Estimating graph parameters with random walks

An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up to a bounded factor in $O\left(t_{\mathrm{rel}}^{3/4}\sqrt{m/d}\right)$ steps, where $t_{\mathrm{rel}}$ is the relaxation time of the lazy random walk on $G$ and $d$ is the minimum degree in $G$. Alternatively, $m$ can be estimated in $O\left(t_{\mathrm{unif}} +t_{\mathrm{rel}}^{5/6}\sqrt{n}\right)$, where $n$ is the number of vertices and $t_{\mathrm{unif}}$ is the uniform mixing time on $G$. The number of vertices $n$ can then be estimated up to a bounded factor in an additional $O\left(t_{\mathrm{unif}}\frac{m}{n}\right)$ steps. Our algorithms are based on counting the number of intersections of random walk paths $X,Y$, i.e. the number of pairs $(t,s)$ such that $X_t=Y_s$. This improves on previous estimates which only consider collisions (i.e., times $t$ with $X_t=Y_t$). We also show that the complexity of our algorithms is optimal, even when restricting to graphs with a prescribed relaxation time. Finally, we show that, given either $m$ or the mixing time of $G$, we can compute the "other parameter" with a self-stopping algorithm

Dynamical systems with heavy-tailed random parameters

Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of an ellipticity condition. More precisely, we classify these systems according to their type and --- in the recurrent case --- provide with sharp conditions quantifying the nature of recurrence by establishing which moments of passage times exist and which do not exist. The problem is tackled by mapping the random dynamical systems into Markov chains on $\mathbb{R}$ with heavy-tailed innovation and then using powerful methods stemming from Lyapunov functions to map the resulting Markov chains into positive semi-martingales.Comment: 24 page

Uncertainty under a multivariate nested-error regression model with logarithmic transformation

Assuming a multivariate linear regression model with one random factor, we consider the parameters defined as exponentials of mixed effects, i.e., linear combinations of fixed and random effects. Such parameters are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. We derive bias-corrected empirical predictors of such parameters. A second order approximation for the mean crossed product error of the predictors of two of these parameters is obtained, and an estimator is derived from it. The mean squared error is obtained as a particular case

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI

Approximation of probability density functions for PDEs with random parameters using truncated series expansions

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored

Selecting random parameters in discrete choice experiment for environmental valuation: A simulation experiment

This paper examines the various tests commonly used to select random parameters in choice modelling. The most common procedures for selecting random parameters are: the Lagrange Multiplier test as proposed by McFadden and Train (2000), the t-statistic of the deviation of the random parameter and the log-likelihood ratio test. The identification of random parameters in other words the recognition of preference heterogeneity among population is based on the fact that an individual makes a choice depending on her/his: tastes, perceptions and experiences. A simulation experiment was carried out based on a real choice experiment and the results indicated that the power of these three tests depends importantly on the spread and type of the tested parameter distribution.choice experiment, simulation, preference heterogeneity, random parameter logit, tests for selecting random parameters