Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase--space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical r\^ole of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non--standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.Comment: 20 pages, 13 figure

Matrix methods for Pad\'e approximation: numerical calculation of poles, zeros and residues

A representation of the Pad\'e approximation of the $Z$-transform of a signal as a resolvent of a tridiagonal matrix $J$ is given. Several formulas for the poles, zeros and residues of the Pad\'e approximation in terms of the matrix $J$ are proposed. Their numerical stability is tested and compared. Methods for computing forward and backward errors are presented

Discrete structure of the brain rhythms

Neuronal activity in the brain generates synchronous oscillations of the Local Field Potential (LFP). The traditional analyses of the LFPs are based on decomposing the signal into simpler components, such as sinusoidal harmonics. However, a common drawback of such methods is that the decomposition primitives are usually presumed from the onset, which may bias our understanding of the signal's structure. Here, we introduce an alternative approach that allows an impartial, high resolution, hands-off decomposition of the brain waves into a small number of discrete, frequency-modulated oscillatory processes, which we call oscillons. In particular, we demonstrate that mouse hippocampal LFP contain a single oscillon that occupies the $\theta$-frequency band and a couple of $\gamma$-oscillons that correspond, respectively, to slow and fast $\gamma$-waves. Since the oscillons were identified empirically, they may represent the actual, physical structure of synchronous oscillations in neuronal ensembles, whereas Fourier-defined "brain waves" are nothing but poorly resolved oscillons.Comment: 17 pages, 9 figure

Parametric and semiparametric estimation of ordered response models with sample selection and individual-specific thresholds

This paper provides a set of new Stata commands for parametric and semiparametric estimation of an extended version of ordered response models that accounts for both sample selection problems and heterogeneity in the thresholds for the latent variable. The standard estimator of ordered response models is therefore generalized along three directions. First, we account for the presence of endogenous selectivity effects that may lead to inconsistent estimates of the model parameters. Second, we control for both observed and unobserved heterogeneity in response scales by allowing the thresholds to depend on a set of covariates and a random individual effect. Finally, we consider two alternative specifications of the model, one parametric and one semiparametric. In the former, the error terms are assumed to follow a multivariate Gaussian distribution and the model parameters are estimated via maximum likelihood. In the latter, the distribution function of the error terms is instead approximated by following Gallant and Nychka (1997), and the model parameters are estimated via pseudo–maximum likelihood. After discussing identification and estimation issues, we present an empirical application using the second wave of the Survey on Health, Ageing and Retirement in Europe (SHARE). Specifically, we estimate an ordered response model for self-reported health on different domains by accounting for both sample selection bias due to survey nonresponse and reporting bias in the self-assessments of health.