visibility networks for time series

Visibility Complex Networks for Chaotic TimeSeries

Georgi D. Gospodinov

Rachel L. Maitra

Applied Mathematics DepartmentWentworth Institute of Technology

June 11, 2014

Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

Introduction: Visibility Complex Networks

Temporal sequences of measurements or observations (timeseries) are the basic elements for investigating naturalphenomena

Time series analysis aims at understanding the dynamics ofstochastic or chaotic processes

Methods have been proposed to transform a single time seriesto a complex network so that the dynamics of the process canbe understood by investigating the topological properties ofthe network

The recently developed method of Visibility Graphs transformsa time series into a complex network which inherits severalproperties of the time series in its structure

Basic Definitions

Definition

The visibility criterion for mapping a time series into a network isdefined as follows. Two arbitrary data (ta, ya) and (tb, yb) in thetime series are visible if any other data (tc , yc) such thatta < tb < tc fulfills

yc < ya + (yb − ya)tc − tatb − ta

Visibility algorithm for a time series.

Basic Definitions

Definition

The visibility criterion for mapping a time series into a network isdefined as follows. Two arbitrary data (ta, ya) and (tb, yb) in thetime series are visible if any other data (tc , yc) such thatta < tb < tc fulfills

yc < ya + (yb − ya)tc − tatb − ta

Visibility algorithm for a time series.Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

Basic Invariant Properties

a) Original time serieswith visibility links

b) Translation of thedata

c) Vertical rescaling

d) Horizontal rescaling

e) Addition of a lineartrend to the data

Note: The visibilitygraph remainsinvariant in all of theabove cases

Basic Properties

the associated visibility graph is connected

periodic, random, and fractal time series map into motif-like,exponential, and scale-free visibility graphs, respectively

the visibility algorithm has been used to estimate the Hurstexponent in fractional Brownian series via the linearrelationship between the Hurst exponent and the the exponentof the power law degree distribution in the scale-freeassociated visibility graph

the visibility algorithm has been applied to analyze time seriesin different contexts, from dynamics, atmospheric sciences, tofinance

the visibility algorithm decomposes time series in aconcatenation of graph motifs, and in this sense acts as ageometric transform

Basic Properties

Degree Distribution of Scale-Free HVG

5 10 15 20 25 k

ln@PHkLDLinear Fit to ln@PHkLD for x-Henon Time Seriesof Length Ranging from 215 to 223 Points

223 Points, l=0.333222 Points, l=0.323221 Points, l=0.320220 Points, l=0.304219 Points, l=0.305218 Points, l=0.313217 Points, l=0.285216 Points, l=0.280215 Points, l=0.301

The Horizontal Visibility Algorithm

the general visibility algorithm was introduced above

the horizontal visibility algorithm is a special case of thegeneral visibility algorithm: ya and yb are visible if ya, yb > ycfor all c such that a < c < b

Random Time Series: HVG Degree Distribution

(left) First 250 values of R(t), where R is a random series of107 data values extracted from U[0, 1]

(right) Degree distribution P(k) of the visibility graphassociated with R(t) (plotted in semilog)

The tail is clearly exponential, a behavior due to data withlarge values (rare events), which are the hubs.

Theorem

Consider a time series that consists of a periodic orbit of period T .The mean degree of an horizontal visibility graph associated to aninfinite periodic series of period T (with no repeated values withina period) is

k ≡ #edges

#nodes= 4

(1− 1

Theorem

Given a sequence {xi} generated by a continuous probability densityf (x), the degree distribution of the associated HVG is

P(k) =1

)k−2

, k ≥ 2

Theorem

Consider a time series that consists of a periodic orbit of period T .The mean degree of an horizontal visibility graph associated to aninfinite periodic series of period T (with no repeated values withina period) is

k ≡ #edges

#nodes= 4

(1− 1

Theorem

Given a sequence {xi} generated by a continuous probability densityf (x), the degree distribution of the associated HVG is

P(k) =1

)k−2

, k ≥ 2

Random Time Series: HVG Properties

Adjacency matrix ofthe HVG of 103

random series data

The adjacency matrixis predominantly filledaround the maindiagonal

A sparse structure,reminiscent of theSmall-World model

Mean path lengthscales logarithmically,implying the HVG to arandom time series isSmall-World

random series data

HVG Degree Distribution of Chaotic Time Series

(solid line) random series

(squares) time series of 106 points extracted from the Logistic mapxn+1 = µxn(1− xn) in the chaotic region µ = 4

(black triangles) {xn} time series from the Henon map(xn+1, yn+1) = (yn + 1− ax2

n , bxn) with (a = 1.4, b = 0.3)

HVGs: Conjectured Distinction of Chaotic vs. Stochastic

HVGs: Counterexamples (Part I)

HVGs: Counterexamples (Part II)

HVGs: Counterexamples (Part III)

Figure : λ values for the HVG degree distribution of chaotic time series(over 300 chaotic systems plotted, with a degree-11 polynomial fit). Theshaded region shows the range of inflection point values depending onthe different linear fit, and the dotted line shows the λ value foruncorrelated random time series.

Introduction: Shannon-Fisher Information Plane

The Shannon-Fisher information plane (SF) is a planarrepresentation in which the horizontal and vertical axes arefunctionals of the PDF: the Shannon Entropy and the FisherInformation Measure, respectively.

A way to represent in the same information plane global andlocal aspects of the PDFs associated to the studied system

The proposed PDFs here are obtained through the horizontalvisibility graph methodology

Given a continuous probability distribution function (PDF), itsShannon entropy is a measure of “global” character that it isnot too sensitive to strong changes in the distribution takingplace on small regions of the PDF’s support

Fisher’s Information Measure constitutes a measure of thegradient content of the PDF, thus being quite sensitive evento tiny localized perturbations

Definitions: Shannon Entropy, Fisher Measure, NormalizedShannon Entropy

HVG-PDF Setup

HVG-PDF Examples

Shannon-Fisher Plane with HVG-PDFs: Chaotic vs.Stochastic

Shannon-Fisher Plane with HVG-PDFs: Zoom

Shannon-Fisher Plane: Chaotic vs. Stochastic

Figure : Shannon-Fisher values for the HVG degree distribution ofchaotic and stochastic time series.

Shannon-Fisher Plane with HVG-PDFs: Zoom

Shannon-Lambda Plane: Chaotic vs. Stochastic

Figure : Shannon − λ values for the HVG degree distribution of chaoticand stochastic time series.

Multi-Dimensional Visibility Graphs

Component Visibility Graphs

Multi-Dimensional Visibility Graphs

Magnitude Visibility Graphs

Magnitude Visibility Graphs Criterion

Shannon-Fisher Plane: Chaotic vs. Stochastic

Current and Future Work

Average degree for multi-dimensional time series

Degree distribution for uncorrelated random multi-dimensionaltime series

Filtration of VGs of multidimensional time series

Multi-dimensional dynamical VGs

Application of dynamical VGs to Shannon-Fisher analysis

Network cluster visibility

Data analysis, Manifold learning, Deep learning of hierarchicaldata through VGs

visibility networks for time series

Science

observations time series

chaotic time series

original time series

single time series

visibility criterion

visibility links georgi

visibility algorithm

gospodinov rachel