212 research outputs found
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 -transform of a signal
as a resolvent of a tridiagonal matrix is given. Several formulas for the
poles, zeros and residues of the Pad\'e approximation in terms of the matrix
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 -frequency band and a
couple of -oscillons that correspond, respectively, to slow and fast
-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.
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