Global Optimization of Redescending Robust Estimators

[EN] Robust estimation has proved to be a valuable alternative to the least squares estimator for the cases where the dataset is contaminated with outliers. Many robust estimators have been designed to be minimally affected by the outlying observations and produce a good fit for the majority of the data. Among them, the redescending estimators have demonstrated the best estimation capabilities. It is little known, however, that the success of a robust estimation method depends not only on the robust estimator used but also on the way the estimator is computed. In the present paper, we show that for complicated cases, the predominant method of computing the robust estimator by means of an iteratively reweighted least squares scheme may result in a local optimum of significantly lower quality than the global optimum attainable by means of a global optimization method. Further, the sequential use of the proposed global robust estimation proves to successfully solve the problem of M-split estimation, that is, the determination of parameters of different functional models implicit in the data.Baselga Moreno, S.; Klein, I.; Sampaio Suraci, S.; Castro De Oliveira, L.; Tomio Matsuoka, M.; Francisco Rofatto, V. (2021). Global Optimization of Redescending Robust Estimators. Mathematical Problems in Engineering. 2021:1-13. https://doi.org/10.1155/2021/9929892S113202

Modeling the thermal structure and magnetic properties of the crust of active regions with application to the Rio Grande rift

Experiments in Curie depth estimation from long wavelength magnetic anomalies are summarized. The heart of the work is equivalent-layer-type magnetization models derived by inversion of high-elevation, long wavelength magnetic anomaly data. The methodology is described in detail in the above references. A magnetization distribution in a thin equivalent layer at the Earth's surface having maximum detail while retaining physical significance, and giving rise to a synthetic anomaly field which makes a best fit to the observed field in a least squares sense is discussed. The apparent magnetization contrast in the equivalent layer is approximated using an array of dipoles distributed in equal area at the Earth's surface. The dipoles are pointed in the direction of the main magnetic field, which carries the implicit assumption that crustal magnetization is dominantly induced or viscous. The determination of the closest possible dipole spacing giving a stable inversion to a solution having physical significance is accomplished by plotting the standard deviation of the solution parameters against their spatial separation for a series of solutions

Essays on spatial autoregressive models with increasingly many parameters

Much cross-sectional data in econometrics is blighted by dependence across units. A solution to this problem is the use of spatial models that allow for an explicit form of dependence across space. This thesis studies problems related to spatial models with increasingly many parameters. A large proportion of the thesis concentrates on Spatial Autoregressive (SAR) models with increasing dimension. Such models are frequently used to model spatial correlation, especially in settings where the data are irregularly spaced. Chapter 1 provides an introduction and background material for the thesis. Chapter 2 develops consistency and asymptotic normality of least squares and instrumental variables (IV) estimates for the parameters of a higher-order spatial autoregressive (SAR) model with regressors. The order of the SAR model and the number of regressors are allowed to approach infinity with sample size, and the permissible rate of growth of the dimension of the parameter space relative to sample size is studied. An alternative to least squares or IV is to use the Gaussian pseudo maximum likelihood estimate (PMLE), studied in Chapter 3. However, this is plagued by finitesample problems due to the implicit definition of the estimate, these being exacerbated by the increasing dimension of the parameter space. A computationally simple Newton type step is used to obtain estimates with the same asymptotic properties as those of the PMLE. Chapters 4 and 5 of the thesis deal with spatial models on an equally spaced, d dimensional lattice. We study the covariance structure of stationary random fields defined on d-dimensional lattices in detail and use the analysis to extend many results from time series. Our main theorem concerns autoregressive spectral density estimation. Stationary random fields on a regularly spaced lattice have an infinite autoregressive representation if they are also purely non-deterministic. We use truncated versions of the AR representation to estimate the spectral density and establish uniform consistency of the proposed spectral density estimate

(WP 2010-11) The Benefits of Environmental Improvement: Estimates From Space-time Analysis

This paper develops estimates of environmental improvement based on a two-stage hedonic price analysis of the single family housing market in the Puget Sound region of Washington State. The analysis — which focuses specifically on several EPA-designated environmental hazards and involves 226,918 transactions for 177,303 unique properties that took place between January 2001 and September 2009 — involves four steps: (i) ten hedonic price functions are estimated year-by-year, one for each year of the 2000s; (ii) the hedonic estimates are used to compute the marginal implicit price of distance from air release, superfund, and toxic release sites; (iii) the marginal implicit prices, which vary through time, are used to estimate a series of implicit demand functions describing the relationship between the price of distance and the quantity consumed; and, finally (iv) the demand estimates are compared to those obtained in other research and then used evaluate the potential scale of benefits associated with some basic environmental improvement scenarios. Overall, the analysis provides further evidence that it is possible to develop a structural model of implicit demand within a single housing market and suggests that the benefits of environmental improvement are substantial

Dynamic Estimation of Rigid Motion from Perspective Views via Recursive Identification of Exterior Differential Systems with Parameters on a Topological Manifold

We formulate the problem of estimating the motion of a rigid object viewed under perspective projection as the identification of a dynamic model in Exterior Differential form with parameters on a topological manifold. We first describe a general method for recursive identification of nonlinear implicit systems using prediction error criteria. The parameters are allowed to move slowly on some topological (not necessarily smooth) manifold. The basic recursion is solved in two different ways: one is based on a simple extension of the traditional Kalman Filter to nonlinear and implicit measurement constraints, the other may be regarded as a generalized "Gauss-Newton" iteration, akin to traditional Recursive Prediction Error Method techniques in linear identification. A derivation of the "Implicit Extended Kalman Filter" (IEKF) is reported in the appendix. The ID framework is then applied to solving the visual motion problem: it indeed is possible to characterize it in terms of identification of an Exterior Differential System with parameters living on a C0 topological manifold, called the "essential manifold". We consider two alternative estimation paradigms. The first is in the local coordinates of the essential manifold: we estimate the state of a nonlinear implicit model on a linear space. The second is obtained by a linear update on the (linear) embedding space followed by a projection onto the essential manifold. These schemes proved successful in performing the motion estimation task, as we show in experiments on real and noisy synthetic image sequences

Hierarchical Implicit Models and Likelihood-Free Variational Inference

Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure. Next, we develop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. Key to LFVI is specifying a variational family that is also implicit. This matches the model's flexibility and allows for accurate approximation of the posterior. We demonstrate diverse applications: a large-scale physical simulator for predator-prey populations in ecology; a Bayesian generative adversarial network for discrete data; and a deep implicit model for text generation.Comment: Appears in Neural Information Processing Systems, 201

Least squares estimation of joint production functions by the differential evolution method of global optimization

Most of the studies relating to estimation of joint production functions have noted two difficulties: first that allocation of inputs to different outputs is not known, and the second that a method of estimation cannot have more than one dependent variable, which necessitates construction of a composite output transformation function. This study has conducted some simulation experiments on joint estimation of the CES, the Transcendental and the Nerlove-Ringstad functions. Allocation parameters of inputs across the products have been introduced. Estimation has been done jointly, but without constructing a composite macro production function or an output transformation function. We use nonlinear least squares based on the Differential Evolution method of global optimization that permits fitting multiple production functions simultaneously.