تجزیه و تحلیل کمی وکیفی فرآیندهای تصادفی بر اساس داده های اندازه گیری: نظریه و کاربرد داده مصنوعی

Analysis of stochastic processes governed by the Langevin equation is discussed. The analysis is based on a general method for non-parametric estimation of deterministic and random terms of the Langevin equation directly from given data. Separate estimation of the terms corresponds to decomposition of process dynamics into deterministic and random components. Such decomposition provides a basis for qualitative and quantitative analysis of process dynamics. In Part I, the following analysis possibilities are described and illustrated using various synthetic datasets: (1) qualitative inspection of the estimated terms presented as fields, (2) reconstruction of the deterministic and stochastic evolution of the process and (3) approximation of the deterministic term by an analytical function and quantitative treatment of the equations obtained. In Part II, these analysis possibilities are applied to experimental datasets from metal cutting and laser-beam welding.

In recent decades, processes which generate non-periodic data have been studied intensively. Interest in these processes was fuelled mainly by the theory of deterministic chaos, which showed that non-periodic, even chaotic data can result from a non-linear deterministic process with only a few active degrees of freedom [1, 2]. Numerous analysis methods have been developed to extract meaningful information about the process from its chaotic data [3]. These methods require the data to be generated by a deterministicprocess, and allow only for negligible measurement noise uncorrelated to the process dynamics. Applicability of these methods to the analysis of data from a stochastic process of which noise is an integral part is limited. However, all experimental data are to some extent noisy, and it is usually di $ cult to distinguish between noisy chaotic data and stochastic data, which may also be corrupted by measurement noise. The problem is illustrated in Figure 1 using data from a forced oscillator. The phase portrait of the oscillations appears complicated, the time series of the displacement is non-periodic, and the associated power spectrum is broad. Since all these properties are also typical of chaotic data, one might assume that the oscillations are chaotic, and employ the analysis methods inspired by chaos theory [3]. In he present case, this assumption would be wrong because the data in fact result from a stochastic process, i.e., from the randomly forced van der Pol oscillator in a limit cycle regime (see section A.1 for details). Therefore, methods suited to stochastic data should be employed instead A general method for non-parametricestimation of the deterministicand random terms of the Langevin equation (1) has already been proposed [4, 5], and applied to syntheticand experimental datasets from medicine and engineering [6, 7]. The aim of this article is to present several possibilities the method o ! ers for qualitative and quantitative analyses of stochastic data. For this purpose, the method is " rst reviewed brie # y and shows how both the deterministicand random terms of equation (1) can be estimated from data, and inspected qualitatively. Since the terms in fact form a model of the process they can be employed to reconstruct either the deterministic or the stochastic evolution of the process. If equations are needed for the model, the deterministicterm can be approximated by an analytical function which can be further analyzed quantitatively. These analysis possibilities are illustrated by examples which include synthetic datasets from (1) the stochastic van der Pol oscillator, (2) a stochastic process exhibiting the sub-critical Hopf bifurcation and (3) the stochastic Lorenz system in a chaotic regime.

Stochastic processes governed by the Langevin equation are analyzed. Analysis is based on estimation of the deterministicand random terms of the Langevin equation directly from data. The estimation method is non-parametricin the sense that no functional form need be assumed in advance for the estimated terms. The main disadvantage of the estimation method is that it requires rather densely sampled data [14]. There are approaches to parameter estimation in stochastic di ! erential equations which are less demanding in terms of the sampling rate of the data than the method discussed in this article, but they are parametric and depend crucially on the assumed functional form [15, 16]. It, therefore, seems that oversampling is the price to be paid for a non-parametric estimate. The estimated deterministicand random terms are presented as " elds and inspected visually. Inspection of the deterministic term can yield information about the average direction and velocity of the deterministic # ow, whereas information about the amplitude and the direction of noise can be extracted from the random term. In the examples presented, the terms were estimated in the state space spanned by all state variables. In experimental situations, when not all state variables are measurable, the terms can be estimated in space reconstructed from the time series of measured variables related to the process dynamics [3]. While most of the information provided by the estimated terms is preserved in the reconstructed space, information about the direction of noise is lost, since it is not possible to determine which state variable is originally in # uenced by noise.The estimated terms of the Langevin equation constitute a model which is used to reconstruct the deterministic and stochastic trajectories of the process. The deterministic trajectories show the hypothetical process evolution in the absence of random noise. Although remotely similar, the deterministic reconstruction is not equivalent to " ltering dynamic noise, since the process would evolve di ! erently under the same deterministiclaws if noise was present. The stochastic trajectories show a realistic process evolution which could actually be observed. They can be employed as surrogates in various situations when the length of the recorded original trajectories is insu $ cient for a particular task [7]. Quantitative analysis of stochastic processes is based on approximation of the estimated deterministic term by an analytical function. In the present study a polynomial is used, and the corresponding coe $ cients are obtained by a least-squares " t. The polynomial is employed to generate the approximate deterministic trajectories of the process and to assess their linear stability. The order of the approximating polynomial should be selected to be consistent with the estimated deterministic term. Consistence can be checked by comparing the deterministic trajectories reconstructed using the estimated deterministic term with those obtained by integrating the approximating polynomial. Several orders should be examined to " nd the most appropriate one. The Lorenz example reveals that overestimating the order of approximation preserves qualitative and, to a large extent, also quantitative properties of the system, although the polynomial coe $ cients do not agree with the theoretical ones. Moreover, sensitivity of the approximation and the amount of data required both increase with the order of approximating polynomial. On the other hand, the sub-critical bifurcation example shows that underestimating the order of approximation does not entirely capture the process properties. Note, " nally, that the polynomial may not necessarily be the correct choice for the approximating function. In all examples presented, noise amplitude g was constant, which means that noise was of the additive type. As shown in reference [17], the same formulae (2) for estimating the deterministic and random terms apply also for the multiplicative type of noise caused by dependence of the noise amplitude g on the process state X ( t ). Some of the analysis possibilities discussed in this paper have already been applied to various experimental datasets from medicine [6, 18] and engineering [6, 19]. In Part 2 of the paper [20] these analysis possibilities are used to analyze experimental data from metal cutting and laser-beam welding.