In many multivariate statistical techniques, a set of linear functions of the original p variables is produced. One of the more dif cult aspects of these techniques is the interpretation of the linear functions, as these functions usually have nonzero coef cients on all p variables. A common approach is to effectively ignore (treat as zero) any coef cients less than some threshold value, so that the function becomes simple and the interpretation becomes easier for the users. Such a procedure can be misleading. There are alternatives to principal component analysis which restrict the coef cients to a smaller number of possible values in the derivationof the linear functions,or replace the principalcomponentsby "principal variables." This article introduces a new technique, borrowing an idea proposed by Tibshirani in the context of multiple regression where similar problems arise in interpreting regression equations. This approach is the so-called LASSO, the "least absolute shrinkage and selection operator," in which a bound is introduced on the sum of the absolute values of the coef cients, and in which some coef cients consequently become zero. We explore some of the propertiesof the new technique,both theoreticallyand using simulation studies, and apply it to an example.

Empirical orthogonal functions (EOFs) are widely used in climate research to identify dominant patterns of variability and to reduce the dimensionality of climate data. EOFs, however, can be difficult to interpret. Rotated empirical orthogonal functions (REOFs) have been proposed as more physical entities with simpler patterns than EOFs. This study presents a new approach for finding climate patterns with simple structures that overcomes the problems encountered with rotation. The method achieves simplicity of the patterns by using the main properties of EOFs and REOFs simultaneously. Orthogonal patterns that maximise variance subject to a constraint that induces a form of simplicity are found. The simplified empirical orthogonal function (SEOF) patterns, being more 'local', are constrained to have zero loadings outside the main centre of action. The method is applied to winter Northern Hemisphere (NH) monthly mean sea level pressure (SLP) reanalyses over the period 1948-2000. The 'simplified' leading patterns of variability are identified and compared to the leading patterns obtained from EOFs and REOFs.

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The complexity inherent in climate data makes it necessary to introduce more than one statistical tool to the researcher to gain insight into the climate system. Empirical orthogonal function (EOF) analysis is one of the most widely used methods to analyze weather/climate modes of variability and to reduce the dimensionality of the system. Simple structure rotation of EOFs can enhance interpretability of the obtained patterns but cannot provide anything more than temporal uncorrelatedness. In this paper, an alternative rotation method based on independent component analysis (ICA) is considered. The ICA is viewed here as a method of EOF rotation. Starting from an initial EOF solution rather than rotating the loadings toward simplicity, ICA seeks a rotation matrix that maximizes the independence between the components in the time domain. If the underlying climate signals have an independent forcing, one can expect to find loadings with interpretable patterns whose time coefficients have properties that go beyond simple noncorrelation observed in EOFs. The methodology is presented and an application to monthly means sea level pressure (SLP) field is discussed. Among the rotated (to independence) EOFs, the North Atlantic Oscillation (NAO) pattern, an Arctic Oscillation-like pattern, and a Scandinavian-like pattern have been identified. There is the suggestion that the NAO is an intrinsic mode of variability independent of the Pacific.

Abstract. Two data analysis problems, the orthonormal Procrustes problem and the Penrose regression problem, are reconsidered in this paper. These problems are known in the literature for their importance as well as di culty. This work presents a way to calculate the projected gradient and the projected Hessian explicitly. One immediate result of this calculation is the complete characterization of the rst order and the second order optimality conditions for both problems. Another application is the natural formulation of a continuous steepest descent o w that can serve as a globally convergent numerical method. Applications are demonstrated by n umerical examples.