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Register a new userCorrespondence between temporal correlations in time series, inverse problems, and the Spherical Model
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In this paper we employ methods from Statistical Mechanics to model temporal correlations in time series. We put forward a methodology based on the Maximum Entropy principle to generate ensembles of time series constrained to preserve part of the temporal structure of an empirical time series of interest. We show that a constraint on the lag-one autocorrelation can be fully handled analytically, and corresponds to the well known Spherical Model of a ferromagnet. We then extend such a model to include constraints on more complex temporal correlations by means of perturbation theory, showing that this leads to substantial improvements in capturing the lag-one autocorrelation in the variance. We apply our approach on synthetic data, and illustrate how it can be used to formulate expectations on the future values of a data generating process.
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Methods connecting dynamical systems and graph theory have attracted increasing interest in the past few years, with applications ranging from a detailed comparison of different kinds of dynamics to the characterisation of empirical data. Here we investigate the effects of the (multi)fractal properties of a time signal, common in sequences arising from chaotic or strange attractors, on the topology of a suitably projected network. Relying on the box counting formalism, we map boxes into the nodes of a network and establish analytic expressions connecting the natural measure of a box with its degree in the graph representation. We single out the conditions yielding to the emergence of a scale-free topology, and validate our findings with extensive numerical simulations.
An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function $T(z)$ gives the spanning tree constant when evaluated at $z=1$, while giving he lattice green function when differentiated. It is known that for the infinite square lattice the partition function $Z(K)$ of the Ising model evaluated at the critical temperature $K=K_c$ is related to $T(1)$. Here we show that this idea in fact generalizes to all real temperatures. We prove that $ ( Z(K) {rm sech~} 2K ~!)^2 = k expbig[ T(k) big] $, where $k= 2 tanh(2K) {rm sech}(2K)$. The identical Mahler measure connects the two seemingly disparate quantities $T(z)$ and $Z(K)$. In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to non-planar lattices.
Recent observation studies have revealed that earthquakes are classified into several different categories. Each category might be characterized by the unique statistical feature in the time series, but the present understanding is still limited due to their nonlinear and nonstationary nature. Here we utilize complex network theory to shed new light on the statistical properties of earthquake time series. We investigate two kinds of time series, which are magnitude and inter-event time (IET), for three different categories of earthquakes: regular earthquakes, earthquake swarms, and tectonic tremors. Following the criterion of visibility graph, earthquake time series are mapped into a complex network by considering each seismic event as a node and determining the links. As opposed to the current common belief, it is found that the magnitude time series are not statistically equivalent to random time series. The IET series exhibit correlations similar to fractional Brownian motion for all the categories of earthquakes. Furthermore, we show that the time series of three different categories of earthquakes can be distinguished by the topology of the associated visibility graph. Analysis on the assortativity coefficient also reveals that the swarms are more intermittent than the tremors.
The problem of the equivalence of the spherical and mean spherical models, which has been thoroughly studied and understood in equilibrium, is considered anew from the dynamical point of view during the time evolution following a quench from above to below the critical temperature. It is found that there exists a crossover time $t^* sim V^{2/d}$ such that for $t < t^*$ the two models are equivalent, while for $t > t^*$ macroscopic discrepancies arise. The relation between the off equilibrium response function and the structure of the equilibrium state, which usually holds for phase ordering systems, is found to hold for the spherical model but not for the mean spherical one. The latter model offers an explicit example of a system which is not stochastically stable.
We study roughness probability distribution functions (PDFs) of the time signal for a critical interface model, which is known to provide a good description of Barkhausen noise in soft ferromagnets. Starting with time ``windows of data collection much larger than the systems internal ``loading time (related to demagnetization effects), we show that the initial Gaussian shape of the PDF evolves into a double-Gaussian structure as window width decreases. We supply a physical explanation for such structure, which is compatible with the observed numerical data. Connections to experiment are suggested.
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