Statistical Learning, Sustainability and Impact Evaluation

We consider the case of a multivariate random vector that obeys a linear mixed model when the vector itself lies in a lower-dimensional subspace. This situation suggests that this subspace can be modeled by the probabilistic (random effects) principal components. Because of this, the random vector follows at the same time two different models. We employ a linear predictor adjusted by the residual part of the probabilistic principal components that results not explained by the linear model. The new predictor can be regarded as the vector of scores by considering the loading matrix of probabilistic principal components, enhanced by the linear mixed model. Due to the concomitant application of both models, the positive definiteness of the isotropic error variance matrix determines the maximum number of the principal components to retain. The application to the official Italian well-being data shows some features of the method.