@InProceedings{CampanharoRamo:2016:HuExEs,
author = "Campanharo, Adriana Susana Lopes de Oliveira and Ramos, Fernando
Manuel",
affiliation = "{Universidade Estadual Paulista (UNESP)} and {Instituto Nacional
de Pesquisas Espaciais (INPE)}",
title = "Hurst exponent estimation of self-affine time series through a
complex network approach",
year = "2016",
organization = "International Conference on Nonlinear Science and Complexity, 6.",
keywords = "nonlinear dynamics and complex systems, time series analysis,
complex networks, Hurst exponent, quantile graphs.",
abstract = "Many natural signals present a fractal-like structure and are
characterized by two parameters, \β, the power-spectrum
power-law exponent, and H, the Hurst exponent [1]. For time series
with a self-affine structure, like fractional Gaussian noises
(fGn) and fractional Brownian motions (fBm), the Hurst exponent H
is one of the key parameters. Over time, researchers accumulated a
large number of time series analysis techniques, ranging from
time-frequency methods, such as Fourier and wavelet transforms [2,
3], to nonlinear methods, such as phase-space embeddings, Lyapunov
exponents, correlation dimensions and entropies [4]. These
techniques allow researchers to summarize the characteristics of a
time series into compact metrics, which can then be used to
understand the dynamics or predict how the system will evolve with
time [5]. Obviously, these measures do not preserve all of the
properties of a time series, so there is considerable research
toward developing novel metrics that capture additional
information or quantify time series in new ways [5, 6, 7]. One of
the most interesting advances is mapping a time series into a
network, based on the concept of transition probabilities [5].
This study has demonstrated that distinct features of a time
series can be mapped onto networks (here called quantile graph or
QG) with distinct topological properties. This finding suggests
that network measures can be used to differentiate properties of
fractal-like time series. In spite of the large number of
applications of complex networks methods in the study time series,
usually involving the classification of dynamical systems or the
identification of dynamical transitions [8], establishing a link
between a network measure and H remains an open question [1].
Recently, a linear relationship between the exponent of the power
law degree distribution of visibility graphs and H has been
established for noises and motions [9,10]. Here, we show an
alternative approach for the computation of the Hurst exponent
[1]. This new approach is based on a generalization of the method
introduced in Ref. [5], in which time series quantiles are mapped
into nodes of a graph. In this approach, a quantile graph is
obtained as follows: The values of a given time series is
coarse-grained into Q quantiles q1, q2,,qQ. A map M from a time
series to a network assigns each quantile qi to a node ni in the
corresponding network. Two nodes ni and nj are connected with a
weighted arc ni, nj, wij k whenever two values x(t) and x(t + k)
belong respectively to quantiles qi and qj, with t = 1, 2, . . .
,T and the time differences k = 1, . . . , kmax < T. Weights wij k
are given by the number of times a value in quantile qi at time t
is followed by a point in quantile qj at time t+k, normalized by
the total number of transitions. Repeated transitions through the
same arc increase the value of the corresponding weight. With
proper normalization, the weighted adjacency matrix becomes a
Markov transition matrix. The resulting network is weighted,
directed and connected. Besides, the QG method is numerically
simple and has only one free parameter, Q, the number of
quantiles/nodes [1, 5]. The QG method for estimating the Hurst
exponent was applied to fBm with different H values. Based on the
QG method described above, H was then computed directly as the
power-law scaling exponent of the mean jump length performed by a
random walker on the QG, for different time differences between
the time series data points [1]. Results were compared to the
exact H values used to generate the motions and showed a good
agreement. For a given time series length, estimation error
depends basically on the statistical framework used for
determining the exponent of a power-law model [1]. Therefore, the
QG method permits to quantify features such as long-range
correlations or anticorrelations associated with the signals
underlying dynamics, expanding the traditional tools of time
series analysis in a new and useful way [1,5].",
conference-location = "S{\~a}o Jos{\'e} dos Campos, SP",
conference-year = "16-20 May",
language = "en",
urlaccessdate = "28 abr. 2024"
}