A0293
Title: Approximation of bivariate densities with compositional splines
Authors: Stanislav Skorna - Palacky University Olomouc (Czech Republic) [presenting]
Karel Hron - Palacky University (Czech Republic)
Jitka Machalova - Palacky University (Czech Republic)
Jana Burkotova - Palacky University (Czech Republic)
Abstract: Multivariate densities occur often as a result of the aggregation of data in many applications. They are used to analyze the association structure and to further process them using methods of functional data analysis. For the purpose of further statistical analysis, proper spline (continuous) representation of the input discrete data is crucial. Bayes Hilbert spaces methodology enables to capture specific features of probability density functions and to construct so-called compositional splines which respect their decomposition into interactive and independent parts. Centered log-ratio, a key tool of this methodology, enables to represent the original densities (and compositional spline as as their estimates) in the standard $L^{2}$ space by $ZB$-spline representation. The resulting spline functions fulfill zero-integral condition, which must be taken into account already when building the basis of the $ZB$-spline representation. Basis can be built using standard $B$-spline basis with implemented zero-integral constraint or using the $ZB$-spline basis, which satisfies the zero-integral constraint automatically. We focus on the latter case, provide a detailed simulation study and apply the resulting spline representation for descriptive analysis of geochemical density data.
