Power of the Spacing test for Least-Angle Regression

Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac-Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the spacing test to the frame of unknown noise variance. More precisely, we investigate the power of the spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the spacing test for LARS and the Pearson's chi-squared test (goodness of fit).Comment: 22 pages, 8 figure

A robust partial least squares method with applications

Partial least squares regression (PLS) is a linear regression technique developed to relate many regressors to one or several response variables. Robust methods are introduced to reduce or remove the effect of outlying data points. In this paper we show that if the sample covariance matrix is properly robustified further robustification of the linear regression steps of the PLS algorithm becomes unnecessary. The robust estimate of the covariance matrix is computed by searching for outliers in univariate projections of the data on a combination of random directions (Stahel-Donoho) and specific directions obtained by maximizing and minimizing the kurtosis coefficient of the projected data, as proposed by Peña and Prieto (2006). It is shown that this procedure is fast to apply and provides better results than other procedures proposed in the literature. Its performance is illustrated by Monte Carlo and by an example, where the algorithm is able to show features of the data which were undetected by previous methods

Nonlinear aerodynamic global model indentification using Gram-Schmidt orthogonalisation13;

This paper discusses a simple technique to identify global models for nonlinear aerodynamic force and moment coefficients of aircraft using multivariate orthogonal functions. Classical Gram-Schmidt procedure and Predicted Squared Error metric are used to generate the orthogonal functions. Global models for the F-16 aircraft are identified from a simplified subsonic (Mach less than 0.6) wind tunnel database available in open literature. The identified models are compared with those found in literature for the same wind tunnel database and conclusions are drawn.