Kernel Two-Sample and Independence Tests for Nonstationary Random Processes

Two-sample and independence tests with the kernel-based mmd and hsic have shown remarkable results on i.i.d. data and stationary random processes. However, these statistics are not directly applicable to nonstationary random processes, a prevalent form of data in many scientific disciplines. In this work, we extend the application of mmd and hsic to nonstationary settings by assuming access to independent realisations of the underlying random process. These realisations—in the form of nonstationary time-series measured on the same temporal grid—can then be viewed as i.i.d. samples from a multivariate probability distribution, to which mmd and hsic can be applied. We further show how to choose suitable kernels over these high-dimensional spaces by maximising the estimated test power with respect to the kernel hyperparameters. In experiments on synthetic data, we demonstrate superior performance of our proposed approaches in terms of test power when compared to current state-of-the-art functional or multivariate two-sample and independence tests. Finally, we employ our methods on a real socioeconomic dataset as an example application.

Stress Testing and Systemic Risk Measures Using Elliptical Conditional Multivariate Probabilities

Systemic risk, in a complex system with several interrelated variables, such as a financial market, is quantifiable from the multivariate probability distribution describing the reciprocal influence between the system’s variables. The effect of stress on the system is reflected by the change in such a multivariate probability distribution, conditioned to some of the variables being at a given stress’ amplitude. Therefore, the knowledge of the conditional probability distribution function can provide a full quantification of risk and stress propagation in the system. However, multivariate probabilities are hard to estimate from observations. In this paper, I investigate the vast family of multivariate elliptical distributions, discussing their estimation from data and proposing novel measures for stress impact and systemic risk in systems with many interrelated variables. Specific examples are described for the multivariate Student-t and the multivariate normal distributions applied to financial stress testing. An example of the US equity market illustrates the practical potentials of this approach.

Exponential ReLU DNN Expression of Holomorphic Maps in High Dimension

For a parameter dimension $$d\in {\mathbb {N}}$$ d ∈ N , we consider the approximation of many-parametric maps $$u: [-\,1,1]^d\rightarrow {\mathbb R}$$ u : [ - 1 , 1 ] d → R by deep ReLU neural networks. The input dimension d may possibly be large, and we assume quantitative control of the domain of holomorphy of u: i.e., u admits a holomorphic extension to a Bernstein polyellipse $${{\mathcal {E}}}_{\rho _1}\times \cdots \times {{\mathcal {E}}}_{\rho _d} \subset {\mathbb {C}}^d$$ E ρ 1 × ⋯ × E ρ d ⊂ C d of semiaxis sums $$\rho _i>1$$ ρ i > 1 containing $$[-\,1,1]^{d}$$ [ - 1 , 1 ] d . We establish the exponential rate $$O(\exp (-\,bN^{1/(d+1)}))$$ O ( exp ( - b N 1 / ( d + 1 ) ) ) of expressive power in terms of the total NN size N and of the input dimension d of the ReLU NN in $$W^{1,\infty }([-\,1,1]^d)$$ W 1 , ∞ ( [ - 1 , 1 ] d ) . The constant $$b>0$$ b > 0 depends on $$(\rho _j)_{j=1}^d$$ ( ρ j ) j = 1 d which characterizes the coordinate-wise sizes of the Bernstein-ellipses for u. We also prove exponential convergence in stronger norms for the approximation by DNNs with more regular, so-called “rectified power unit” activations. Finally, we extend DNN expression rate bounds also to two classes of non-holomorphic functions, in particular to d-variate, Gevrey-regular functions, and, by composition, to certain multivariate probability distribution functions with Lipschitz marginals.

Multivariate probability distribution for some intact rock properties

A multivariate probability distribution model for nine parameters of intact rocks, including unit weight (γ), porosity (n), L-type Schmidt hammer hardness (RL), Shore scleroscope hardness (Sh), Brazilian tensile strength (σbt), point load strength index (Is50), uniaxial compressive strength (σc), Young’s modulus (E), and P-wave velocity (Vp), is constructed based on the ROCK/9/4069 database that was compiled by the authors. It is shown that the multivariate probability distribution captures the correlation behaviors in the database among the nine parameters. This multivariate distribution model serves as a prior distribution model in the Bayesian analysis and can be updated into the posterior distribution of the design intact rock parameter when multivariate site-specific information is available. In this paper, the parameters for the posterior distribution of the design intact rock parameter are summarized into user-friendly tables so that engineers do not need to conduct the actual Bayesian analysis. Caution should be taken in extrapolating the results of this paper to cases that are not covered by ROCK/9/4069, because the resulting posterior distribution can be misleading.