Functional principal component analysis via regularized Gaussian basis expansions and its application to unbalanced data

Abstract

This paper introduces regularized functional principal component analysis for multidimensional functional data sets, utilizing Gaussian basis functions. An essential point in a functional approach via basis expansions is the evaluation of the matrix for the integral of the product of any two bases (cross product matrix). Advantages of the use of the Gaussian type of basis functions in the functional approach are that its cross product matrix can be easily calculated, and it creates a much more flexible instrument for transforming each individual's observation into a functional form. The proposed method is applied to the analysis of three-dimensional (3D) protein structural data that can be referred to as unbalanced data. It is shown that our method extracts useful information from unbalanced data. Numerical experiments are conducted to investigate the effectiveness of our method via Gaussian basis functions, comparing to the method based on B-splines. On performing regularized functional principal component analysis with B-splines, we also derive the exact form of its cross product matrix. The numerical results show that our methodology is superior to that based on B-splines for unbalanced data.

Functional zoning for air quality

“…The choice of B-splines is classic for nonperiodic data, but recently Kayano and Konishi (2009) have proposed to use Gaussian basis expansion instead of B-splines since they deal with unbalanced data (time series observed at possibly different time points). Our pollutant data are already balanced because, as we said before, through the aggregation functions (provided by air quality EU directives) we obtain daily values.…”

A note on variable selection in functional regression via random subspace method

“…The formula of cross product matrix for Gaussian basis is presented for example in Matsui and Konishi (2011). For computing the cross product matrix for the B-spline basis, the procedure in Kayano and Konishi (2009) can be used. The cross product matrix for these as well as other bases can be approximated by using the function inprod from the R package fda (R Core Team 2017; Ramsay et al 2009Ramsay et al , 2017.…”

An Application of Functional Multivariate Regression Model to Multiclass Classification

“…In the next Section, this relation is used for multiclass classification for multivariate functional data. Other results of such type and their usage are presented, for instance, in Kayano and Konishi (2009), Matsui and Konishi (2011), Matsui (2014, Górecki et al (2015) and Collazos et al (2016).…”