Predicting precision matrices for color matching problem

In this paper, as data, ellipsoids in a color coordinate called the Commission Internationale de l’Eclairage (CIE)-Lab system are given as data for 19 colors. Each ellipsoid is a region where all points are visually recognized as the same color at the center of the coordinate system. Our aim here is to predict the shape of an ellipsoid whose center is given by a new color. We proposed two prediction methods of positive definite matrices determining ellipsoids. The first one is a nonparametric method with Gaussian kernel. The prediction is provided as a weighted sum of positive definite matrices corresponding to 19 ellipsoids in the training data. The second one is to use a matrix-valued regression model applied to a logarithm of positive definite matrices where explanatory variables are three elements of color centers. The best result was obtained by the nonparametric methods with three bandwidth parameters. The log normal regression had a weaker performance, but even so the model estimation was easily carried out.