Biomathematics
Department of Mathematics and Statistics
Texas Tech University
 | Sunday
1
11 AM
MA 010
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Yuchi Qiu
Department of Mathematics and Statistics, University of California -- Irvine
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-->Abstract pdf
Quasiperiodically driven dynamical systems are nonlinear systems which are driven by some periodic source with multiple base-frequencies. Such systems abound in nature, and are present in data collected from sources such as astronomy and traffic data. Such dynamics decomposes into two components - the driving quasiperiodic source with generating frequencies; and the driven nonlinear dynamics. Analysis of the quasiperiodic part presents the same challenges as classical Harmonic analyis. On the other hand, the nonlinear part bears all the aspects of chaotic dynamics, and possibly carry stochastic perturbations. We present a kernel-based method which provides a robust means to learn both these components. It uses a combination of a kernel based Harmonic analysis and kernel based interpolation technique, to discover these two parts. The technique performs reliably in several real world systems, ranging from analyzing the human heart to traffic data.
Abstract pdf
A stochastic differential equation model is derived for the evolution of mountain elevations. The derivation is based on several assumptions about tectonic and erosion processes in mountain elevation dynamics. At any given time, the model yields a CIR-type probability distribution for mountain heights. As data are often available for mountains of greatest elevation in a mountainous region, the tail of the CIR distribution is studied and compared with mountain height data for the highest mountains in the region. The stochastic model indicates that the tail distribution is proportional to the product of a power of height and an exponential function of height. Specifically, for mountain height h, the model tail density is proportional to $h^{b-1} exp(-a h)$ where a and b are constants. The resulting inverse distribution function of the tail probability density leads to a function that relates rank in height to the corresponding mountain height. This function indicates how mountain heights in a region are related and provides, for example, a decreasing sequence of theoretical mountain heights in the region. The derived inverse distribution function is tested against mountain height data sets for several mountainous regions in the British Isles, Continental Europe, Northern Africa, and North America. The derived inverse cumulative distribution function provides an excellent fit to the mountain height data ranked by height.
Abstract pdf