# ICS 269, Spring 2005: Theory Seminar

## 13 May 2005:

Ivan Mizera

Department of Mathematical and Statistical Sciences

University of Alberta (Edmonton, Canada)

The talk will start by explaining certain episodes on a way from the
halfspace depth in multivariate location ("the Tukey depth") through depth
in general data-analytic situations (models?) toward the psychedelic
experience of a new notion of depth in the location-scale model,
Location-Scale Depth, and its most tractable version, the Student
depth. The latter has a couple of entertaining theoretical and
computational properties, stemming from the fact that it is nothing
but the bivariate halfspace depth interpreted in the Poincar\'e plane
model of the Lobachevski geometry - in particular, invariance with
respect to the M\"obius group and favorable time complexities of
algorithms. The practical implications involve a new fancy
location-scale typical value, the Student median, as well as somewhat
extravagant graphical tool for exploring distributional properties of
univariate samples, a sort of cousin to the quantile-quantile plot.

However, perhaps more than those particular accomplishments it may be
worthy to note potential new views on data and questions that the
process raises: the role of invariance (if any) in data analyses;
whether there can be such a thing as median in sophisticated
situations; whether classical rank-based nonparametrics can be
elevated beyond their traditional (essentially) univariate setting;
and last but not least, whether all these hallucinations can be
effectively computed.