2016/08/29 by Esa Ollila, Ollila, Esa, Ilya Soloveychik +5
Mathematics · #Morphological variations and asymmetry #Point processes and geometric inequalities #Advanced Statistical Methods and Models
paper · pdf · doi:10.48550/arxiv.1608.08126
A common assumption when sampling p-dimensional observations from K\ndistinct group is the equality of the covariance matrices. In this paper, we\npropose two penalized M-estimation approaches for the estimation of the\ncovariance or scatter matrices under the broader assumption that they may\nsimply be close to each other, and hence roughly deviate from some positive\ndefinite "center". The first approach begins by generating a pooled\nM-estimator of scatter based on all the data, followed by a penalised\nM-estimator of scatter for each group, with the penalty term chosen so that\nthe individual scatter matrices are shrunk towards the pooled scatter matrix.\nIn the second approach, we minimize the sum of the individual group\nM-estimation cost functions together with an additive joint penalty term\nwhich enforces some similarity between the individual scatter estimators, i.e.\nshrinkage towards a mutual center. In both approaches, we utilize the concept\nof geodesic convexity to prove the existence and uniqueness of the penalized\nsolution under general conditions. We consider three specific penalty functions\nbased on the Euclidean, the Riemannian, and the Kullback-Leibler distances. In\nthe second approach, the distance based penalties are shown to lead to\nestimators of the mutual center that are related to the arithmetic, the\nRiemannian and the harmonic means of positive definite matrices, respectively.\nA penalty based on an ellipticity measure is also considered which is\nparticularly useful for shape matrix estimators. Fixed point equations are\nderived for each penalty function and the benefits of the estimators are\nillustrated in regularized discriminant analysis problem.\n