price series identifies major financial-instability periods and distinguishes them from a stable interval.
- C. Multiplex analysis of financial instabilities.: The study analyzes 35 major NYSE and NASDAQ assets sampled once per minute from 1998 through 2013.Each company contributes approximately O(2 · 10^6) observations.
- C. Multiplex analysis of financial instabilities.: Multiplex mutual-information peaks coincide with the .com bubble in 1998–1999 and the mortgage-subprime crisis in 2007–2012.The interval 2001–2007 is described as comparatively stable and seemingly unsynchronized.
- C. Multiplex analysis of financial instabilities.: Standard symbolization of the same financial series does not single out crises as effectively as multiplex mutual information.The symbolized signal is described as more erratic and less informative.
- C. Multiplex analysis of financial instabilities.: During financial instability, layer-network spanning trees contain a large hub, whereas stable-period trees distribute degree more evenly.The crisis hub can connect to as much as 50% of the other nodes.
III. DISCUSSION
The discussion presents multiplex visibility graphs as a flexible alternative for multivariate time-series analysis, while limiting the paper's analysis to information flow among variables.
- III. DISCUSSION: Multiplex structural measures capture dynamical phases and changes in mutual information among layers in high-dimensional coupled chaotic maps.The method avoids problems associated with standard symbolization procedures in the reported analyses.
- III. DISCUSSION: For financial series, multiplex measures distinguish stability from crises and can support decision making.The paper describes the approach as applicable when the underlying dynamics are poorly understood or unknown.
- III. DISCUSSION: The analysis focuses on information flow among variables rather than the full range of possible multiplex-visibility-graph properties.This scope boundary follows from the paper's stated focus on one particular aspect of the approach.