Wavelet Functional ANOVA, Bayesian False Discovery Rate, and Longitudinal Measurements of Oxygen Pressure in Rats

In conventional statistical practice, an observation is usually a number or a vector. But in many situations, observed values are curves or vectors of curves. Prototypical examples are growth curves (e.g., measurements of height and weight in children at particular age times), brain potentials, and a variety of responses in biological, chemical, and geophysical measurements. A vibrant research in this area is summarized in the monograph by Ramsay and Silverman (1997). Two characteristics are common to any functional data analysis (FDA): a strong link with the multivariate statistical paradigm and the need for regularization. The strong link with multivariate statistics arises from the fact that methods, such as principal component analysis, multivariate linear modeling, canonical correlation analysis, etc. can be applied within the functional data analysis framework. Function values , where is continuous, are observed at discrete time points. In the process of analysis, the integrals are replaced by discrete quadrature-based approximations, methods and the final conclusions are usually in terms of weighted multivariate analysis measures. Regularization in FDA consists of assuming a particular class of smooth functions for the estimators. The implementation is carried out in various ways. For example, one can penalize roughness as part of the fitting criteria or use particular representations that have inherent smoothness (e.g., splines, wavelets, neural networks). Often there are links between magnitudes of coefficients in a particular representation and the regularity features of the represented objects (wavelets, pursuit methods). In making an inference in longitudinal functional data models, two problems are of particular concern for the inference maker. The most important problem is dimensionality. Explanatory variables are sampled curves, and