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Principal component analysis (PCA) is widely used in data processing and dimensionality reduction. However, PCA suffers from the fact that each principal component is a linear combination of all the original variables, thus it is often difficult to interpret the results. We introduce a new method called sparse principal component analysis (SPCA) using the lasso (elastic net) to produce modified principal components with sparse loadings. We first show that PCA can be formulated as a regression-type optimization problem; sparse loadings are then obtained by imposing the lasso (elastic net) constraint on the regression coefficients. Efficient algorithms are proposed to fit our SPCA models for both regular multivariate data and gene expression arrays. We also give a new formula to compute the total variance of modified principal components. As illustrations, SPCA is applied to real and simulated data with encouraging results. [ABSTRACT FROM AUTHOR]

Principal curves are smooth one-dimensional curves that pass through the middle of a p-dimensional data set, providing a nonlinear summary of the data. They are nonparametric, and their shape is suggested by the data. The algorithm for constructing principal curves starts with some prior summary, such as the usual principal-component line. The curve in each successive iteration is a smooth or local average of the p-dimensional points, where the definition of local is based on the distance in arc length of the projections of the points onto the curve found in the previous iteration. In this article principal curves are defined, an algorithm for their construction is given, some theoretical results arc presented, and the procedure is compared to other generalizations of principal components. Two applications illustrate the use of principal curves. The first describes how the principal-curve procedure was used to align the magnets of the Stanford linear collider. The collider uses about 950 magnets in a roughly circular arrangement to bend electron and positron beams and bring them to collision. After construction, it was found that some of the magnets had ended up significantly out of place. As a result, the beams had to be bent too sharply and could not be focused. The engineers realized that the magnets did not have to be moved to their originally planned locations, but rather to a sufficiently smooth are through the middle of the existing positions. This arc was found using the principal curve procedure. In the second application, two different assays for gold content in several samples of computer chip waste appear to show some systematic differences that are blurred by measurement error. The classical approach using linear errors in variables regression can detect systematic linear differences but is not able to account for nonlinearities. When the first linear principal component is replaced with a... [ABSTRACT FROM AUTHOR]

Generalized additive models have the form eta (x) = alpha + SIGMA f[sub j](x[sub j]), where eta might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models eta(x) = beta'x, where eta(x) = g(mu(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions f[sub j](x[sub j]) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximate hypothesis tests. The local scoring algorithm is analogous to the iterative reweighted least squares algorithm for solving likelihood and nonlinear regression equations. At each iteration, an adjusted dependent variable is formed and an additive regression model is fit using the backfitting algorithm. The backfitting algorithm cycles through the variables and estimates each coordinated function by smoothing the partial residuals. [ABSTRACT FROM AUTHOR]

Discusses an alternative algorithm in which the smooth functions are estimated using smoothing spline. The generalized cross-validation (GSV) based techniques used to select the smoothing parameters and number of terms; Structure of the automatic smoothing spline projection pursuit (ASP) algorithm; Examples comparing ASP with the standard supersmoother-based PPR algorithm of Friedman.