%0 Conference Proceedings
%4 sid.inpe.br/mtc-m21b/2016/06.20.17.48
%2 sid.inpe.br/mtc-m21b/2016/06.20.17.48.04
%T Hurst exponent estimation of self-affine time series through a complex network approach
%D 2016
%A Campanharo, Adriana Susana Lopes de Oliveira,
%A Ramos, Fernando Manuel,
%@affiliation Universidade Estadual Paulista (UNESP)
%@affiliation Instituto Nacional de Pesquisas Espaciais (INPE)
%@electronicmailaddress andriana@ibb.unesp.br
%@electronicmailaddress fernando.ramos@inpe.br
%B International Conference on Nonlinear Science and Complexity, 6
%C São José dos Campos, SP
%8 16-20 May
%K nonlinear dynamics and complex systems, time series analysis, complex networks, Hurst exponent, quantile graphs.
%X 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].
%@language en