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Register a new userA fluctuation theorem for time-series of signal-response models with the backward transfer entropy
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The irreversibility of trajectories in stochastic dynamical systems is linked to the structure of their causal representation in terms of Bayesian networks. We consider stochastic maps resulting from a time discretization with interval tau of signal-response models, and we find an integral fluctuation theorem that sets the backward transfer entropy as a lower bound to the conditional entropy production. We apply this to a linear signal-response model providing analytical solutions, and to a nonlinear model of receptor-ligand systems. We show that the observational time tau has to be fine-tuned for an efficient detection of the irreversibility in time-series.
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We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series to those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima (WTMM) method, and show that the results are equivalent.
The performance of the multifractal detrended analysis on short time series is evaluated for synthetic samples of several mono- and multifractal models. The reconstruction of the generalized Hurst exponents is used to determine the range of applicability of the method and the precision of its results as a function of the decreasing length of the series. As an application the series of the daily exchange rate between the U.S. dollar and the euro is studied.
Resume: Le principal objet de cette communication est de faire une retro perspective succincte de lutilisation de lentropie et du principe du maximum dentropie dans le domaine du traitement du signal. Apr`es un bref rappel de quelques definitions et du principe du maximum dentropie, nous verrons successivement comment lentropie est utilisee en separation de sources, en modelisation de signaux, en analyse spectrale et pour la resolution des probl`emes inverses lineaires. Mots cles : Entropie, Entropie croisee, Distance de Kullback, Information mutuelle, Estimation spectrale, Probl`emes inverses Abstract: The main object of this work is to give a brief overview of the different ways the entropy has been used in signal and image processing. After a short introduction of different quantities related to the entropy and the maximum entropy principle, we will study their use in different fields of signal processing such as: source separation, model order selection, spectral estimation and, finally, general linear inverse problems. Keywords : Entropy, Relative entropy, Kullback distance, Mutual information, Spectral estimation, Inverse problems.
Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, $alpha$, that determines the strength of the correlation of the noise. To predict $alpha$ the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the $alpha$ value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same $alpha$ parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.
Exponential Random Graph Models (ERGMs) have gained increasing popularity over the years. Rooted into statistical physics, the ERGMs framework has been successfully employed for reconstructing networks, detecting statistically significant patterns in graphs, counting networked configurations with given properties. From a technical point of view, the ERGMs workflow is defined by two subsequent optimization steps: the first one concerns the maximization of Shannon entropy and leads to identify the functional form of the ensemble probability distribution that is maximally non-committal with respect to the missing information; the second one concerns the maximization of the likelihood function induced by this probability distribution and leads to its numerical determination. This second step translates into the resolution of a system of $O(N)$ non-linear, coupled equations (with $N$ being the total number of nodes of the network under analysis), a problem that is affected by three main issues, i.e. accuracy, speed and scalability. The present paper aims at addressing these problems by comparing the performance of three algorithms (i.e. Newtons method, a quasi-Newton method and a recently-proposed fixed-point recipe) in solving several ERGMs, defined by binary and weighted constraints in both a directed and an undirected fashion. While Newtons method performs best for relatively little networks, the fixed-point recipe is to be preferred when large configurations are considered, as it ensures convergence to the solution within seconds for networks with hundreds of thousands of nodes (e.g. the Internet, Bitcoin). We attach to the paper a Python code implementing the three aforementioned algorithms on all the ERGMs considered in the present work.
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