Presented by Kojo Essuman Ackah Spring 2018 STA 6557 Project

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Presentation transcript:

Presented by Kojo Essuman Ackah Spring 2018 STA 6557 Project Statistical Analysis on High-Dimensional Spheres and Shape Spaces By Ian L. Dryden (University of Nottingham) (Annals. Of Statistics 33, p.1643-1665) Presented by Kojo Essuman Ackah Spring 2018 STA 6557 Project

Contents Applications on High-Dimensional Data Bingham Distribution Practical application: Brain shape model Discussion Conclusion

Applications on High-Dimensional Data High-dimensional observation Xp on unit sphere in p real dimensions . Model Xp as and observation tends to a generalized function. Obtain appropriate probability distributions and statistical inference.

Bingham Distribution Modeling on ,we need useful non-uniform distributions. The Bingham family ( ) of distributions on has probability measure where , , , is the volume measure on the sphere. If , then the mode of the distribution is . is regarded as the (j -1) st PC.

Bingham Distribution For large p and under some assumptions, (*) Let be a random sample with Bingham distribution, the MLE of is approximately by (*), which has eigenvectors corresponding to eigenvalues .

Bingham Distribution ,using spectral decomposition, and , then and . The MLE for the mode of the distribution is ,and is an estimate of the concentration of the mode. The sample eigenvector is the (j-1) st sample PC with estimated variance .

Practical application: Brain shape model A sample of 74 magnetic resonance of adult brains is taken. Magnetic resonance imaging (MRI) is primarily used in medical imaging to visualize the structure and function of the body. Shape of modal (the most frequent value in a distribution) cortical surface of the brain is of interest.

Practical application: Brain shape model Restrict the analysis to the p=62,501 points on the cortical surface along a hemisphere of rays which radiate from the origin at a central landmark. Goal: obtain an estimate of the modal cortical shape and the principal components of shape variability for the dataset.

Practical application: Brain shape model The measurement taken for the ith brain (i=1,…,n) are , which are the lengths of the rays. Remove the scale information by taking . =0.99885 shows the data are extremely concentrated, with a high contribution from the first eigenvector.

Plots of Modal Cortical Surface This PC explains of the variability about the mode.

Plots of Modal Cortical Surface This PC explains of the variability about the mode.

Discussion The noise models considered in the paper should have further applications in addition to those in high-dimensional directional data analysis and shape analysis. For example, the work could be used to model noise in (high-dimensional) images where the parameters of the noise process would depend on the particular imaging modality and the object(s) in the image.

Conclusion An improved analysis would be to locate points at more accurate points of biological homology, then the mean shape and PC would give more accurate estimates of population properties of the cortical surfaces. It is of interest to extend the work to other manifolds, such as the Stiefel manifold of orthonormal frames and the Grassmann manifold. Watson (1983) provides some asymptotic high- dimensional results, but the study of probability distributions in the continuous limit requires further developments.