Department of Mathematics and Statistics Graduate Projects
http://hdl.handle.net/1974/6158
2016-12-05T14:38:18ZDegeneracy of velocity constraints in rigid body systems
http://hdl.handle.net/1974/14975
Degeneracy of velocity constraints in rigid body systems
Brggs, Jonny
The equations governing the dynamics of rigid body systems with velocity constraints are singular at degenerate configurations in the constraint distribution. In this report, we describe the causes of singularities in the constraint distribution of interconnected rigid body systems with smooth configuration manifolds. A convention of defining primary velocity constraints in terms of orthogonal complements of one-dimensional subspaces is introduced. Using this convention, linear maps are defined and used to describe the space of allowable velocities of a rigid body. Through the definition of these maps, we present a condition for non-degeneracy of velocity constraints in terms of the one dimensional subspaces defining the primary velocity constraints. A method for defining the constraint subspace and distribution in terms of linear maps is presented. Using these maps, the constraint distribution is shown to be singular at configuration where there is an increase in its dimension.
2016-09-27T00:00:00ZRepresentation of numbers by quaternary quadratic forms
http://hdl.handle.net/1974/14974
Representation of numbers by quaternary quadratic forms
Kar, Arpita
2016-09-27T00:00:00ZAn R-package for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial Likelihood
http://hdl.handle.net/1974/14973
An R-package for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial Likelihood
Zhang, Siwei
When we study the variables that a ffect survival time, we usually estimate their eff ects by the Cox regression model. In biomedical research, e ffects of the covariates are often modi ed by a biomarker variable. This leads to covariates-biomarker interactions.
Here biomarker is an objective measurement of the patient characteristics at baseline.
Liu et al. (2015) has built up a local partial likelihood bootstrap model to estimate
and test this interaction e ffect of covariates and biomarker, but the R code developed
by Liu et al. (2015) can only handle one variable and one interaction term and can
not t the model with adjustment to nuisance variables. In this project, we expand
the model to allow adjustment to nuisance variables, expand the R code to take
any chosen interaction terms, and we set up many parameters for users to customize
their research. We also build up an R package called "lplb" to integrate the complex
computations into a simple interface.
We conduct numerical simulation to show that the new method has excellent fi nite
sample properties under both the null and alternative hypothesis. We also applied the
method to analyze data from a prostate cancer clinical trial with acid phosphatase
(AP) biomarker.
2016-09-27T00:00:00ZOn The Dirichlet Distribution
http://hdl.handle.net/1974/14972
On The Dirichlet Distribution
Lin, Jiayu
The Dirichlet distribution is a multivariate generalization of the Beta distribution. It is an important multivariate continuous distribution in probability and statistics. In this report, we review the Dirichlet distribution and study its properties, including statistical and information-theoretic quantities involving this distribution. Also, relationships between the Dirichlet distribution and other distributions are discussed. There are some different ways to think about generating random variables with a Dirichlet distribution. The stick-breaking approach and the PĆ³lya urn method are discussed.
In Bayesian statistics, the Dirichlet distribution and the generalized Dirichlet distribution can both be a conjugate prior for the Multinomial distribution. The Dirichlet distribution has many applications in different fields. We focus on the unsupervised learning of a finite mixture model based on the Dirichlet distribution. The Initialization Algorithm and Dirichlet Mixture Estimation Algorithm are both reviewed for estimating the parameters of a Dirichlet mixture. Three experimental results are shown for the estimation of artificial histograms, summarization of image databases and human skin detection.
2016-09-27T00:00:00Z