The von Mises-Fisher Procrustes model in functional Magnetic Resonance Imaging data
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The Procrustes method allows to align matrices into a common space using similarity transformation. However, it is an ill-posed problem, i.e., it doesn't return a unique solution about the optimal transformation, losing results interpretability. For that, we extend the Perturbation model, which rephrases the Procrustes method as a statistical model, defining the von Mises-Fisher Procrustes model. The extension is focused on specifying a regularization term, using a proper prior distribution for the orthogonal matrix parameter of the Perturbation model. The von Mises-Fisher distribution is then utilized to insert prior information about the final common space structure. Thanks to that, we resolve the no-uniqueness problem of the Procrustes method, getting an interpretable estimator for the orthogonal matrix transformation. Being a conjugate prior, the posterior parameter is a sort of weighted average between the maximum likelihood and prior estimator. Application on functional Magnetic Resonance Imaging data shows an improvement in the group-level analysis in terms of inference and interpretability of the results. In this case, the prior information used is the three-dimensional voxel coordinates. It permits the construction of the location matrix parameter of the von Mises-Fisher distribution as a similarity euclidean matrix. In this way, we can exploit the idea that the orthogonal matrix must combine spatially close variables, i.e., voxels. The resulting orthogonal estimators reflect the three-dimensional structure of the voxel's space, as the final group-analysis results.
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