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A Nonparametric Bootstrap Comparison of Variances of Robust Regression Estimators.
Kalina, Jan; Tobišková, Nicole; Tichavský, Jan
2019 - English
While various robust regression estimators are available for the standard linear regression model, performance comparisons of individual robust estimators over real or simulated datasets seem to be still lacking. In general, a reliable robust estimator of regression parameters should be consistent and at the same time should have a relatively small variability, i.e. the variances of individual regression parameters should be small. The aim of this paper is to compare the variability of S-estimators, MM-estimators, least trimmed squares, and least weighted squares estimators. While they all are consistent under general assumptions, the asymptotic covariance matrix of the least weighted squares remains infeasible, because the only available formula for its computation depends on the unknown random errors. Thus, we take resort to a nonparametric bootstrap comparison of variability of different robust regression estimators. It turns out that the best results are obtained either with MM-estimators, or with the least weighted squares with suitable weights. The latter estimator is especially recommendable for small sample sizes.
Keywords:
robustness; linear regression; outliers; bootstrap; least weighted squares
Fulltext is available at external website.
A Nonparametric Bootstrap Comparison of Variances of Robust Regression Estimators.
While various robust regression estimators are available for the standard linear regression model, performance comparisons of individual robust estimators over real or simulated datasets seem to be ...
Implicitly weighted robust estimation of quantiles in linear regression
Kalina, Jan; Vidnerová, Petra
2019 - English
Estimation of quantiles represents a very important task in econometric regression modeling, while the standard regression quantiles machinery is well developed as well as popular with a large number of econometric applications. Although regression quantiles are commonly known as robust tools, they are vulnerable to the presence of leverage points in the data. We propose here a novel approach for the linear regression based on a specific version of the least weighted squares estimator, together with an additional estimator based only on observations between two different novel quantiles. The new methods are conceptually simple and comprehensible. Without the ambition to derive theoretical properties of the novel methods, numerical computations reveal them to perform comparably to standard regression quantiles, if the data are not contaminated by outliers. Moreover, the new methods seem much more robust on a simulated dataset with severe leverage points.
Keywords:
regression quantiles; robust regression; outliers; leverage points
Fulltext is available at external website.
Implicitly weighted robust estimation of quantiles in linear regression
Estimation of quantiles represents a very important task in econometric regression modeling, while the standard regression quantiles machinery is well developed as well as popular with a large number ...
A Robustified Metalearning Procedure for Regression Estimators
Kalina, Jan; Neoral, A.
2019 - English
Metalearning represents a useful methodology for selecting and recommending a suitable algorithm or method for a new dataset exploiting a database of training datasets. While metalearning is potentially beneficial for the analysis of economic data, we must be aware of its instability and sensitivity to outlying measurements (outliers) as well as measurement errors. The aim of this paper is to robustify the metalearning process. First, we prepare some useful theoretical tools exploiting the idea of implicit weighting, inspired by the least weighted squares estimator. These include a robust coefficient of determination, a robust version of mean square error, and a simple rule for outlier detection in linear regression. We perform a metalearning study for recommending the best linear regression estimator for a new dataset (not included in the training database). The prediction of the optimal estimator is learned over a set of 20 real datasets with economic motivation, while the least squares are compared with several (highly) robust estimators. We investigate the effect of variable selection on the metalearning results. If the training as well as validation data are considered after a proper robust variable selection, the metalearning performance is improved remarkably, especially if a robust prediction error is used.
Keywords:
model choice; computational statistics; robustness; variable selection
Available in digital repository of the ASCR
A Robustified Metalearning Procedure for Regression Estimators
Metalearning represents a useful methodology for selecting and recommending a suitable algorithm or method for a new dataset exploiting a database of training datasets. While metalearning is ...
Absolute Value Mapping
Rohn, Jiří
2019 - English
We prove a necessary and sufficient condition for an absolute value mapping to be bijective. This result simultaneously gives a characterization of unique solvability of an absolute value equation for each right-hand side.
Keywords:
absolute value mapping; bijectivity; interval matrix; regularity; absolute value equation; unique solvability
Available in a digital repository NRGL
Absolute Value Mapping
We prove a necessary and sufficient condition for an absolute value mapping to be bijective. This result simultaneously gives a characterization of unique solvability of an absolute value equation for ...
On the Optimal Initial Conditions for an Inverse Problem of Model Parameter Estimation - a Complementarity Principle
Matonoha, Ctirad; Papáček, Š.
