the best rank-r approximation of X with respect to the Frobenius norm. We write ∆r = X − Tr(X) for the ’residual’. In general, Tr(M) wil be used to denote the best rank-r approximation of a matrix M . Further, PM denotes the orthogonal projection on the subspace spanned by the columns of M , and we write M− for the Moore-Penrose pseudoinverse of a matrix M . The i-th column of M is denoted by M...

Data envelopment analysis (DEA) is an axiomatic, mathematical programming approach to productive efficiency analysis and performance measurement. This paper shows that DEA can be interpreted as a nonparametric least squares regression subject to shape constraints on production frontier and sign constraints on residuals. Thus, DEA can be seen as a nonparametric counter-part of the corrected ordi...

The semiparametric accelerated failure time model relates the logarithm of the failure time linearly to the covariates while leaving the error distribution unspecified. The present paper describes simple and reliable inference procedures based on the least-squares principle for this model with right-censored data. The proposed estimator of the vectorvalued regression parameter is an iterative s...

Total least squares (TLS) is a data modelling technique which can be used for many types of statistical analysis, e.g. a regression. In the regression setup, both dependent and independent variables are considered to be measured with errors. Thereby, the TLS approach in statistics is sometimes called an errors-invariables (EIV) modelling and, moreover, this type of regression is usually known a...

This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high-dimensional stochastic space in the context of limited amount of information. The proposed approach involves a procedure for the selection of an approximation basis and th...

Partial least squares (PLS) regression is a powerful and frequently applied technique in multivariate statistical process control when the process variables are highly correlated. Selection of the number of latent variables to build a representative model is an important issue. A metric frequently used by chemometricians for the determination of the number of latent variables is that of Wold’s ...

Partial least squares is a popular method for soft modelling in industrial applications. This paper introduces the basic concepts and illustrates them with a chemometric example. An appendix describes the experimental PLS procedure of SAS/STAT software.

A locally regularized orthogonal least squares (LROLS) algorithm is proposed for constructing parsimonious or sparse regression models that generalize well. By associating each orthogonal weight in the regression model with an individual regularization parameter, the ability for the orthogonal least squares model selection to produce a very sparse model with good generalization performance is g...

To characterize color values measured by color devices (e.g., scanners, color copiers, and color cameras) in a deviceindependent fashion these values must be transformed to colorimetric tristimulus values. The measured RGB 3-vectors are not a linear transformation away from such colorimetric vectors, however, but still the best transformation between these two data sets, or between RGB values m...

We compute a sparse solution to the classical least-squares problem minx ‖Ax−b‖2, where A is an arbitrary matrix. We describe a novel algorithm for this sparse least-squares problem. The algorithm operates as follows: first, it selects columns from A, and then solves a least-squares problem only with the selected columns. The column selection algorithm that we use is known to perform well for t...