Multivariate linear systems for learning from data

Learning from data means a process of information extraction from finite size samples in order to estimate an unknown dependency. A series of problems can be modeled in terms of a system that computes according to an unknown rule a response (output) for each input. The paper provides a series of results concerning the learning from data a linear regressive model in a multivariate framework. The parameter estimates of the regressive model are determined using the maximum likelihood principle and the adaptive learning algorithms are derived using the gradient ascent technique. In the second section of the paper the parameters of the linear regressive model are determined by minimizing the arithmetic mean of square errors and an adaptive learning scheme of gradient descent type is also considered. We consider a probabilistic approach in the third section for modeling the effects of both the latent variables and noise. The cumulative effects of latent variables and noise are modeled in terms of multivariate Gaussian repartitions.