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Partial least squares (PLS) regression on an L2-continuous stochastic process is an extension of the finite set case of predictor variables. The PLS components existence as eigenvectors of some operator and convergence properties of the PLS approximation are proved. The results of an application to stock-exchange data will be compared with those obtained by other methods.

It does not seem usual to perform a linear regression when the number of predictors is infinite. However, it is the case when one tries to predict a response variable Y thanks to the observation of a time-dependent variable Xt, for any t∈[0,T] (for example, (Xt)t∈[0,T] can represent temperature curves observed in n places and Y the amount of crops). Theoretically, this can be expressed by the regression of Y on the process (Xt)t∈[0,T]. The aim of this paper is to adapt the PLS regression when the set of predictor variables forms a stochastic process Fig. 1. The problems brought about by the classical linear regression on a process—the indetermination of the regression coefficients (Ramsay and Dalzell, 1991; Ramsay and Silverman, 1997; Saporta, 1981) or the choice of the principal components of (Xt)t∈[0,T] as predictor variables ( Deville, 1978; Saporta, 1981; Aguilera et al., 1998)—get satisfactory solutions within this framework, the main characteristics which are derived from those of the Escoufier operator associated with the process (Xt)t∈[0,T].PLS regression on a stochastic process is an extension of the finite case (finite set of predictors) developed by Wold et al. (1984), Tenenhaus et al. (1995) and Cazes (1997) (see also Eldén, 2003; Nguyen and Rocke, 2003). We prove the existence of PLS components as well as some convergence properties towards the classical linear regression. The case Y=(Xt)t∈[T,T+a], a>0, presents an alternative to prediction methods proposed by Aguilera et al. (1998) and Deville (1978). The results of an application on stock exchange data are compared with those obtained by other methods.

The PLS regression on a stochastic process offers an alternative to regression on principal components. It gives a solution to the problems of multicollinearity of predictors and when the number of observations is smaller than the number of predictor variables, which is often the case in this context.