Quantifying nonlinear variation in manifolds
Speaker: Rima Izem , Statistics Department, Harvard university
Date: April 21 2005
Time: 1:00PM to 2:00PM
Location: 32-D451
Host: Polina Golland, CSAIL, MIT
Contact: Polina Golland, x38005, polina@csail.mit.edu
Scientists in an increasing number of fields, including
biology, medicine, and physics, collect samples of curves or
images of common shape. Although these data are discrete, the
processes generating them are continuous. Analyzing variation in
these samples, with the aim of making inferences about the general
population from which the sample is drawn, is often the main
statistical interest. Functional Data Analysis (FDA) methods use
the underlying continuity of the data to analyze the variation.
However, usual FDA methods, such as Principal Components Analysis,
are only effective in analyzing linear variations, and do not
always produce interpretable results.
In this talk, a general model for curves or images of common shape
is considered. We present a new method for analyzing variation
under this model. Our method achieves two important goals. The
first goal is to decompose the variation in the data into
predetermined and interpretable directions of interest, and these
could be linear or non-linear. The second goal is to quantify each
direction by a newly defined ratio of sums of squares, to allow
for a comparison of the contributions to the total variation. The
new ratio of sums of squares quantifies a non-linear direction by
taking into account the curvature of the space of variation. We
discuss, in the general case, consistency of our estimates of
variation, using mathematical tools from differential geometry and
shape statistics.
We successfully applied our method to two different examples of
reaction norm curves in Biology. Our analysis shows that
non-linear components are dominant. Moreover, our decomposition
allows biologists to compare the prevalence of different genetic
tradeoffs in a population and to quantify the effect of selection
on evolution.
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