**1606**

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**The scalar-valued score functions of continuous probability distribution**

Fabián, Zdeněk

2019 - English
In this report we give theoretical basis of probability theory of continuous random variables based on scalar valued score functions. We maintain consistently the following point of view: It is not the observed value, which is to be used in probabilistic and statistical considerations, but its 'treated form', the value of the scalar-valued score function of distribution of the assumed model. Actually, the opinion that an observed value of random variable should be 'treated' with respect to underlying model is one of main ideas of the inference based on likelihood in classical statistics. However, a vector nature of Fisher score functions of classical statistics does not enable a consistent use of this point of view. Instead, various inference functions are suggested and used in solutions of various statistical problems. Inference function of this report is the scalar-valued score function of distribution.
Keywords:
*Shortcomings of probability theory; Scalar-valued score functions; Characteristics of continous random variables; Parametric estimation; Transformed distributions; Skew-symmetric distributions*
Available at various institutes of the ASCR
The scalar-valued score functions of continuous probability distribution

In this report we give theoretical basis of probability theory of continuous random variables based on scalar valued score functions. We maintain consistently the following point of view: It is not ...

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**Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator**

Gergelits, Tomáš; Mardal, K.-A.; Nielsen, B. F.; Strakoš, Z.

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:
*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*
Available in digital repository of the ASCR
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 ...

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**A Logical Characteristic of Read-Once Branching Programs**

Žák, Stanislav

2019 - English
We present a mathematical model of the intuitive notions such as the knowledge or the information arising at different stages of computations on branching programs (b.p.). The model has two appropriate properties: i) The ”knowledge” arising at a stage of computation in question is derivable from the ”knowledge” arising at the previous stage according to the rules of the model and according to the local arrangement of the b.p. ii) The model confirms the intuitively well-known fact that the knowledge arising at a node of a computation depends not only on it but in some cases also on a ”mystery” information. (I. e. different computations reaching the same node may have different knowledge(s) arisen at it.) We prove that with respect to our model no such information exists in read-once b.p.‘s but on the other hand in b. p.‘s which are not read-once such information must be present. The read-once property forms a frontier. More concretely, we may see the instances of our models as a systems S = (U,D) where U is a universe of knowledge and D are derivation rules. We say that a b.p. P is compatible with a system S iff along each computation in P S derives F (false) or T (true) at the end correctly according to the label of the reached sink. This key notion modifies the classic paradigm which takes the computational complexity with respect to different classes of restricted b.p.‘s (e.g. read-once b.p.‘s, k-b.p.‘s, b.p.‘s computing in limited time etc.). Now, the restriction is defined by a subset of systems and only these programs are taken into account which are compatible with at least one of the chosen systems. Further we understand the sets U of knowledge(s) as a sets of admissible logical formulae. It is clear that more rich sets U‘s imply the large restrictions on b.p.‘s and consequently the smaller complexities of Boolean functions are detected. More rich logical equipment implies stronger computational effectiveness. Another question arises: given a set of Boolean functions (e.g. codes of some graphs) what logical equipment is optimal from the point of complexity?
Keywords:
*branching programs; computational complexity; logic*
Available in digital repository of the ASCR
A Logical Characteristic of Read-Once Branching Programs

We present a mathematical model of the intuitive notions such as the knowledge or the information arising at different stages of computations on branching programs (b.p.). The model has two ...

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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 ...

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**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 ...

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**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 ...

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**Rozhodování za neurčitosti: Pohled matematika na plánované hospodářství**

Rohn, Jiří

2019 - Czech
V práci jsou popsány hlavní výsledky neoficiálního ekonomicko-matematického výzkumu provedeného v letech 1973-1980 pracovníky Ekonomicko-matematické laboratoře Ekonomického ústavu ČSAV a MFF (J. Bouška, J. Rohn a B. Kalendovský).
Keywords:
*Leontěvův model; intervalová data; zaručené řešení; neexistence; matice 28 x 28*
Available in digital repository of the ASCR
Rozhodování za neurčitosti: Pohled matematika na plánované hospodářství

V práci jsou popsány hlavní výsledky neoficiálního ekonomicko-matematického výzkumu provedeného v letech 1973-1980 pracovníky Ekonomicko-matematické laboratoře Ekonomického ústavu ČSAV a MFF (J. ...

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**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 ...

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**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.
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 ...

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**A Hybrid Method for Nonlinear Least Squares that Uses Quasi-Newton Updates Applied to an Approximation of the Jacobian Matrix**

Lukšan, Ladislav; Vlček, Jan

2019 - English
In this contribution, we propose a new hybrid method for minimization of nonlinear least squares. This method is based on quasi-Newton updates, applied to an approximation A of the Jacobian matrix J, such that AT f = JT f. This property allows us to solve a linear least squares problem, minimizing ∥Ad+f∥ instead of solving the normal equation ATAd+JT f = 0, where d ∈ Rn is the required direction vector. Computational experiments confirm the efficiency of the new method.
Keywords:
*nonlinear least squares; hybrid methods; trust-region methods; quasi-Newton methods; numerical algorithms; numerical experiments*
Available at various institutes of the ASCR
A Hybrid Method for Nonlinear Least Squares that Uses Quasi-Newton Updates Applied to an Approximation of the Jacobian Matrix

In this contribution, we propose a new hybrid method for minimization of nonlinear least squares. This method is based on quasi-Newton updates, applied to an approximation A of the Jacobian matrix J, ...

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