High-dimensional statistics for point and jump processes
Final Report Abstract
Whenever the evolution of a time dependent system is influenced by random phenomena, stochastic processes are used for mathematical modelling. Since an important feature of data sets in modern applications is high dimensionality, statistical methods for stochastic processes need to be tailored to high-dimensional data. Moreover, it is of utmost importance tobe able to estimate the underlying uncertainties especially in applications from natural science. In the first part of this project novel regression approaches have been developed which allow us to circumvent the curse of dimensionality via dimension reduction techniques in general and via deep neural networks in particular. A focus was on Bayesian methods which also allow for uncertainty quantification. To achieve a scalable Markov chain Monte Carlo based method, a corrected stochastic Metropolis-Hastings algorithm has been proposed. This method is computationally feasible for large samples and it satisfies an optimal bound for the prediction risk as well as uncertainty guarantees. In an on-going interdisciplinary cooperation this method will be applied to uncertainty quantification in particle physics applications. In a second part the estimation of the jump distribution of multi-dimensional Levy process has been studied. Based on discrete observations the multivariate Levy density has been estimated via a spectral approach. Allowing for low- and high-frequency observations, rates of convergence have been proved and numerical experiments confirm the theoretical findings. The proposed method is robust to various dependence structures which may lead to singular jump distributions. The combination of both parts to obtain numerically efficient estimation methods for highdimensional Levy processes is subject of on-going research which will heavily build on the achievements of this project.
Publications
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A PAC-Bayes oracle inequality for sparse neural networks
M. F. Steffen & M. Trabs
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Dimensionality Reduction and Wasserstein stability for kernel regression
S. Eckstein; A. Iske & M. Trabs
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Estimating a multivariate Lévy density based on discrete observations
M. F. Steffen
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PAC-Bayes bounds for high-dimensional multi-index models with unknown active dimensio
M. F. Steffen
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Statistical guarantees for stochastic Metropolis-Hastings
S. Bieringer; G. Kasieczka; M.F. Steffen & M. Trabs
