An econophysics approach to analyse uncertainty in financial markets: an application to the Portuguese stock market

In recent years there has been a closer interrelationship between several scientific areas trying to obtain a more realistic and rich explanation of the natural and social phenomena. Among these it should be emphasized the increasing interrelationship between physics and financial theory. In this field the analysis of uncertainty, which is crucial in financial analysis, can be made using measures of physics statistics and information theory, namely the Shannon entropy. One advantage of this approach is that the entropy is a more general measure than the variance, since it accounts for higher order moments of a probability distribution function. An empirical application was made using data collected from the Portuguese Stock Market.Comment: 8 pages, 2 figures, presented in the conference Next Sigma-Phi 200

Estimates for the large time behavior of the Landau equation in the Coulomb case

This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation $D(f)$ by a weighted relative Fisher information of $f$ with respect to the associated Maxwellian distribution, which leads to a variant of Cercignani's conjecture thanks to a logarithmic Sobolev inequality. Secondly, we prove the propagation of polynomial and stretched exponential moments with an at most linearly growing in time rate. As an application of these estimates, we show the convergence of any ($H$- or weak) solution to the Landau equation with Coulomb potential to the associated Maxwellian equilibrium with an explicitly computable rate, assuming initial data with finite mass, energy, entropy and some higher $L^1$-moment. More precisely, if the initial data have some (large enough) polynomial $L^1$-moment, then we obtain an algebraic decay. If the initial data have a stretched exponential $L^1$-moment, then we recover a stretched exponential decay

Efficient Coding of Local 2D Shape

Efficient coding provides a concise account of key early visual properties, but can it explain higher-level visual function such as shape perception? If curvature is a key primitive of local shape representation, efficient shape coding predicts that sensitivity of visual neurons should be determined by naturally-occurring curvature statistics, which follow a scale-invariant power-law distribution. To assess visual sensitivity to these power-law statistics, we developed a novel family of synthetic maximum-entropy shape stimuli that progressively match the local curvature statistics of natural shapes, but lack global structure. We find that humans can reliably identify natural shapes based on 4th and higher-order moments of the curvature distribution, demonstrating fine sensitivity to these naturally-occurring statistics. What is the physiological basis for this sensitivity? Many V4 neurons are selective for curvature and analysis of population response suggests that neural population sensitivity is optimized to maximize information rate for natural shapes. Further, we find that average neural response in the foveal confluence of early visual cortex increases as object curvature converges to the naturally-occurring distribution, reflecting an increased upper bound on information rate. Reducing the variance of the curvature distribution of synthetic shapes to match the variance of the naturally-occurring distribution impairs the linear decoding of individual shapes, presumably due to the reduction in stimulus entropy. However, matching higher-order moments improves decoding performance, despite further reducing stimulus entropy. Collectively, these results suggest that efficient coding can account for many aspects of curvature perception

Entanglement, quantum randomness, and complexity beyond scrambling

Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that "hides" initially localized quantum information. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular R\'enyi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the R\'enyi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests R\'enyi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the R\'enyi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published versio

Data-Driven Statistical Reduced-Order Modeling and Quantification of Polycrystal Mechanics Leading to Porosity-Based Ductile Damage

Predicting the process of porosity-based ductile damage in polycrystalline metallic materials is an essential practical topic. Ductile damage and its precursors are represented by extreme values in stress and material state quantities, the spatial PDF of which are highly non-Gaussian with strong fat tails. Traditional deterministic forecasts using physical models often fail to capture the statistics of structural evolution during material deformation. This study proposes a data-driven statistical reduced-order modeling framework to provide a probabilistic forecast of the deformation process leading to porosity-based ductile damage, with uncertainty quantification. The framework starts with computing the time evolution of the leading moments of specific state variables from full-field polycrystal simulations. Then a sparse model identification algorithm based on causation entropy, including essential physical constraints, is used to discover the governing equations of these moments. An approximate solution of the time evolution of the PDF is obtained from the predicted moments exploiting the maximum entropy principle. Numerical experiments based on polycrystal realizations show that the model can characterize the time evolution of the non-Gaussian PDF of the von Mises stress and quantify the probability of extreme events. The learning process also reveals that the mean stress interacts with higher-order moments and extreme events in a strongly nonlinear and multiplicative fashion. In addition, the calibrated moment equations provide a reasonably accurate forecast when applied to the realizations outside the training data set, indicating the robustness of the model and the skill for extrapolation. Finally, an information-based measurement shows that the leading four moments are sufficient to characterize the crucial non-Gaussian features throughout the entire deformation history

Information geometric duality of ϕ\phi-deformed exponential families

In the world of generalized entropies---which, for example, play a role in physical systems with sub- and super-exponential phasespace growth per degree of freedom---there are two ways for implementing constraints in the maximum entropy principle: linear- and escort constraints. Both appear naturally in different contexts. Linear constraints appear e.g. in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of $\phi$-deformed exponential families that correspond to linear constraints (as studied by J. Naudts), with those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of $(c,d)$-entropy that covers all situations that are compatible with the first three Shannon-Khinchin axioms, and that include Shannon, Tsallis, Anteneodo-Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality, and mention that the escort distributions arising in the two dualities are generally different and only coincide for the case of the Tsallis deformation.Comment: 9 pages, 2 figure