Elsevier

Physics Letters A

Volume 373, Issue 39, 21 September 2009, Pages 3507-3512
Physics Letters A

Kernel-based regression of drift and diffusion coefficients of stochastic processes

https://doi.org/10.1016/j.physleta.2009.07.073Get rights and content

Abstract

To improve the estimation of drift and diffusion coefficients of stochastic processes in case of a limited amount of usable data due to e.g. non-stationarity of natural systems we suggest to use kernel-based instead of histogram-based regression. We propose a method for bandwidth selection and compare it to a widely used cross-validation method. Kernel-based regression reveals an enhanced ability to estimate drift and diffusion especially for a small amount of data. This allows one to improve resolvability of changes in complex dynamical systems as evidenced by an exemplary analysis of electroencephalographic data recorded from a human epileptic brain.

Introduction

The complex behavior of natural systems often stems from a large number of degrees of freedom. Considering time series of a suitable observable ξ of such systems the influence of these degrees of freedom can be modeled as random fluctuations by the Itō stochastic integral equation [1], [2]:ξ(t)=ξ(t0)+t0tdtD(1)(ξ(t),t)+t0tdtD(2)(ξ(t),t)dW(t), where D(1)(ξ,t) and D(2)(ξ,t) describe the deterministic and stochastic part of the motion, respectively. dW is a Wiener process with dW=Γdt where Γ is a random variable with zero mean and Γ(t)Γ(t)=2δ(tt), which is commonly denoted as fluctuating force or noise.

Equivalently such systems can be described by the underlying transition probability, whose evolution can be described by the Fokker–Planck equation:tp(x,t|x,t)=[xD(1)(x,t)+x2D(2)(x,t)]p(x,t|x,t), x and x denote values that can be taken by ξ. Historically, the coefficients D(1)(x,t) and D(2)(x,t) of the Fokker–Planck equation are called drift and diffusion, respectively. Their definitionD(1)(x,t)=limτ01τ(ξ(t+τ)ξ(t))|ξ(t)=x,D(2)(x,t)=12limτ01τ(ξ(t+τ)ξ(t))2|ξ(t)=x reveals the possibility to investigate unknown systems by estimating these coefficients from empirical data with non-parametric regression analysis [3], [4], [5], [6]. The limit τ0 has to be extrapolated or τ has to be chosen small enough so that the error arising with neglecting the limit becomes negligible. Using this approach a variety of complex systems have been examined such as turbulent flows [3], rough surfaces [7], [8], financial exchange rates [9], [10], [11], stochastic dynamics of metal cutting [12], traffic flow [13], climate changes [14], [15], cosmic microwave background radiation [16], conformational dynamics of biomolecules [17], cardiac [18], [19], [20] and movement dynamics [21], [22], and the human epileptic brain [23], [24]. These investigations needed several 105 data points to reliably estimate drift and diffusion from time series data. As the estimation requires the ensemble mean to be replaced by the temporal mean, phenomena occurring on smaller time scales could not be resolved. In Ref. [25] an averaging technique has been proposed to estimate drift and diffusion in short and non-stationary data sets. This technique, however, is based on a number of assumptions that may not be generally valid. Most of the aforementioned investigations made use of the so-called regressogram [26], which bases on histograms. Other approaches include state space based nearest neighbor techniques [17], which can be considered as a step toward kernel-based methods. However, the crucial parameter of this technique, i.e., the number of nearest neighbors, had to be chosen heuristically.

Kernel-based methods show a higher convergence rate in the limit of a large number of data points with respect to density estimation and regression than histogram-based methods [27], [28]. This motivates the idea that kernel-based methods could also be able to give better estimates for low amounts of data, thus enabling the detection of changes in the dynamics on smaller time scales. The crucial part of all kernel-based methods, however, lies within an appropriate selection of a smoothing parameter – referred to as bandwidth.

The purpose of this Letter is therefore twofold. First, we compare quantitatively and qualitatively histogram-based regression (HBR) with kernel-based regression (KBR). For the quantitative comparison we investigate time series from a stochastic system with well-known properties and compare the best achievable regression obtained by KBR with the one obtained by HBR depending on the number of data points N. Thereby we assess to what extent KBR is able to reliably estimate drift and diffusion even with a reduced amount of data. For the qualitative comparison we analyze time series from the Lorenz oscillator in two different chaotic regimes and evaluate whether the regimes can be distinguished with drift and diffusion coefficients estimated by HBR and KBR, respectively. Second, we propose an alternative bandwidth selection method and compare it to a widely used cross validation algorithm as the data driven determination of a useful bandwidth is essential for KBR.

As an exemplary analysis of field data we apply KBR together with the proposed algorithm to electroencephalographic data recorded from an epilepsy patient and assess whether changes in brain dynamics on small time scales can be detected by means of drift and diffusion analysis.

Section snippets

Histogram-based regression

Consider a measurement of a scalar time series {xj} with j=1,,N, where N denotes the number of data points, and yj is the increment of xj, i.e., yj=xj+τxj. HBR estimates Dˆ(i) with finite time step τ byDˆ(1)(x,τ,n)=1τΔtj,=1N,nI(xB)I(xjB)yjj,=1N,nI(xB)I(xjB),Dˆ(2)(x,τ,n)=12τΔtj,=1N,nI(xB)I(xjB)yj2j,=1N,nI(xB)I(xjB) where I is the indicator function, and the range [xmin,xmax] is subdivided into n (equally large) equidistant bins B=[xmin+b,xmin+(+1)b[, with the bin

Comparison of regression methods and bandwidth selection algorithms

The minimal values of E shown in Fig. 1 reveal that KBR returned significantly better drift estimates of the stochastic oscillator for all considered numbers of data points N than HBR. Even for N=105 KBR outperformed HBR. More important though is the finding that half to less than a third of the data points were needed for KBR to achieve a performance that compares to that of HBR.

While for large N the bandwidth selection algorithms did not seem to differ significantly, Fig. 2 shows that the

Conclusions

We proposed to use kernel-based regression to estimate drift and diffusion coefficients from empirical data in order to counteract the shortcomings of previous approaches that occur when only few data are available. We could show that a reduction of the amount of data needed to estimate drift and diffusion from a stochastic model system with well-known properties is possible up to a factor of three in the considered range of number of data points. Through a comparison of kernel-based with

Acknowledgements

We acknowledge fruitful discussions with Jens Prusseit. This work was supported by the Deutsche Forschungsgemeinschaft (LE 660/4-1).

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