Variance Of Eigenvalues Research Articles

Statistical artifacts in diffusion tensor MRI (DT-MRI) caused by background noise.

This work helps elucidate how background noise introduces statistical artifacts in the distribution of the sorted eigenvalues and eigenvectors in diffusion tensor MRI (DT-MRI) data. Although it was known that sorting eigenvalues (principal diffusivities) by magnitude introduces a bias in their sample mean within a homogeneous region of interest (ROI), here it is shown that magnitude sorting also introduces a significant bias in the variance of the sample mean eigenvalues. New methods are presented to calculate the mean and variance of the eigenvectors of the diffusion tensor, based on a dyadic tensor representation of eigenvalue-eigenvector pairs. Based on their use it is shown that sorting eigenvalues by magnitude also introduces a bias in the mean and the variance of the sample eigenvectors (principal directions). This required the development of new methods to calculate the mean and variance of the eigenvectors of the diffusion tensor, based on a dyadic tensor representation of eigenvalue-eigenvector pairs. Moreover, a new approach is proposed to order these pairs within an ROI. To do this, a correspondence between each principal axis of the diffusion ellipsoid, an eigenvalue-eigenvector pair, and a dyadic tensor constructed from it is exploited. A measure of overlap between principal axes of diffusion ellipsoids in different voxels is defined that employs projections between these dyadic tensors. The optimal eigenvalue assignment within an ROI maximizes this overlap. Bias in the estimate of the mean and of the variance of the eigenvalues and of their corresponding eigenvectors is reduced in DT-MRI experiments and in Monte Carlo simulations of such experiments. Improvement is most significant in isotropic regions, but some is also observed in anisotropic regions. This statistical framework should enhance our ability to characterize microstructure and architecture of healthy tissue, and help to assess its changes in development, disease, and degeneration. Mitigating these artifacts should also improve the characterization of diffusion anisotropy and the elucidation of fiber-tract trajectories in the brain and in other fibrous tissues. Magn Reson Med 44:41-50, 2000. Published 2000 Wiley-Liss, Inc.

Pca stability studied by the bootstrap and the infinitesimal jackknife method

General results are given about the asymptotic convergence of the Infinitesimal Jackknife estimation method. The development until the order four are given. The Infinitesimal Jackknife method is used to give stability index of the results of a PCA. The exact variance of a bootstrapped variance matrix is given. Infinitesimal Jackknife estimates are obtained for the bias and variances of eigenvalues, eigenvectors and stability index for a PCA on a variance or a correlation matrix. The Infinitesimal Jackknife method is compared with the bootstrap on real data

DYNAMICS AND EIGENVALUE ANALYSIS OF A RECTANGULAR PLATE WITH STOCHASTIC PROPERTIES

Determination of the mean and variance of eigenvalues of simply supported rectangular plates with stochastic properties is presented by applying the random perturbation techniques of Boyce. Then, determination of the mean and variance of the displacement of a simply supported uniform rectangular plate with uniformly distributed viscous damping having a random value to a distributed random excitation is studied. The technique of Boyce is extended to the inhomogeneous equation for the plate in the latter case. It was found that for free vibration, the distribution of the variance of natural frequency is obtained in exponential form for the given exponential correlation functions of the material properties.