Lubomír Kubáček

Statistical properties of linearization of the Arrhenius equation via the logarithmic transformation

Abstract The temperature dependence of a rate constant is usually described by the Arrhenius equation. It can be transformed into a linear model by the logarithmic transformation of the dependent variable. The parameter vector in the linearized model is then often estimated by the ordinary least squares estimator. Theoretical bias and the covariance matrix of the parameter vector in the linearized model are given for this method if the original errors are normally distributed, independent and have a constant variance. The expressions are verified by computer simulation of a numerical example. This true covariance matrix of the parameter vector is compared with that if the change of distribution of errors is neglected. The results are also compared with those obtained by the generalized least squares estimator.

The present approach to the study of the least-squares method

SummaryThe present approach to the study of the least squares method which lies in introducing the basic principle of this method into various functional metric spaces is dealt with.

Methodological Note on the Fractal Analysis of Texts

Abstract The experimental procedure concerning the text exploration is demonstrated step by step on an illustrative example. Our methodological note involves, besides other things, detailed linguistic, statistical, numerical and fractal analyses. It can be regarded as an instructive text for further linguistic experiments in this field.This paper is dedicated to Gabriel Altmann

Advanced statistical evaluation of complex formation constant from electrophoretic data

A new method for the estimation of complex formation constants is presented. The method is based on electrophoretically measured effective mobilities and applied to the estimation of the complex formation constant of interaction between nitrogen heterocyclic bases and cyclodextrines. The calculation of constants is based on the linearization of the dependence between effective mobility and the cyclodextrine concentration and the application of an advanced statistical evaluation procedure. Complex formation constants 14.8 and 63.2 l mol−1 were obtained for the interaction of pyridinium and benzylaminopurinium with dimethyl-β-cyclodextrine (DM-β-CD), respectively. Constants in the order of magnitude 101–102 l mol−1 were obtained for some other purine derivatives. The proposed procedure, in connection with the mathematical software for matrix operations, is rather simple and gives much more valuable outputs than commonly used concepts.

How the design of an experiment influences the nonsensitiveness regions in models with variance components

Nonsensitiveness regions for estimators of linear functions, for confidence ellipsoids, for the level of a test of a linear hypothesis on parameters and for the value of the power function are investigated in a linear model with variance components.The influence of the design of an experiment on the nonsensitiveness regions mentioned is numerically demonstrated and discussed on an example.

Statistical aspects of eliminating systematic effects

РезюмеПрuве¶rt;ен меmо¶rt; обрaбоmкu наблюо¶rt;енuŭ с сuсmемamuческuмu ошuбкамuгaрaнmuрующuŭ uх оnmuмaлъную элuмuнaцuю в резулъmamaх. Докaзaно чmо нuкaкuм ¶rt;ругuм меmо¶rt;ом элuмuнaцuu сuсmемamuческuх влuянuŭ не возможно nолучumя лучщuе резульmamы.SummaryA procedure for processing measurements subject to systematic effects is derived such that it ensures their optimum elimination in the results of the processing. It is proved that no other method of eliminating systematic effects yields better results.

Universal model for adjusting observed values

SummaryFundamental models of the calculus of observations. Regularity and singularity of models. Universal model. Unbiased estimable and unbiased unestimable functions of the model parameters.

On the stochastic relations between the points of a geodetic network

SummaryThree important characteristics of the stochastic relation between the position of one point of a geodetic network and the positions of the remaining points are investigated: a) Change of the ellipse of confidence of the position of a point of the network to the ellipse of confidence of the position of a point of the network, relative to any other point of the network. (This characteristic is called the change of the global ellipse of confidence to the relative ellipse). b) The regression between the positional vector of the investigated point and the positional vectors of some group of other points of the network and c) the so-called total correlation between any two points of the network. The latter characteristic is a question of determining a characteristic direction for each pair of investigated points, which would have the property that the correlation coefficient between the random displacement of the first point in its characteristic direction and the random displacement of the second point in its characteristic direction is maximum.

Design of Experiment for Measurement of Langevin Function

Design of Experiment for Measurement of Langevin Function The presented study focuses on a confrontation of the theory of regression models and theory of experiment with the real situation of determining properties of magnetic (nano)materials. Their magnetic properties can be deduced by measuring their magnetization, being the fundamental magnetic quantity of an arbitrary (nano)material. The results of the magnetization measurements determine the unknown parameters of a known nonlinear function that characterizes the (nano)material under investigation. Knowledge of the values of the uknkown parameters enables to decide whether the (nano)material is suitable or not for a particular application. Thus, in this work, we present a possible approach how to estimate the unknown parameters of the nonlinear function by the regression models, taking into account a relevant linearization criterion. Then, we suggest an appropriate design for the measurement to get better estimators of the parameters.

Multivariate regression model with constraints

The aim of the paper is to present explicit formulae for parameter estimators and confidence regions in multivariate regression model with different kind of constraints and to give some comments to it. The covariance matrix of observation is either totally known, or some unknown parameters of it must be estimated, or the covariance matrix is totally unknown.