Georgia Institute of Technology College of Sciences

In this talk, I will present two examples of the application of wavelet analysis to the understanding of mild Traumatic Brain Injury (mTBI). First, the talk will focus on how wavelet-based features can be used to define important characteristics of blast-related acceleration and pressure signatures, and how these can be used to drive a Naïve Bayes classifier using wavelet packets. Later, some recent progress on the use of wavelets for data-driven clustering of brain regions and the characterization of functional network dynamics related to mTBI will be discussed. In particular, because neurological time series such as the ones obtained from an fMRI scan belong to the class of long term memory processes (also referred to as 1/f-like processes), the use of wavelet analysis guarantees optimal theoretical whitening properties and leads to better clusters compared to classical seed-based approaches.

In this era of "big data", Mathematics as it applies to human behavior is becoming a much more relevant and penetrable topic of research. This holds true even for some of the less desirable forms of human behavior, such as crime. In this talk, I will discuss the mathematical modeling of crime on two different "scales", as well as the results of experiments that are being performed to test the usefulness and accuracy of these models. First, I will present a data-driven model of crime hotspots at the scale of neighborhoods -- adapted from literature on earthquake predictions -- along with the results of this model's application within the LAPD. Second, I will describe a game-theoretic model of crime and punishment at the scale of a society, and compare the model to results of lab-based economic experiments performed by myself and collaborators.

We discuss various exponential time differencing (ETD) schemes designed to handle nonlinear parabolic systems. The ETD schemes use certain Pade approximations of the matrix exponential function. These ETD schemes have potential to be implemented in parallel and their performance is very robust with respect to the type of PDE. They are unconditionally stable and computationally very fast due to the technique of computing the nonlinear part explicitly. To handle the problem of irregular initial or boundary data an adaptive ETD scheme is utilized, which adds sufficient damping of spurious oscillations. We discuss algorithm development, theory and applications.

Slow Feature Analysis (SFA) is a method for extracting slowly varying features from input signals. In this talk, we generalize SFA to vector-valued functions of multivariables and apply it to the problem of blind source separation, in particular image separation. When the sources are correlated, we apply the following technique called decorrelation filtering: use a linear filter to decorrelate the sources and their derivatives, then apply the separating matrix obtained on the filtered sources to the original sources. We show that if the filtered sources are perfectly separated by this matrix, then so are the original sources.We show how to numerically obtain such a decorrelation filter by solving a nonlinear optimization problem. This technique can also be applied to other linear separation methods, whose output signals are uncorrelated, such as ICA.This is joint work with Laurenz Wiskott (Proceedings of the 13th IEEE International Conference in Computer Vision, ICCV 2011, Barcelona, Spain).