Smoothness, degrees of freedom and Liapunov exponents of a time series

We propose a set of tests addressing the issue of determining whether the generating law of a time series is a stochastic process or a chaotic dynamics. In the latter case, we test the smoothness and find the number of degrees of freedom of the underlying dynamics. We propose an adaptation of Eckmann and Ruelle algorithm for the computation of the Liapunov exponents of a time series. This algorithm computes efficiently the whole Liapunov spectrum of the observed dynamics, avoiding the problem of the spurious exponents

Reduction of noise of large amplitude through adaptive neighbourhoods

We propose a noise reduction algorithm based on adaptive neighbourhood selection able to obtain high levels of noise reduction for chaotic vector time series corrupted by observational noises with a noise to signal ratio of up to 300%

Error covariance matrix estimation of noisy and dynamically coupled time series.

We estimate the covariance matrix of the errors in several dynamically coupled time series corrupted by measurement errors. We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. The estimation of the covariance matrix of the errors is made using a noise reduction algorithm that efficiently exploits the information contained jointly in the dynamically coupled noisy time series. The method is particularly powerful for short length time series with high uncertainties

A zero-one half law for porosity of measures

We prove that the upper porosity of any Radon probability measure is either 0 or 1/

Attainable values for upper porosities of measures

In this paper we introduce two definitions of upper porosity of a measure which range from 0 to (1/2) and from 0 to 1 respectively, and prove that actually the first porosity only can take the extreme values 0 or (1/2), and the second one takes either the value 0 or the values (1/2) or 1. The other main result of this paper says that any measure μ which does not satisfy the doubling condition μ-a.e. has a maximal porosity

Noise reduction by recycling dynamically coupled time series

We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. We show that much of the information lost due to measurement noise in a target time series can be recovered with a noise reduction algorithm by crossing the time series with another time series with which it is dynamically coupled. The method is particularly useful for reduction of measurement noise in short length time series with high uncertainties

Geometric noise reduction for multivariate time series

We propose an algorithm for the reduction of observational noise in chaotic multivariate time series. The algorithm is based on a maximum likelihood criterion, and its goal is to reduce the mean distance of the points of the cleaned time series to the attractor. We give evidence of the convergence of the empirical measure associated with the cleaned time series to the underlying invariant measure, implying the possibility to predict the long run behavior of the true dynamics

Convergence of the Eckmann and Ruelle algorithm for the estimation of Liapunov exponents

We analyze the convergence conditions of the Eckmann and Ruelle algorithm (E.R.A. for the sequel) used to estimate the Liapunov exponents, for the tangent map, of an ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence which are related to those ensuring the convergence to the tangent map of the best linear L^{p}-fittings of the action of a mapping f on small balls. Under such conditions, we show how to use E.R.A. to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of E.R.A. for the computation of Liapunov exponents in smooth manifolds which allows us to avoid the problem of detecting the spurious exponents. We prove, for a Borel measurable dynamics f, the existence of Liapunov exponents for the function Sr(x), mapping each point x to the matrix of the best linear Lp-fitting of the action of f on the closed ball of radius r centered at x, and we show how to use E.R.A. to get reliable estimates of the Liapunov exponents of Sr. We also propose a test for checking the differentiability of an empirically observed dynamics

Degrees of freedom of a time series

We give a formal proof that if f is a smooth dynamics on a d-dimensional smooth manifold and μ is an ergodic and exact dimensional measure with Hausdorff dimension dimμ>d-1, then the number d of degrees of freedom of the dynamics can be recovered from the observation of an orbit. We implement, with this purpose, an algorithm based on the analysis of the microstructure of μ. We show how a correct estimation of d permits the computation of the Liapunov spectrum with a high accuracy avoiding the issue of the spurious exponents

La investigación desde el proyecto arquitectónico

En este artículo se presenta el beneficio que le puede aportar la investigación al Diseño Arquitectónico y se brinda un método claro que permita formar investigadores a través del desarrollo de un Proyecto Arquitectónico. Se muestra el resultado de una investigación que, después de tener claridad frente al significado del Proyecto Arquitectónico y el Proyecto de Investigación y luego de un análisis de la norma que cobija el ejercicio y la realización de un Estado del Arte (en donde se estudió cómo se aborda la investigación en el diseño arquitectónico), propuso una metodología que aportara a la formación de investigadores a través del proyecto arquitectónico