Statistics & Probability
The Science of Data Collection & Analysis
Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.
My research focuses on developing novel methodology to solve statistical problems raised from neuroimaging data, including fMRI, sMRI, DTI, and EEG. Generally, the functional neuroimaging data are of spatially and temporally high-dimensions. Analyzing these data includes various statistical topics: time series analysis, dimension reduction, classification, variable selection, longitudinal data analysis, covariance estimation, etc. I am also interested in variable selection methods for repeatedly measured data. I have collaborated with many scientists in various fields, including veterinary science, psychiatry, radiology, neurology, immunology, and biomedical engineering.
I use techniques from the fields of dynamical systems, stochastic processes, probability and statistics to develop and analyze mathematical models of biological systems. I use those models to address questions that arise in population ecology, evolution, epidemiology (infectious diseases) and immunology. Recently, I've begun working with methods for fitting nonlinear dynamic models to time series data. Using these models as statistical models presents a number of challenges, as parameter estimators for these models are not guaranteed to be as statistically well-behaved as, for example, estimators for classical linear models. In addition to parameter estimation for dynamic models, I also use approximation methods that exploit the deeper connections between deterministic models and their stochastic counterparts, as these two modeling frameworks can both be useful in applications.
My main research interests include theory and applications of stable, geometric stable, and other heavy-tail random variables and stochastic processes. A stable variable has the property of stability: the sum of n copies of X has the same type of distribution as X. More general notions of stability include cases when the number of variables n is itself a random variable and/or or when the variables are combined by operations other than adding. A heavy-tail random variable is one that has a non-negligible probability of resulting in a value relatively far from the center of the distribution. I have worked on applications of stable and related distributions in actuarial science, economics, financial mathematics, as well as other areas. My other research interests include computational statistics, characterizations of probability distributions, and stochastic simulation.
My research interests include probability, statistics, stochastic modeling and interdisciplinary work. In particular, I study the limit theory for random and deterministic sums of random quantities and estimation for heavy tailed distributions. Stochastic modeling and interdisciplinary work cover finance and insurance, hydrology and water resources, atmospheric science and climate, environmental science and biostatistics. Current research projects include statistical estimation for heavy tailed hydrology data, climate and hydrological extremes in the US, and clean water issues in Nevada and California.
My research is driven by a desire to understand the roles of stochasticity, structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability and statistics, dynamical systems, and random graph theory to shed light on biological issues while generating new mathematical questions. In particular, I study stochastic processes on networks with applications in neuroscience and stochastic models in genetics.
Currently, I am working on optimal reduction techniques for complex ion channel gating models, which can be represented as a stochastic (Markov) process on a graph. I am also looking at the relative contributions of network structure and node dynamics in determining the collective dynamics of a network, thinking specifically about neuronal networks involved in sleep-wake regulation. Lastly, I am broadly interested in mathematical and statistical applications in population and evolutionary genetics.
My research focuses on theoretical and applied statistical analysis of complex (non-linear) dynamical systems, with emphasis on spatio-temporal pattern formation and development of extreme events. Specifically, I work on multiscale methods of time series analysis, heavy-tailed random processes, and spatial statistics. This choice is predicated by the essential common properties of the observed complex systems: they tend to evolve in multiple spatio-temporal scales; and have observables that exhibit absence of characteristic size, long-range correlations in space-time, and not-negligible probability of assuming extremely large values. The underlying methods of analysis include those of hierarchical aggregation and its inverse - branching processes.
Examples of the observed systems relevant to my research include the Earth's lithosphere which generates destructive earthquakes, its atmosphere that produces El-Ninos, stock-markets subject to financial crashes, etc. My current applications and ongoing collaborations are in Solid Earth geophysics (seismology, geodynamics), climate dynamics, computational finance, biology, and hydrology.