A number of people have asked for a talk on Topological Data Analysis so here it is, courtesy of Steve Ellis.
About the talk:
Data analytic methods possessing the following three features are desirable:
(1) The method describes "high-order dependence" among variables. (2) It does so with few preconceptions. And (3) it can handle at least dozens, maybe hundreds of variables.
However, if approached in a naive fashion, data analysis having these three features triggers a "combinatorial explosion": The output from the analysis can include thousands, maybe millions of numbers. Few methods exist possessing all three features yet which avoid the combinatorial explosion. Ellis has devised a data analytic method he calls "Concurrence Topology (CT)" which does so.
CT takes an apparently radically new approach to solving this problem. It starts by translating data into a "filtration", a series of "shapes". The shapes in the series are called "frames." A filtration is like a building. The frames are like floors of the building. But while the floors of a building are two-dimensional, the frames of a filtration can have dimension much higher than two.
A filtration can have holes that are like elevator shafts in a building. Such holes indicate relatively weak or negative association among the variables. CT uses computational algebraic topology to describe the pattern of holes. Normally, there are no more than a few dozen holes, so CT avoids the combinatorial explosion. Often one can identify small groups of variables that are closely associated with a given hole. This process facilitates interpretation of the hole.
A limitation of CT is that, so far, it only works with binary data. But quantitative data can always be binarized.
Steve Ellis wrote software in R (available upon request) implementing Concurrence Topology. A paper, written by Arno Klein and Ellis, introducing CT and demonstrating it on fMRI data has been accepted by a topology journal.
Pizza begins at 6:15, announcements, giveaways and the talk start at 7, followed by the local bar.