Least-squares approximation of a space distribution for a given covariance and latent sub-space

In this paper, a new method to approximate a data set by another data set with constrained covariance matrix is proposed. The method is termed Approximation of a DIstribution for a given COVariance (ADICOV). The approximation is solved in any projection subspace, including that of Principal Component Analysis (PCA) and Partial Least Squares (PLS). Given the direct relationship between covariance matrices and projection models, ADICOV is useful to test whether a data set satisfies the covariance structure in a projection model. This idea is broadly applicable in chemometrics. Also, ADICOV can be used to simulate data with a specific covariance structure and data distribution. Some applications are illustrated in an industrial case of study.