On the Co-Inertia(-PLS) analysis with linear constraints

This paper presents a theoretical framework based on Co-Inertia Analysis (Chessel and Mercier, in: Lebreton, Asselain (eds) Biométrie et Environnement, Masson, Paris, 1993), which has been enhanced by incorporating the Restricted Eigenvalue Problem principle (Rao in Linear statistical inference and its application, Wiley, New York, 1973). This modified framework includes external information as a linear restrictions on parameters from both sets of variables, introducing two new constrained methods. One method is an extension of Wold’s two-block “Mode A” Partial Least Squares (Wegelin in A survey of partial least squares (PLS) methods with emphasis on the two-block case, 2000; Wold, in: Joreskog, Wold (eds) Systems under indirect observations: part II, North-Holland, Amsterdam, 1982; in: Kotz, Johnson (eds) Encyclopedia of statistical sciences, Wiley, New York, 1985), while the other applies linear constraints to the coefficients of Partial Least Squares Regression (Höskuldsson in J Chemom 2:211–228, 1988; Tenenhaus in La Régression PLS: Théorie et pratique, 1998) that is a popular technique that generalizes and combines features from principal component analysis and multiple regression. The paper also discusses the mathematical and geometrical properties of these approaches.