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The Equation |x| - |Ax| = b
Rohn, Jiří
We formulate conditions on A and b under which the double absolute value equation |x| - |Ax| = b possesses in each orthant a unique solution which, moreover, belongs to the interior of that orthant.
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Globální implicitní funkce
Rohn, Jiří
Tento text pochází z roku 1973 a nebyl dosud zveřejněn. Jeho hlavním výsledkem je věta o existenci a jednoznačnosti globální implicitní funkce v Rn. Tomuto výsledku předchází řada pomocných tvrzení.
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Some modifications of the limited-memory variable metric optimization methods
Vlček, Jan ; Lukšan, Ladislav
Several modifications of the limited-memory variable metric (or quasi-Newton) line search methods for large scale unconstrained optimization are investigated. First the block version of the symmetric rank-one (SR1) update formula is derived in a similar way as for the block BFGS update in Vlˇcek and Lukˇsan (Numerical Algorithms 2019). The block SR1 formula is then modified to obtain an update which can reduce the required number of arithmetic operations per iteration. Since it usually violates the corresponding secant conditions, this update is combined with the shifting investigated in Vlˇcek and Lukˇsan (J. Comput. Appl. Math. 2006). Moreover, a new efficient way how to realize the limited-memory shifted BFGS method is proposed. For a class of methods based on the generalized shifted economy BFGS update, global convergence is established. A numerical comparison with the standard L-BFGS and BNS methods is given.
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Introduction to statistical inference based on scalar-valued scores
Fabián, Zdeněk
In the report we maintain consistently the following point of view: Given a continuous model, there are not the observed values, which are to be used in probabilistic and statistical considerations, but their ”treated forms”,the values of the scalar-valued score function corresponding to the model. Based on this modified concept of the score function, we develop theory of score random variables, study their geometry and define their new characteristics, finite even in cases of heavy-tailed models. A generalization for parametric families provides a new approach to parametric point estimation.
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A New Look to Information and Uncertainty of Continuous Distributions
Fabián, Zdeněk
We define information and uncertainty function of a family of continuous distributions. Their values are relative information and uncertainty of an observation from the given parametric family, their mean values are the generalized Fisher information and a new measure of variability, the score variance. In a series of examples we show why to use new concepts instead of the differential entropy.
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Score correlation for skewed distributions
Fabián, Zdeněk
Based on the new concept of the scalar-valued score function of continuous distributions we introduce the score correlation coefficient ”tai-lored” to the assumed probabilistic model and study its properties by means of simulation experiments. It appeared that the new correlation method is useful for enormously skewed distributions.
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Scalar-Valued Score Functions and their use in Parametric Estimation
Fabián, Zdeněk
In the paper we describe and explain a new direction in probabilistic and statistical reasoning, the approach based on scalar-valued score functions of continuous random variables. We show basic properties of score functions of standard distributions, generalize the approach for parametric families and show how to use them for solutions of problems of parametric statistics.
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Spatio-Spectral EEG Patterns in the Source-Reconstructed Space and Relation to Resting-State Networks: An EEG-fMRI Study
Jiříček, Stanislav ; Koudelka, V. ; Mantini, D. ; Mareček, R. ; Hlinka, Jaroslav
In this work, we present and evaluate a novel EEG-fMRI integration approach combining a spatio-spectral decomposition method and a reliable source localization technique. On the large 72 subjects resting- state hdEEG-fMRI data set we tested the stability of the proposed method in terms of both extracted spatio-spectral patterns(SSPs) as well as their correspondence to the BOLD signal. We also compared the proposed method with the spatio-spectral decomposition in the electrode space as well as well-known occipital alpha correlate in terms of the explained variance of BOLD signal. We showed that the proposed method is stable in terms of extracted patterns and where they correlate with the BOLD signal. Furthermore, we show that the proposed method explains a very similar level of the BOLD signal with the other methods and that the BOLD signal in areas of typical BOLD functional networks is explained significantly more than by a chance. Nevertheless, we didn’t observe a significant relation between our source-space SSPs and the BOLD ICs when spatio-temporally comparing them. Finally, we report several the most stable source space EEG-fMRI patterns together with their interpretation and comparison to the electrode space patterns.
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A Measure of Variability WIthin Parametric Families of Continuous Distributions
Fabián, Zdeněk
A continuous probability measure on an open interval of the real line induces in it a unique geometry, "center of gravity" of which is the typical value of the distribution. In the paper is identified a score variance as a finite measure of variability of distributions with respect to the typical value and discussed its properties and methods of estimation. Itroducing a generalized Rao distance in the sample space one can appraise the precision of the estimate of the typical value.
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