We introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non-dynamical setting. Given a loss function ${l}$, we detail the prior and posterior items, their consequences and exhibit several examples. Taking $\Theta$ as the set of parameters and $Y$ as the set of data (which usually provides {random samples}), a general IFS is a measurable map $\tau:\Theta\times Y \to Y$, which can be interpreted as a family of maps $\tau_\theta:Y\to Y,\,\theta\in\Theta$. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of {information.} We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.
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This paper presents a new algorithm for generating random inverse-Wishart matrices that directly generates the Cholesky factor of the matrix without computing the factorization. Whenever parameterized in terms of a precision matrix $\Omega=\Sigma^{-1}$, or its Cholesky factor, instead of a covariance matrix $\Sigma$, the new algorithm is more efficient than the current standard algorithm.
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
Most of the literature on causality considers the structural framework of Pearl and the potential-outcome framework of Neyman and Rubin to be formally equivalent, and therefore interchangeably uses the do-notation and the potential-outcome subscript notation to write counterfactual outcomes. In this paper, we superimpose the two causal frameworks to prove that structural counterfactual outcomes and potential outcomes do not coincide in general -- not even in law. More precisely, we express the law of the potential outcomes in terms of the latent structural causal model under the fundamental assumptions of causal inference. This enables us to precisely identify when counterfactual inference is or is not equivalent between approaches, and to clarify the meaning of each kind of counterfactuals.
Models of complex technological systems inherently contain interactions and dependencies among their input variables that affect their joint influence on the output. Such models are often computationally expensive and few sensitivity analysis methods can effectively process such complexities. Moreover, the sensitivity analysis field as a whole pays limited attention to the nature of interaction effects, whose understanding can prove to be critical for the design of safe and reliable systems. In this paper, we introduce and extensively test a simple binning approach for computing sensitivity indices and demonstrate how complementing it with the smart visualization method, simulation decomposition (SimDec), can permit important insights into the behavior of complex engineering models. The simple binning approach computes first-, second-order effects, and a combined sensitivity index, and is considerably more computationally efficient than Sobol' indices. The totality of the sensitivity analysis framework provides an efficient and intuitive way to analyze the behavior of complex systems containing interactions and dependencies.
We present the conditional determinantal point process (DPP) approach to obtain new (mostly Fredholm determinantal) expressions for various eigenvalue statistics in random matrix theory. It is well-known that many (especially $\beta=2$) eigenvalue $n$-point correlation functions are given in terms of $n\times n$ determinants, i.e., they are continuous DPPs. We exploit a derived kernel of the conditional DPP which gives the $n$-point correlation function conditioned on the event of some eigenvalues already existing at fixed locations. Using such kernels we obtain new determinantal expressions for the joint densities of the $k$ largest eigenvalues, probability density functions of the $k^\text{th}$ largest eigenvalue, density of the first eigenvalue spacing, and more. Our formulae are highly amenable to numerical computations and we provide various numerical experiments. Several numerical values that required hours of computing time could now be computed in seconds with our expressions, which proves the effectiveness of our approach. We also demonstrate that our technique can be applied to an efficient sampling of DR paths of the Aztec diamond domino tiling. Further extending the conditional DPP sampling technique, we sample Airy processes from the extended Airy kernel. Additionally we propose a sampling method for non-Hermitian projection DPPs.
This paper introduces a mathematical framework for explicit structural dynamics, employing approximate dual functionals and rowsum mass lumping. We demonstrate that the approach may be interpreted as a Petrov-Galerkin method that utilizes rowsum mass lumping or as a Galerkin method with a customized higher-order accurate mass matrix. Unlike prior work, our method correctly incorporates Dirichlet boundary conditions while preserving higher order accuracy. The mathematical analysis is substantiated by spectral analysis and a two-dimensional linear benchmark that involves a non-linear geometric mapping. Our results reveal that our approach achieves accuracy and robustness comparable to a traditional Galerkin method employing the consistent mass formulation, while retaining the explicit nature of the lumped mass formulation.
A new mechanical model on noncircular shallow tunnelling considering initial stress field is proposed in this paper by constraining far-field ground surface to eliminate displacement singularity at infinity, and the originally unbalanced tunnel excavation problem in existing solutions is turned to an equilibrium one of mixed boundaries. By applying analytic continuation, the mixed boundaries are transformed to a homogenerous Riemann-Hilbert problem, which is subsequently solved via an efficient and accurate iterative method with boundary conditions of static equilibrium, displacement single-valuedness, and traction along tunnel periphery. The Lanczos filtering technique is used in the final stress and displacement solution to reduce the Gibbs phenomena caused by the constrained far-field ground surface for more accurte results. Several numerical cases are conducted to intensively verify the proposed solution by examining boundary conditions and comparing with existing solutions, and all the results are in good agreements. Then more numerical cases are conducted to investigate the stress and deformation distribution along ground surface and tunnel periphery, and several engineering advices are given. Further discussions on the defects of the proposed solution are also conducted for objectivity.
This article investigates a local discontinuous Galerkin (LDG) method for one-dimensional and two-dimensional singularly perturbed reaction-diffusion problems on a Shishkin mesh. During this process, due to the inability of the energy norm to fully capture the behavior of the boundary layers appearing in the solutions, a balanced norm is introduced. By designing novel numerical fluxes and constructing special interpolations, optimal convergences under the balanced norm are achieved in both 1D and 2D cases. Numerical experiments support the main theoretical conclusions.
We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of H\"older regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of H\"older exponent. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.
Differential geometric approaches are ubiquitous in several fields of mathematics, physics and engineering, and their discretizations enable the development of network-based mathematical and computational frameworks, which are essential for large-scale data science. The Forman-Ricci curvature (FRC) - a statistical measure based on Riemannian geometry and designed for networks - is known for its high capacity for extracting geometric information from complex networks. However, extracting information from dense networks is still challenging due to the combinatorial explosion of high-order network structures. Motivated by this challenge we sought a set-theoretic representation theory for high-order network cells and FRC, as well as their associated concepts and properties, which together provide an alternative and efficient formulation for computing high-order FRC in complex networks. We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. Crucially, our representation theory reveals previous computational bottlenecks and also accelerates the computation of FRC. As a consequence, our findings open new research possibilities in complex systems where higher-order geometric computations are required.
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