Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator
Gergelits, Tomáš; Mardal, K.-A.; Nielsen, B. F.; Strakoš, Z.
2019 - anglický
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.
Klíčová slova:
second order elliptic PDEs; preconditioning by the inverse Laplacian; eigenvalues of the discretized preconditioned problem; nodal values of the coefficient function; Hall’s theorem; convergence of the conjugate gradient method
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator
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 ...
A Nonparametric Bootstrap Comparison of Variances of Robust Regression Estimators.
Kalina, Jan; Tobišková, Nicole; Tichavský, Jan
2019 - anglický
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.
Klíčová slova:
robustness; linear regression; outliers; bootstrap; least weighted squares
Dokument je dostupný na externích webových stránkách.
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 - anglický
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.
Klíčová slova:
regression quantiles; robust regression; outliers; leverage points
Dokument je dostupný na externích webových stránkách.
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 - anglický
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.
Klíčová slova:
model choice; computational statistics; robustness; variable selection
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
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 ...
On the Optimal Initial Conditions for an Inverse Problem of Model Parameter Estimation - a Complementarity Principle
Matonoha, Ctirad; Papáček, Š.
2019 - anglický
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.
Klíčová slova:
parameter identification; bleaching pattern; initial boundary value problem; sensitivity measure
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
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 ...
Nonparametric Bootstrap Techniques for Implicitly Weighted Robust Estimators
Kalina, Jan
2018 - anglický
The paper is devoted to highly robust statistical estimators based on implicit weighting, which have a potential to find econometric applications. Two particular methods include a robust correlation coefficient based on the least weighted squares regression and the minimum weighted covariance determinant estimator, where the latter allows to estimate the mean and covariance matrix of multivariate data. New tools are proposed allowing to test hypotheses about these robust estimators or to estimate their variance. The techniques considered in the paper include resampling approaches with or without replacement, i.e. permutation tests, bootstrap variance estimation, and bootstrap confidence intervals. The performance of the newly described tools is illustrated on numerical examples. They reveal the suitability of the robust procedures also for non-contaminated data, as their confidence intervals are not much wider compared to those for standard maximum likelihood estimators. While resampling without replacement turns out to be more suitable for hypothesis testing, bootstrapping with replacement yields reliable confidence intervals but not corresponding hypothesis tests.
Klíčová slova:
robust statistics; econometrics; correlation coefficient; multivariate data
Dokument je dostupný na externích webových stránkách.
Nonparametric Bootstrap Techniques for Implicitly Weighted Robust Estimators
The paper is devoted to highly robust statistical estimators based on implicit weighting, which have a potential to find econometric applications. Two particular methods include a robust correlation ...
Robust Metalearning: Comparing Robust Regression Using A Robust Prediction Error
Peštová, Barbora; Kalina, Jan
2018 - anglický
The aim of this paper is to construct a classification rule for predicting the best regression estimator for a new data set based on a database of 20 training data sets. Various estimators considered here include some popular methods of robust statistics. The methodology used for constructing the classification rule can be described as metalearning. Nevertheless, standard approaches of metalearning should be robustified if working with data sets contaminated by outlying measurements (outliers). Therefore, our contribution can be also described as robustification of the metalearning process by using a robust prediction error. In addition to performing the metalearning study by means of both standard and robust approaches, we search for a detailed interpretation in two particular situations. The results of detailed investigation show that the knowledge obtained by a metalearning approach standing on standard principles is prone to great variability and instability, which makes it hard to believe that the results are not just a consequence of a mere chance. Such aspect of metalearning seems not to have been previously analyzed in literature.
Klíčová slova:
metalearning; robust regression; outliers; robust prediction error
Dokument je dostupný na externích webových stránkách.
Robust Metalearning: Comparing Robust Regression Using A Robust Prediction Error
The aim of this paper is to construct a classification rule for predicting the best regression estimator for a new data set based on a database of 20 training data sets. Various estimators considered ...
An adaptive recursive multilevel approximate inverse preconditioning: Computation of the Schur complement
Kopal, Jiří; Rozložník, Miroslav; Tůma, Miroslav
2017 - anglický
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
An adaptive recursive multilevel approximate inverse preconditioning: Computation of the Schur complement
Exact Inference In Robust Econometrics under Heteroscedasticity
Kalina, Jan; Peštová, Barbora
2017 - anglický
The paper is devoted to the least weighted squares estimator, which is one of highly robust estimators for the linear regression model. Novel permutation tests of heteroscedasticity are proposed. Also the asymptotic behavior of the permutation test statistics of the Goldfeld-Quandt and Breusch-Pagan tests is investigated. A numerical experiment on real economic data is presented, which also shows how to perform a robust prediction model under heteroscedasticity. Theoretical results may be simply extended to the context of multivariate quantiles
Klíčová slova:
heteroscedasticity; robust statistics; regression; diagnostic tools; economic data
Dokument je dostupný na externích webových stránkách.
Exact Inference In Robust Econometrics under Heteroscedasticity
The paper is devoted to the least weighted squares estimator, which is one of highly robust estimators for the linear regression model. Novel permutation tests of heteroscedasticity are proposed. Also ...
On the Optimization of Initial Conditions for a Model Parameter Estimation
Matonoha, Ctirad; Papáček, Š.; Kindermann, S.
2017 - anglický
The design of an experiment, e.g., the setting of initial conditions, strongly influences the accuracy of the process of determining model parameters from data. The key concept relies on the analysis of the sensitivity of the measured output with respect to the model parameters. Based on this approach we optimize an experimental design factor, the initial condition for an inverse problem of a model parameter estimation. Our approach, although case independent, is illustrated at the FRAP (Fluorescence Recovery After Photobleaching) experimental technique. The core idea resides in the maximization of a sensitivity measure, which depends on the initial condition. Numerical experiments show that the discretized optimal initial condition attains only two values. The number of jumps between these values is inversely proportional to the value of a diffusion coefficient D (characterizing the biophysical and numerical process). The smaller value of D is, the larger number of jumps occurs.
Klíčová slova:
FRAP; sensitivity analysis; optimal experimental design; parameter estimation; finite differences
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
On the Optimization of Initial Conditions for a Model Parameter Estimation
The design of an experiment, e.g., the setting of initial conditions, strongly influences the accuracy of the process of determining model parameters from data. The key concept relies on the analysis ...
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