Standardization of multivariate Gaussian mixture models and background adjustment of PET images in brain oncology
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Given observations from a multivariate Gaussian mixture model plus outliers, this paper addresses the question of how to standardize the mixture to a standard multivariate normal distribution, so that the outliers can be detected using a statistical test. This question is motivated by an image analysis problem in brain oncology of detecting changes between a post-treatment Positron Emission Tomography (PET) scan, where background adjustment is necessary to reduce confounding by tissue-dependent changes not related to the disease. When modeling the voxel intensities for the two scans as a bivariate Gaussian mixture, background adjustment translates into standardizing the mixture at each voxel, while tumor lesions present themselves as outliers to be detected. For general multivariate Gaussian mixtures, we show theoretically and numerically that the tail distribution of the standardized scores is favorably close to standard normal in a wide range of scenarios while being conservative at the tails, validating voxelwise hypothesis testing based on standardized scores. To address standardization in spatially heterogeneous data, we propose a spatial and robust multivariate expectation-maximization (EM) algorithm, where prior class membership probabilities are provided by transformation of spatial probability template maps and the estimation of the class mean and covariances are robust to outliers. Simulations in both univariate and bivariate cases suggest that standardized scores with soft assignment have tail probabilities that are either very close to or more conservative than standard normal. The proposed methods are applied to a real data set from a PET phantom experiment, yet they are generic and can be used in other contexts.
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