Testing fluvial erosion models using the transient response of bedrock rivers to tectonic forcing in the Apennines, Italy

The transient response of bedrock rivers to a drop in base level can be used to discriminate between competing fluvial erosion models. However, some recent studies of bedrock erosion conclude that transient river long profiles can be approximately characterized by a transport‐limited erosion model, while other authors suggest that a detachment‐limited model best explains their field data. The difference is thought to be due to the relative volume of sediment being fluxed through the fluvial system. Using a pragmatic approach, we address this debate by testing the ability of end‐member fluvial erosion models to reproduce the well‐documented evolution of three catchments in the central Apennines (Italy) which have been perturbed to various extents by an independently constrained increase in relative uplift rate. The transport‐limited model is unable to account for the catchments’response to the increase in uplift rate, consistent with the observed low rates of sediment supply to the channels. Instead, a detachment‐limited model with a threshold corresponding to the field‐derived median grain size of the sediment plus a slope‐dependent channel width satisfactorily reproduces the overall convex long profiles along the studied rivers. Importantly, we find that the prefactor in the hydraulic scaling relationship is uplift dependent, leading to landscapes responding faster the higher the uplift rate, consistent with field observations. We conclude that a slope‐ dependent channel width and an entrainment/erosion threshold are necessary ingredients when modeling landscape evolution or mapping the distribution of fluvial erosion rates in areas where the rate of sediment supply to channels is low

R\'enyi Divergence and Kullback-Leibler Divergence

R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by R\'enyi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the R\'enyi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of R\'enyi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of $\sigma$-algebras and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results.Comment: To appear in IEEE Transactions on Information Theor

Asymptotic distribution of conical-hull estimators of directional edges

Nonparametric data envelopment analysis (DEA) estimators have been widely applied in analysis of productive efficiency. Typically they are defined in terms of convex-hulls of the observed combinations of $\mathrm{inputs}\times\mathrm{outputs}$ in a sample of enterprises. The shape of the convex-hull relies on a hypothesis on the shape of the technology, defined as the boundary of the set of technically attainable points in the $\mathrm{inputs}\times\mathrm{outputs}$ space. So far, only the statistical properties of the smallest convex polyhedron enveloping the data points has been considered which corresponds to a situation where the technology presents variable returns-to-scale (VRS). This paper analyzes the case where the most common constant returns-to-scale (CRS) hypothesis is assumed. Here the DEA is defined as the smallest conical-hull with vertex at the origin enveloping the cloud of observed points. In this paper we determine the asymptotic properties of this estimator, showing that the rate of convergence is better than for the VRS estimator. We derive also its asymptotic sampling distribution with a practical way to simulate it. This allows to define a bias-corrected estimator and to build confidence intervals for the frontier. We compare in a simulated example the bias-corrected estimator with the original conical-hull estimator and show its superiority in terms of median squared error.Comment: Published in at http://dx.doi.org/10.1214/09-AOS746 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stochastic Optimization with Variance Reduction for Infinite Datasets with Finite-Sum Structure

Stochastic optimization algorithms with variance reduction have proven successful for minimizing large finite sums of functions. Unfortunately, these techniques are unable to deal with stochastic perturbations of input data, induced for example by data augmentation. In such cases, the objective is no longer a finite sum, and the main candidate for optimization is the stochastic gradient descent method (SGD). In this paper, we introduce a variance reduction approach for these settings when the objective is composite and strongly convex. The convergence rate outperforms SGD with a typically much smaller constant factor, which depends on the variance of gradient estimates only due to perturbations on a single example.Comment: Advances in Neural Information Processing Systems (NIPS), Dec 2017, Long Beach, CA, United State

Approximate Profile Maximum Likelihood

We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspired by observations gleaned from exactly solvable cases, we look for an approximate PML solution, which, intuitively, clumps comparably frequent symbols into one symbol. This amounts to lower-bounding a certain matrix permanent by summing over a subgroup of the symmetric group rather than the whole group during the computation. We extensively experiment with the approximate solution, and find the empirical performance of our approach is competitive and sometimes significantly better than state-of-the-art performance for various estimation problems

The Power and Limitations of Uniform Samples in Testing Properties of Figures

We investigate testing of properties of 2-dimensional figures that consist of a black object on a white background. Given a parameter epsilon in (0,1/2), a tester for a specified property has to accept with probability at least 2/3 if the input figure satisfies the property and reject with probability at least 2/3 if it does not. In general, property testers can query the color of any point in the input figure. We study the power of testers that get access only to uniform samples from the input figure. We show that for the property of being a half-plane, the uniform testers are as powerful as general testers: they require only O(1/epsilon) samples. In contrast, we prove that convexity can be tested with O(1/epsilon) queries by testers that can make queries of their choice while uniform testers for this property require Omega(1/epsilon^{5/4}) samples. Previously, the fastest known tester for convexity needed Theta(1/epsilon^{4/3}) queries

Conditional Nonparametric Frontier Models for Convex and Non Convex Technologies: a Unifying Approach

The explanation of productivity differentials is very important to identify the economic conditions that create inefficiency and to improve managerial performance. In literature two main approaches have been developed: one-stage approaches and two-stage approaches. Daraio and Simar (2003) propose a full nonparametric methodology based on conditional FDH and conditional order-m frontiers without any convexity assumption on the technology. On the one hand, convexity has always been assumed in mainstream production theory and general equilibrium. On the other hand, in many empirical applications, the convexity assumption can be reasonable and sometimes natural. Leading by these considerations, in this paper we propose a unifying approach to introduce external-environmental variables in nonparametric frontier models for convex and non convex technologies. Developing further the work done in Daraio and Simar (2003) we introduce a conditional DEA estimator, i.e., an estimator of production frontier of DEA type conditioned to some external-environmental variables which are neither inputs nor outputs under the control of the producer. A robust version of this conditional estimator is also proposed. These various measures of efficiency provide also indicators of convexity. Illustrations through simulated and real data (mutual funds) examples are reported.Convexity, External-Environmental Factors, Production Frontier, Nonparametric Estimation, Robust Estimation.