Quantifying chaos from time-series data through Lyapunov exponents: a computational data science approach

La aparición de la teoría del caos ha supuesto un cambio de paradigma en la ciencia. Los sistemas caóticos son sistemas dinámicos deterministas no lineales que pueden comportarse como un movimiento aparentemente errático e irregular. Tengan en cuenta que sí pudiésemos caracterizar un sistema caótico en algún sentido nos permitiría evidenciar que detrás de los mismos existe un sistema generador determinista a pesar de mostrar un comportamiento aparentemente aleatorio. Este hecho nos permitiría aprovechar ese carácter determinista para poder hacer predicciones y controlar las variables de estos sistemas dinámicos deterministas (caóticos). Los métodos y técnicas que persiguen contrastar la hipótesis del caos tratan de estimar los llamados exponentes de Lyapunov como una forma de caracterizar un sistema caótico. Hoy en día, cuantificar el caos a partir de datos de series temporales mediante este tipo de medidas cuantitativas de manera rigurosa está lejos de ser un ejercicio trivial y plantea una serie de retos teóricos y prácticos..

DChaos: an R package for chaotic time series analysis

Chaos theory has been hailed as a revolution of thoughts and attracting ever-increasing attention of many scientists from diverse disciplines. Chaotic systems are non-linear deterministic dynamic systems which can behave like an erratic and apparently random motion. A relevant field inside chaos theory is the detection of chaotic behavior from empirical time-series data. One of the main features of chaos is the well-known initial-value sensitivity property. Methods and techniques related to testing the hypothesis of chaos try to quantify the initial-value sensitive property estimating the so-called Lyapunov exponents. This paper describes the main estimation methods of the Lyapunov exponent from time series data. At the same time, we present the DChaos library. R users may compute the delayed-coordinate embedding vector from time series data, estimates the best-fitted neural net model from the delayed-coordinate embedding vectors, calculates analytically the partial derivatives from the chosen neural nets model. They can also obtain the neural net estimator of the Lyapunov exponent from the partial derivatives computed previously by two different procedures and four ways of subsampling by blocks. To sum up, the DChaos package allows the R users to test robustly the hypothesis of chaos in order to know if the data-generating process behind time series behaves chaotically or not. The package’s functionality is illustrated by examples

R-adaptación de la asignatura de métodos econométricos en economía y finanzas del Grado de Estadística Aplicada

El objetivo fundamental de este proyecto es el de adaptar todo el contenido práctico de la asignatura de Métodos Econométricos de Economía y Finanzas de cuarto curso del Grado de Estadística Aplicada de la Facultad de Estudios Estadísticos al software libre R (dejando de usar el software comercial Eviews). Para ello se han preparado una serie de materiales y recursos didácticos que se han puesto a disposición de los alumnos en el Campus Virtual. También hemos integrado en el curso el uso de la plataforma de autoaprendizaje de R "DataCamp for classroom", y se ha preparado diverso material específico de autoevaluación en el aula mediante las plataforma “Kahood”

Chaotic signals inside some tick-by-tick financial time series

It has been more than four decades since ideas from chaos began appearing in the literature showing that it is possible to design economic models in regime of chaotic behaviour from a theoretical point of view. However there is no clear evidence that economic time series behave chaotically. So far researchers have found substantial evidence for nonlinearity but relatively weak evidence for chaos. In this paper we propose a possible explanation to this ”chaos model-data paradox”. Our main motivation is that chaos is elusive in financial datasets because of loss of information that occurs when daily quotes are used. This could hinder the detection of chaos in those time series. Chaotic systems are sensitive to initial conditions, so temporal dependence is lost as the chaotic time series are sampled at too long-time intervals, appearing as independent even though they come from a (chaotic) dynamical system. In the case of financial time series, which quotes are continuously traded on markets, the daily sampling may be too long. In order to avoid this problem high-frequency data can be used to detect chaos in financial time series. We have found evidence of chaotic signals inside the 14 tick-by-tick time series considered about some top currency pairs from the Foreign Exchange Market (FOREX). Notice that we do not intend to generalize this finding to all financial series or even to all FOREX series. The main interest of our paper is to illustrate that by choosing a tick-by-tick frequency (instead of a daily one), and with the purpose of preserving the dynamic dependence on the time series, we could find chaos. At least in the 14 specific currency pairs analyzed and during the time intervals considered. Hence we propose take into account all the information available in the financial markets (full sample information on FX rates) instead of daily data. This kind of time series entails several difficulties due to the need to process a huge quantity of information and regarding the reconstruction of the attractor from tick-by-tick time series which are unevenly-spaced. In this sense we have had to implemented new algorithms in order to solve such drawbacks. As far as we know these tick-by-tick financial time series have never been tested for chaos so farGoverment of SpainFaculty os Statistical Studies UCMData Analysis in Social and Gender Studies and Equality Policies Research GroupDepto. de Economía Aplicada, Pública y PolíticaFac. de Estudios EstadísticosTRUEpu