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Remark on properties of bases for additive logratio transformations of compositional data

Karel Hron — 2008

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The statistical analysis of compositional data, multivariate data when all its components are strictly positive real numbers that carry only relative information and having a simplex as the sample space, is in the state-of-the-art devoted to represent compositions in orthonormal bases with respect to the geometry on the simplex and thus provide an isometric transformation of the data to an usual linear space, where standard statistical methods can be used (e.g. [2], [4], [5], [9]). However, in some...

Inversion of 3 × 3 partitioned matrices in investigation of the twoepoch linear model with the nuisance parameters

Karel Hron — 2006

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The estimation procedures in the multiepoch (and specially twoepoch) linear regression models with the nuisance parameters that were described in [2], Chapter 9, frequently need finding the inverse of a 3 × 3 partitioned matrix. We use different kinds of such inversion in dependence on simplicity of the result, similarly as in well known Rohde formula for 2 × 2 partitioned matrix. We will show some of these formulas, also methods how to get the other formulas, and then we applicate the formulas in estimation...

Estimation of the first order parameters in the twoepoch linear model

Karel Hron — 2007

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The linear regression model, where the mean value parameters are divided into stable and nonstable part in each of both epochs of measurement, is considered in this paper. Then, equivalent formulas of the best linear unbiased estimators of this parameters in both epochs using partitioned matrix inverse are derived.

Analytical representation of ellipses in the Aitchison geometry and its application

Karel Hron — 2009

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Compositional data, multivariate observations that hold only relative information, need a special treatment while performing statistical analysis, with respect to the simplex as their sample space ([Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London, 1986.], [Aitchison, J., Greenacre, M.: Biplots of compositional data. Applied Statistics 51 (2002), 375–392.], [Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V. (eds): Compositional data analysis in the geosciences:...

On the Equivalence between Orthogonal Regression and Linear Model with Type-II Constraints

Sandra Donevska;  Eva Fišerová;  Karel Hron — 2011

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Orthogonal regression, also known as the total least squares method, regression with errors-in variables or as a calibration problem, analyzes linear relationship between variables. Comparing to the standard regression, both dependent and explanatory variables account for measurement errors. Through this paper we shortly discuss the orthogonal least squares, the least squares and the maximum likelihood methods for estimation of the orthogonal regression line. We also show that all mentioned approaches...

Covariance Structure of Principal Components for Three-Part Compositional Data

Klára Hrůzová;  Karel Hron;  Miroslav Rypka;  Eva Fišerová — 2013

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Statistical analysis of compositional data, multivariate observations carrying only relative information (proportions, percentages), should be performed only in orthonormal coordinates with respect to the Aitchison geometry on the simplex. In case of three-part compositions it is possible to decompose the covariance structure of the well-known principal components using variances of log-ratios of the original parts. They seem to be helpful for the interpretation of these special orthonormal coordinates....

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