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Full-Text Articles in Mathematics
Allometric Extension For Multivariate Regression Models, Thaddeus Tarpey, Christopher T. Ivey
Allometric Extension For Multivariate Regression Models, Thaddeus Tarpey, Christopher T. Ivey
Mathematics and Statistics Faculty Publications
In multivariate regression, interest lies on how the response vector depends on a set of covariates. A multivariate regression model is proposed where the covariates explain variation in the response only in the direction of the first principal component axis. This model is not only parsimonious, but it provides an easy interpretation in allometric growth studies where the first principal component of the log-transformed data corresponds to constants of allometric growth. The proposed model naturally generalizes the two–group allometric extension model to the situation where groups differ according to a set of covariates. A bootstrap test for the model is …
Self-Consistency Algorithms, Thaddeus Tarpey
Self-Consistency Algorithms, Thaddeus Tarpey
Mathematics and Statistics Faculty Publications
The k-means algorithm and the principal curve algorithm are special cases of a self-consistency algorithm. A general self-consistency algorithm is described and results are provided describing the behavior of the algorithm for theoretical distributions, in particular elliptical distributions. The results are used to contrast the behavior of the algorithms when applied to a theoretical model and when applied to finite datasets from the model. The algorithm is also used to determine principal loops for the bivariate normal distribution.
Self-Consistency: A Fundamental Concept In Statistics, Thaddeus Tarpey, Bernard Flury
Self-Consistency: A Fundamental Concept In Statistics, Thaddeus Tarpey, Bernard Flury
Mathematics and Statistics Faculty Publications
The term ''self-consistency'' was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if E[X|Y] = Y almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give …