arXiv subjects

Jitka Machalová

Publications and source records attributed to Jitka Machalová.

6 recordsLinked to original sources

arXiv2026

Penalized likelihood estimation of probability density functions using compositional splines

Probability density functions are commonly estimated through preliminary smoothing or aggregation procedures, e.g., histograms or kernel density estimation, before subsequent functional representation and functional data analyses. Such a two-stage approach can lead to additional approximation bias and weaken the direct connection between the observed data and the underlying distributional structure. In this paper, we propose a penalized maximum likelihood framework for direct estimation of probability density functions from raw observations within the framework of Bayes Hilbert spaces while preserving the compositional geometry of densities. The methodology is based on the centered log-ratio (clr) transformation, an isometric isomorphism between Bayes Hilbert spaces and the standard Lebesgue space of square integrable functions with zero integral, enabling efficient spline representations. The clr transformed densities are represented using ZB-spline basis functions, while their smoothness is controlled through quadratic penalties imposed on the spline coefficients. The proposed framework is developed for univariate and bivariate densities. In the latter case, it naturally incorporates the orthogonal decomposition into independent and interactive parts together with the corresponding geometric marginals. The performance is evaluated in a simulation study involving multiple complex scenarios and compared with kernel smoothing. Finally, the applicability of the framework is illustrated using empirical geochemical data.

stat.ME↗

arXiv2026

Compositional Periodic Spline Approximation for Circular Density Data in Bayes Spaces

This paper proposes a novel framework for the approximation and analysis of circular density data using compositional periodic splines within Bayes spaces with the Hilbert space structure. By applying the centered log-ratio transformation, densities are represented in a subspace of the standard $L^2$ space of real-valued functions, which enables the use of functional data analysis tools while preserving the relative nature of distributions and their periodic structure. A coefficient-based construction of periodic splines with a zero-integral constraint is developed, together with matrix formulations for both smoothing splines and penalized splines, allowing efficient estimation and implementation. The methodology is applied to long-term wind direction data, where it provides smooth and interpretable density estimates and supports further statistical analysis, including functional regression. The results demonstrate the practical relevance of the proposed approach and its potential for extensions to more complex density-valued data.

stat.ME↗

arXiv2024

Approximation of bivariate densities with compositional splines

Reliable estimation and approximation of probability density functions is fundamental for their further processing. However, their specific properties, i.e. scale invariance and relative scale, prevent the use of standard methods of spline approximation and have to be considered when building a suitable spline basis. Bayes Hilbert space methodology allows to account for these properties of densities and enables their conversion to a standard Lebesgue space of square integrable functions using the centered log-ratio transformation. As the transformed densities fulfill a zero integral constraint, the constraint should likewise be respected by any spline basis used. Bayes Hilbert space methodology also allows to decompose bivariate densities into their interactive and independent parts with univariate marginals. As this yields a useful framework for studying the dependence structure between random variables, a spline basis ideally should admit a corresponding decomposition. This paper proposes a new spline basis for (transformed) bivariate densities respecting the desired zero integral property. We show that there is a one-to-one correspondence of this basis to a corresponding basis in the Bayes Hilbert space of bivariate densities using tools of this methodology. Furthermore, the spline representation and the resulting decomposition into interactive and independent parts are derived. Finally, this novel spline representation is evaluated in a simulation study and applied to empirical geochemical data.

stat.ME↗

arXiv2024

Efficient spline orthogonal basis for representation of density functions

Probability density functions form a specific class of functional data objects with intrinsic properties of scale invariance and relative scale characterized by the unit integral constraint. The Bayes spaces methodology respects their specific nature, and the centred log-ratio transformation enables processing such functional data in the standard Lebesgue space of square-integrable functions. As the data representing densities are frequently observed in their discrete form, the focus has been on their spline representation. Therefore, the crucial step in the approximation is to construct a proper spline basis reflecting their specific properties. Since the centred log-ratio transformation forms a subspace of functions with a zero integral constraint, the standard $B$-spline basis is no longer suitable. Recently, a new spline basis incorporating this zero integral property, called $Z\!B$-splines, was developed. However, this basis does not possess the orthogonal property which is beneficial from computational and application point of view. As a result of this paper, we describe an efficient method for constructing an orthogonal $Z\!B$-splines basis, called $Z\!B$-splinets. The advantages of the $Z\!B$-splinet approach are foremost a computational efficiency and locality of basis supports that is desirable for data interpretability, e.g. in the context of functional principal component analysis. The proposed approach is demonstrated on an empirical demographic dataset.

stat.ME↗

arXiv2020

Bivariate Densities in Bayes Spaces: Orthogonal Decomposition and Spline Representation

A new orthogonal decomposition for bivariate probability densities embedded in Bayes Hilbert spaces is derived. It allows one to represent a density into independent and interactive parts, the former being built as the product of revised definitions of marginal densities and the latter capturing the dependence between the two random variables being studied. The developed framework opens new perspectives for dependence modelling (which is commonly performed through copulas), and allows for the analysis of dataset of bivariate densities, in a Functional Data Analysis perspective. A spline representation for bivariate densities is also proposed, providing a computational cornerstone for the developed theory.

math.ST↗

arXiv2019

Some comments on Gao beam model

In this small comment mathematical formulations concerning the nonlinear beam model published in [1] are analyzed. The beam is subjected to vertical and axial loading (at its right end). This nonlinear model can be used to study post-buckling problems. Unfortunately, some inconsistency between pure bending and pure buckling problems was discovered by the authors of this comment. This is concerned with definition of an integral constant, which is not in [1] strictly determined. In this comment there is proposed the adjustment of of this constant which should solve the mentioned troubles.

physics.class-ph↗