2019 - English
This contribution represents an extension of our earlier studies on the paradigmatic example of the inverse problem of the diffusion parameter estimation from spatio-temporal measurements of fluorescent particle concentration, see [6, 1, 3, 4, 5]. More precisely, we continue to look for an optimal bleaching pattern used in FRAP (Fluorescence Recovery After Photobleaching), being the initial condition of the Fickian diffusion equation maximizing a sensitivity measure. As follows, we define an optimization problem and we show the special feature (so-called complementarity principle) of the optimal binary-valued initial conditions.
Keywords:
parameter identification; bleaching pattern; initial boundary value problem; sensitivity measure
Available in digital repository of the ASCR
On the Optimal Initial Conditions for an Inverse Problem of Model Parameter Estimation - a Complementarity Principle
This contribution represents an extension of our earlier studies on the paradigmatic example of the inverse problem of the diffusion parameter estimation from spatio-temporal measurements of ...
Overdetermined Absolute Value Equations
Rohn, Jiří
2019 - English
We consider existence, uniqueness and computation of a solution of an absolute value equation in the overdetermined case.
Keywords:
absolute value equations; overdetermined system
Available in a digital repository NRGL
Overdetermined Absolute Value Equations
We consider existence, uniqueness and computation of a solution of an absolute value equation in the overdetermined case.
Generalization of a Theorem on Eigenvalues of Symmetric Matrices
Rohn, Jiří
2019 - English
We prove that the product of a symmetric positive semide nite matrix and a symmetric matrix has all eigenvalues real.
Keywords:
symmetric matrix; positive semide nite matrix; real spectrum
Available in a digital repository NRGL
Generalization of a Theorem on Eigenvalues of Symmetric Matrices
We prove that the product of a symmetric positive semide nite matrix and a symmetric matrix has all eigenvalues real.
Hybrid Methods for Nonlinear Least Squares Problems
Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan
2019 - English
This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function F(x) = (1=2)fT (x)f(x), where x ∈ Rn and f ∈ Rm, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved in a unified way. Some proofs concerning trust region methods, which are difficult to find in the literature, are also added. In particular, the report contains an analysis of a new simple hybrid method with Jacobian corrections (Section 8) and an investigation of the simple hybrid method for sparse least squares problems proposed previously in [33] (Section 14).
Keywords:
numerical optimization; nonlinear least squares; trust region methods; hybrid methods; sparse problems; partially separable problems; numerical experiments
Available in a digital repository NRGL
Hybrid Methods for Nonlinear Least Squares Problems
This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function F(x) = (1=2)fT (x)f(x), where x ∈ Rn and f ∈ Rm, together with ...
Does a Singular Symmetric Interval Matrix Contain a Symmetric Singular Matrix?
Rohn, Jiří
2019 - English
We consider the conjecture formulated in the title concerning existence of a symmetric singular matrix in a singular symmetric interval matrix. We show by means of a counterexample that it is generally not valid, and we prove that it becomes true under an additional assumption of positive semide niteness of the midpoint matrix. The proof is constructive.
Keywords:
symmetric interval matrix; singularity; positive semide niteness
Available in a digital repository NRGL
Does a Singular Symmetric Interval Matrix Contain a Symmetric Singular Matrix?
We consider the conjecture formulated in the title concerning existence of a symmetric singular matrix in a singular symmetric interval matrix. We show by means of a counterexample that it is ...
Transforming hierarchical images to program expressions using deep networks
Křen, Tomáš
2018 - English
We present a technique describing how to effectively train a neural network given an image to produce a formal description of the given image. The basic motivation of the proposed technique is an intention to design a new tool for automatic program synthesis capable of transforming sensory data (in our case static image, but generally a phenotype) to a formal code expression (i.e. syntactic tree of a program), such that the code (from evolutionary perspective a genotype) evaluates to a value that is similar to the input data, ideally identical. Our approach is partially based on our technique for generating program expressions in the context of typed functional genetic programming. We present promising results evaluating a simple image description language achieved with a deep network combining convolution encoder of images and recurrent decoder for generating program expressions in the sequential prefix notation and propose possible future applications.
Keywords:
deep networks; automatic program synthesis; image processing
Available in a digital repository NRGL
Transforming hierarchical images to program expressions using deep networks
We present a technique describing how to effectively train a neural network given an image to produce a formal description of the given image. The basic motivation of the proposed technique is an ...
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