Event

The Department of Mathematics and Statistics, Faculty of Science and Technology invites you to two talks from visiting Professor Bala Rajaratnam of Stanford University, California, USA:

Talk One: Regularization of positive definite matrices: connections between algebra, graph theory, analysis and statistics.

Tuesday 23rd April, 2013 - 10.00am -Room 313, Natural Sciences Building

Abstract: Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications, including the earth sciences and biomedical sciences.  In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of positive definite matrices which are thresholded. Some interesting and unexpected results will be presented.  If time permits, probabilistic properties of thresholded positive definite matrices and connections to optimization will also be discussed.

Talk Two: A Spatial Modeling Approach to Multiproxy Paleoclimate Reconstructions

Wednesday 24th April, 2013- 10.00 am -Faculty Conference Room, Ground Floor, Natural Sciences Building

Abstract: The study of climate over the Earth’s history is a topic of current interest whose relevance has increased rapidly with the growing concern over climate change. Reconstructing climates of the past (sometimes referred to as the ``hockey stick" problem) has been used to understand whether current climate is anomalous in a millennial context. To this end, various climate field reconstructions (CFR) methods have been proposed to infer past temperature from (paleoclimate) multiproxy networks. We propose a new climate field reconstruction method that aims to use recent advances in statistics, and in particular, high dimensional covariance estimation to tackle this problem. The new CFR method provides a flexible framework for modeling the inherent spatial heterogeneities of high-dimensional spatial fields and at the same time provide the parameter reduction necessary for obtaining precise and well-conditioned estimates of the covariance structure of the field, even when the sample size is much smaller than the number of variables. Our results show that the new method can yield significant improvements over existing methods, with gains uniformly over space. We also show that the new methodology is useful for regional paleoclimate reconstructions, and can yield better uncertainty quantification. We demonstrate that the increase in performance is directly related to recovering the underlying structure in the covariance of the spatial field. We also provide compelling evidence that the new methodology performs well even at spatial locations with few proxies. (The work is based on two papers: joint work with D.Guillot and J. Emile-Geay, and joint work with L. Janson).