Нейросетевое приложение для оценивания характеристической экспоненты процесса Леви на примере распределения Бандорффa-Нильсена

Abstract

Application of the method of principal components and the generalized method of principal components to analyze data is not always reasonable, because the moments of necessary order don’t always exist in the analyzed distribution. At the same time, the interest in Levy processes continues to increase due to their numerous applications, but the principal component method is not applicable to the Levy process is. An important feature of the Levy process, which simplifies the analysis, is that the Levy process is completely defined by a complex-valued function of a real argument. It is a characteristic exponent. To identify the Levy process is to find the estimate of the characteristic exponent in the training set. The property of independence and homogeneity of Levy process increments allows using the increment of the process as a learning sample. The article considers the problem of building a neural network model for estimation of the characteristic exponent at a given interval of the argument. To estimate the characteristic exponent of the Levy process the authors propose the stochastic analogue of the adaptive neural network learning algorithm that uses the potential functions of Lanczos. The learning algorithm is tested on a hyperbolic Bandorff-Nilsen distribution. The hyperbolic distribution is a mix of normal laws, which allows generating a training sample with little effort. As a result the neural network has calculated an estimation of the Levy process characteristic exponent with a satisfactory degree of accuracy.