On a quantitative method to analyze dynamical and measurement noise

Abstract

This letter reports on a new method of analysing experimentally gained time series with respect to different types of noise involved, namely, we show that it is possible to differentiate between dynamical and measurement noise. This method does not depend on previous knowledge of model equations. For the complicated case of a chaotic dynamics spoiled at the same time by dynamical and measurement noise, we even show how to extract from data the magnitude of both types of noise. As a further result, we present a new criterion to verify the correct embedding for chaotic dynamics with dynamical noise.

A Langevin equation for the rates of currency exchange based on the Markov analysis

“…Thus, the probability densities of the fluctuations in the returns r for the rate of exchange of various currencies versus the U.S. dollar satisfy a Fokker-Planck equation, characterized by a drift and a diffusion coefficient, which represent the first two coefficients in the Kramers-Moyal expansion. We computed accurate approximants for the coefficients for the stochastic time series r by using the polynomial ansatz [10][11][12][13][14][15]. We then used a novel method to utilize the returns' data, which contain a degree of stochasticity, and constructed a simple equation that governs the time series.…”

“…As is well-known, a given process with a degree of randomness or stochasticity may have a finite or an infinite Markov time scale [10][11][12]. The proposed method utilizes a set of data for a phenomenon which contain a degree of stochasticity.…”

“…The proposed method utilizes a set of data for a phenomenon which contain a degree of stochasticity. We begin by describing the procedure that leads to the development of an effective Langevin equation based on the (stochastic) data set [11,12]. As the first step, we check whether the returns for the data, as defined above, follow a Markov chain and, if so, measure the Markov time scale t M .…”

“…for given r 1 and r 2 , in terms of, for example, t − t 1 and considering the possible errors in estimating S. Then, t M = t − t 1 for that value of t − t 1 for which either S vanishes, or reach its minimum [12].…”

Stochastic Modelling of Wind Speed Power Production Correlations

“…Since additional measurement noise causes an offset (see [4]), we estimate the coefficients by calculating the slope…”