Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

Linearly Supporting Feature Extraction For Automated Estimation Of Stellar Atmospheric Parameters

We describe a scheme to extract linearly supporting (LSU) features from stellar spectra to automatically estimate the atmospheric parameters $T_{eff}$, log$~g$, and [Fe/H]. "Linearly supporting" means that the atmospheric parameters can be accurately estimated from the extracted features through a linear model. The successive steps of the process are as follow: first, decompose the spectrum using a wavelet packet (WP) and represent it by the derived decomposition coefficients; second, detect representative spectral features from the decomposition coefficients using the proposed method Least Absolute Shrinkage and Selection Operator (LARS)$_{bs}$; third, estimate the atmospheric parameters $T_{eff}$, log$~g$, and [Fe/H] from the detected features using a linear regression method. One prominent characteristic of this scheme is its ability to evaluate quantitatively the contribution of each detected feature to the atmospheric parameter estimate and also to trace back the physical significance of that feature. This work also shows that the usefulness of a component depends on both wavelength and frequency. The proposed scheme has been evaluated on both real spectra from the Sloan Digital Sky Survey (SDSS)/SEGUE and synthetic spectra calculated from Kurucz's NEWODF models. On real spectra, we extracted 23 features to estimate $T_{eff}$, 62 features for log$~g$, and 68 features for [Fe/H]. Test consistencies between our estimates and those provided by the Spectroscopic Sarameter Pipeline of SDSS show that the mean absolute errors (MAEs) are 0.0062 dex for log$~T_{eff}$ (83 K for $T_{eff}$), 0.2345 dex for log$~g$, and 0.1564 dex for [Fe/H]. For the synthetic spectra, the MAE test accuracies are 0.0022 dex for log$~T_{eff}$ (32 K for $T_{eff}$), 0.0337 dex for log$~g$, and 0.0268 dex for [Fe/H].Comment: 21 pages, 7 figures, 8 tables, The Astrophysical Journal Supplement Series (accepted for publication

The wavelet-NARMAX representation : a hybrid model structure combining polynomial models with multiresolution wavelet decompositions

A new hybrid model structure combing polynomial models with multiresolution wavelet decompositions is introduced for nonlinear system identification. Polynomial models play an important role in approximation theory, and have been extensively used in linear and nonlinear system identification. Wavelet decompositions, in which the basis functions have the property of localization in both time and frequency, outperform many other approximation schemes and offer a flexible solution for approximating arbitrary functions. Although wavelet representations can approximate even severe nonlinearities in a given signal very well, the advantage of these representations can be lost when wavelets are used to capture linear or low-order nonlinear behaviour in a signal. In order to sufficiently utilise the global property of polynomials and the local property of wavelet representations simultaneously, in this study polynomial models and wavelet decompositions are combined together in a parallel structure to represent nonlinear input-output systems. As a special form of the NARMAX model, this hybrid model structure will be referred to as the WAvelet-NARMAX model, or simply WANARMAX. Generally, such a WANARMAX representation for an input-output system might involve a large number of basis functions and therefore a great number of model terms. Experience reveals that only a small number of these model terms are significant to the system output. A new fast orthogonal least squares algorithm, called the matching pursuit orthogonal least squares (MPOLS) algorithm, is also introduced in this study to determine which terms should be included in the final model

Forecasting the geomagnetic activity of the Dst Index using radial basis function networks

The Dst index is a key parameter which characterises the disturbance of the geomagnetic field in magnetic storms. Modelling of the Dst index is thus very important for the analysis of the geomagnetic field. A data-based modelling approach, aimed at obtaining efficient models based on limited input-output observational data, provides a powerful tool for analysing and forecasting geomagnetic activities including the prediction of the Dst index. Radial basis function (RBF) networks are an important and popular network model for nonlinear system identification and dynamical modelling. A novel generalised multiscale RBF (MSRBF) network is introduced for Dst index modelling. The proposed MSRBF network can easily be converted into a linear-in-the-parameters form and the training of the linear network model can easily be implemented using an orthogonal least squares (OLS) type algorithm. One advantage of the new MSRBF network, compared with traditional single scale RBF networks, is that the new network is more flexible for describing complex nonlinear dynamical systems