Dr Frederic Ferraty, University of Toulouse, France

Most-predictive design points for functional data predictors

(In coll. with Peter Hall and Phillipe Vieu)

Functional data analysis (FDA) has found application in a great many fields, including biology, chemometrics, econometrics, geophysics, medical sciences, pattern recognization....  For instance a sample of curves or a sample of surfaces is a special case of functional data.  In the example of near infrared (NIR) spectroscopy, X(t) denotes the absorbance of the NIR spectrum at wavelength t.  The observation of X(t) for a discrete but large set of values (or design points) t produces what is called a spectrometric curve.  A standard chemometrical dataset is that where X(t) corresponds to the NIR spectrum of a piece of meat and where a scalar response Y denotes a constituent of the piece of meat (eg, fat or moisture).

Here, we are interested in regressing a scalar response Y on a functional predictor X(t) where t belongs to a discrete but large set I of "design points" (hundreds or thousands).  From now on, one sets X:={X(t);t in I}.  It is of practical interest to know which design points of t, have greatest influence on the response, Y.  In this situation, we propose a method for choosing a small subset of design points to optimize prediction of a response variable, Y.  The selected design points are referred to as the most predictive design points, or covariates, and are computed using information contained in a set of independent observations (X_i,Y_i) of (X,Y).  The algorithm is based on local linear regression, and calculations can be accelerated using linear regression to preselect design points.  Boosting can be employed to further improve predictive performance.  We illustrate the usefulness of our ideas through examples drawn from chemometrics, and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings.