Wavelet-based nonparametric functional mapping of longitudinal curves

Functional mapping based on parametric and nonparametric modeling of functional data can estimate the developmental pattern of genetic effects on a complex dynamic or longitudinal process triggered by quantitative trait loci (QTLs). But existing functional mapping models have a limitation for mapping dynamic QTLs with irregular functional data characterized by many local features, such as peaks. We derive a statistical model for QTL mapping of longitudinal curves of any form based on wavelet shrinkage techniques. The fundamental idea of this model is a repeated splitting of an initial sequence into detail coefficients that quantify local fluctuations at a particular scale and smooth coefficients that quantify remaining low-frequency variation in the signal after the high-frequency detail is removed and, subsequently, QTL mapping with the smooth coefficients extracted from noisy longitudinal data. Compared with conventional full-dimensional functional mapping, wavelet-based nonparametric functional mapping provides consistent results, and better results in some circumstances, and is much more computationally efficient. This wavelet-based model is validated by the analysis of a real example for stem diameter growth trajectories in a forest tree, and its statistical properties are examined through extensive simulation studies. Wavelet-based functional mapping broadens the use of functional mapping to studying an arbitrary form of longitudinal curves and will have many implications for investigating the interplay between gene actions/interactions and developmental pathways in various complex biological processes and networks.