95 research outputs found
Lognormal Distributions and Geometric Averages of Positive Definite Matrices
This article gives a formal definition of a lognormal family of probability
distributions on the set of symmetric positive definite (PD) matrices, seen as
a matrix-variate extension of the univariate lognormal family of distributions.
Two forms of this distribution are obtained as the large sample limiting
distribution via the central limit theorem of two types of geometric averages
of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric
average. These averages correspond to two different geometries imposed on the
set of PD matrices. The limiting distributions of these averages are used to
provide large-sample confidence regions for the corresponding population means.
The methods are illustrated on a voxelwise analysis of diffusion tensor imaging
data, permitting a comparison between the various average types from the point
of view of their sampling variability.Comment: 28 pages, 8 figure
Multiple Testing of Local Maxima for Detection of Peaks in Random Fields
A topological multiple testing scheme is presented for detecting peaks in
images under stationary ergodic Gaussian noise, where tests are performed at
local maxima of the smoothed observed signals. The procedure generalizes the
one-dimensional scheme of Schwartzman et al. (2011) to Euclidean domains of
arbitrary dimension. Two methods are developed according to two different ways
of computing p-values: (i) using the exact distribution of the height of local
maxima (Cheng and Schwartzman, 2014), available explicitly when the noise field
is isotropic; (ii) using an approximation to the overshoot distribution of
local maxima above a pre-threshold (Cheng and Schwartzman, 2014), applicable
when the exact distribution is unknown, such as when the stationary noise field
is non-isotropic. The algorithms, combined with the Benjamini-Hochberg
procedure for thresholding p-values, provide asymptotic strong control of the
False Discovery Rate (FDR) and power consistency, with specific rates, as the
search space and signal strength get large. The optimal smoothing bandwidth and
optimal pre-threshold are obtained to achieve maximum power. Simulations show
that FDR levels are maintained in non-asymptotic conditions. The methods are
illustrated in a nanoscopy image analysis problem of detecting fluorescent
molecules against the image background.Comment: 30 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1203.306
Standardization of multivariate Gaussian mixture models and background adjustment of PET images in brain oncology
In brain oncology, it is routine to evaluate the progress or remission of the
disease based on the differences between a pre-treatment and a post-treatment
Positron Emission Tomography (PET) scan. 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.
In this paper, we address the question of how to standardize the mixture to a
standard multivariate normal distribution, so that the outliers (i.e., tumor
lesions) can be detected using a statistical test. 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 image 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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