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Register a new userA general approach to statistical modeling of physical laws: nonparametric regression
النهج العام لنمذجة الإحصائية لقوانين الفيزياء: انحدار غير معلم
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I. Grabec
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تستخدم التحليل الإحصائي لقوانين الفيزياء التجريبية على دالة توزيع الاحتمالات للمتغيرات المقياسة. ويتم تعبيرها عن طريق تقدير النواة بواسطة البيانات التجريبية. ويتم تحديد النواة بشكل عادل بواسطة تشويش البيانات أثناء تعيين الإعداد التجريبي. وتتم استخراج قانون فيزيائي يربط المتغيرات المقياسة بشكل أفضل من البيانات التجريبية بواسطة محدد المتوسط الشرطي. ويتم توليده مباشرة من محدد النواة ويتوافق مع الانحدار غير المعياري. ويتم إظهار الطريقة المقترحة بواسطة تحليل خريطة العودة للبيانات الكائجة الضوئية. في هذا المثال يتم استخدام الانحدار غير المعياري لتنبؤ قيمة المستقبل للسلسلة الزمنية الكائجة من الحالية. ويتم استخدام خطأ المنبؤ في تعريف جودة المنبؤ ، في حين أن التكرار يتم تعبيره بواسطة المسافة المربعة المتوسطة بين نقاط البيانات. ويتم استخدام كلا من الإحصاءات في تعريف دالة تكلفة المنبؤ الجديدة. من الحد الأدنى لدالة تكلفة المنبؤ ، يتم تقدير عدد البيانات الصحيح في النموذج.
Statistical modeling of experimental physical laws is based on the probability density function of measured variables. It is expressed by experimental data via a kernel estimator. The kernel is determined objectively by the scattering of data during calibration of experimental setup. A physical law, which relates measured variables, is optimally extracted from experimental data by the conditional average estimator. It is derived directly from the kernel estimator and corresponds to a general nonparametric regression. The proposed method is demonstrated by the modeling of a return map of noisy chaotic data. In this example, the nonparametric regression is used to predict a future value of chaotic time series from the present one. The mean predictor error is used in the definition of predictor quality, while the redundancy is expressed by the mean square distance between data points. Both statistics are used in a new definition of predictor cost function. From the minimum of the predictor cost function, a proper number of data in the model is estimated.
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A physical law is represented by the probability distribution of a measured variable. The probability density is described by measured data using an estimator whose kernel is the instrument scattering function. The experimental information and data redundancy are defined in terms of information entropy. The model cost function, comprised of data redundancy and estimation error, is minimized by the creation-annihilation process.
The extraction of a physical law y=yo(x) from joint experimental data about x and y is treated. The joint, the marginal and the conditional probability density functions (PDF) are expressed by given data over an estimator whose kernel is the instrument scattering function. As an optimal estimator of yo(x) the conditional average is proposed. The analysis of its properties is based upon a new definition of prediction quality. The joint experimental information and the redundancy of joint measurements are expressed by the relative entropy. With the number of experiments the redundancy on average increases, while the experimental information converges to a certain limit value. The difference between this limit value and the experimental information at a finite number of data represents the discrepancy between the experimentally determined and the true properties of the phenomenon. The sum of the discrepancy measure and the redundancy is utilized as a cost function. By its minimum a reasonable number of data for the extraction of the law yo(x) is specified. The mutual information is defined by the marginal and the conditional PDFs of the variables. The ratio between mutual information and marginal information is used to indicate which variable is the independent one. The properties of the introduced statistics are demonstrated on deterministically and randomly related variables.
The Photonic Force Microscope (PFM) is an opto-mechanical technique based on an optical trap that can be assumed to probe forces in microscopic systems. This technique has been used to measure forces in the range of pico- and femto-Newton, assessing the mechanical properties of biomolecules as well as of other microscopic systems. For a correct use of the PFM, the force field to measure has to be invariable (homogeneous) on the scale of the Brownian motion of the trapped probe. This condition implicates that the force field must be conservative, excluding the possibility of a rotational component. However, there are cases where these assumptions are not fulfilled Here, we show how to improve the PFM technique in order to be able to deal with these cases. We introduce the theory of this enhanced PFM and we propose a concrete analysis workflow to reconstruct the force field from the experimental time-series of the probe position. Furthermore, we experimentally verify some particularly important cases, namely the case of a conservative or rotational force-field.
In this paper, we consider a surrogate modeling approach using a data-driven nonparametric likelihood function constructed on a manifold on which the data lie (or to which they are close). The proposed method represents the likelihood function using a spectral expansion formulation known as the kernel embedding of the conditional distribution. To respect the geometry of the data, we employ this spectral expansion using a set of data-driven basis functions obtained from the diffusion maps algorithm. The theoretical error estimate suggests that the error bound of the approximate data-driven likelihood function is independent of the variance of the basis functions, which allows us to determine the amount of training data for accurate likelihood function estimations. Supporting numerical results to demonstrate the robustness of the data-driven likelihood functions for parameter estimation are given on instructive examples involving stochastic and deterministic differential equations. When the dimension of the data manifold is strictly less than the dimension of the ambient space, we found that the proposed approach (which does not require the knowledge of the data manifold) is superior compared to likelihood functions constructed using standard parametric basis functions defined on the ambient coordinates. In an example where the data manifold is not smooth and unknown, the proposed method is more robust compared to an existing polynomial chaos surrogate model which assumes a parametric likelihood, the non-intrusive spectral projection.
A theory of additive Markov chains with long-range memory is used for description of correlation properties of coarse-grained literary texts. The complex structure of the correlations in texts is revealed. Antipersistent correlations at small distances, L < 300, and persistent ones at L > 300 define this nontrivial structure. For some concrete examples of literary texts, the memory functions are obtained and their power-law behavior at long distances is disclosed. This property is shown to be a cause of self-similarity of texts with respect to the decimation procedure.
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