Doug Lake (Mathematics). Determining the Motion of an Object through Variable Time-Delay Estimation, Ph.D., Department of Mathematics, May 1991; Present Posi- tion: Research Scientist, Cardiology Department (with Dr. Randall Moormann, MD), University of Virginia Medical School, Charlottesville, Va; (Until Recently: Research Scientist, Army Research Laboratory, Adelphi, Md.);
Weimin Sun (Statistics). Statistical Modeling and Simulation of the Time-Delayed Feedback Regulation of the Hypothalamic-Pituitary-Testicular Gonadal Axis, Ph.D., Division of Statistics, May 1995; Present Position: Chief Statistician, Pathogenesis Inc., Seattle, WA.;
Paula Shorter (Mathematics). Diffusion Processes for Stochastic Global Optimization on a Manifold with Applications in Image Processing, Ph.D., Department of Mathe- matics, Aug. 1996; Present Position: Associate Professor (and Chairman), Department of Mathematics, Physics, and Computer Science, Rockhurst College, Kansas City, Mo.;
Ronghua Yang (Statistics). Maximum Likelihood Estimation Asymptotics for Parameter- Dependent Mixed Effects Models with Applications to Hormone Data, Ph.D., Division of Statistics, May 1997, Present Position: Statistician, Pharmaceutical Research Asso- ciates, Charlottesville, Va.;
Robert Rand (Engineering Physics). A Gibbs-based unsupervised segmentation ap- proach to partitioning hyperspectral imagery for terrain applications, Ph.D., Depart- ment of Engineering Physics, Aug. 2001; Present Position: Research Scientist, Army Topographical Engineering Center, Arlington, Va;
Somesh Chattopadhyay (Statistics). Simultaneous Hormone Pulse Time and Secre- tion/Elimination Estimation: An Alternating Metropolis and Diffusion Scheme, Ph.D., Department of Statistics, Aug. 2001. Present Position: Research Scientist, Food and Drug Administration (FDA), Silver Spring, MD.
Xin Wang (Statistics). Derivation and Implementation of the Asymptotics for Approx- imate Entropy (ApEn) with Applications to Medicine. Ph.D., Department of Statis- tics, May 2009. Present Position: Postdoctoral Fellow, Division of Biostatistics, Mayo Clinic, Rochester, MN
Yan Liu (Statistics), PhD, 2010. Statistical Modeling in Endocrinology: MLE Asymp- totics for Difference and Integral Equation-Based Methods.
Cailin Steiner (4th year PhD student, Statistics, to complete, May 2016)- Statistical estimation of Motor Cortex-Muscle Control, utilizing Transcranial Magnetic Stimula- tion (TMS) (This PhD thesis is part of ongoing joint research (Keenan) with Dr. Mark Hallett, MD, at NIH (NINDS)).
Chenyi Pan (4th year PhD student, Statistics-co-advisor, with Prof. Dan Spitzner, to complete, May 2016) - My part of her research, concerns the establishment of statistical spectral methods (for EEG) to compare the spectral ”signatures” for different brain regions.
Taylor Brown (3rd year PhD student, Statistics, to complete, May 2018) - Novel particle filtering models for the estimating the stochastic dynamics of a large collection of financial instruments.
A Summary of the Ph.D Students’ Theses:
I have had 3 Ph.D. students work on problems in computer vision/object recognition, and 5 on problems in physiological/endocrinological modeling. My first Ph.D. student (D. Lake) addressed the problem of recovering an object’s path of motion from a spatial arrangement of passive non-directional sonar sensors; the motivation was to submarine tracking. The differing time delays from the various sensors, at each point in time, were probabilistically reconciled, from which the path was then estimated. One element of the asymptotic theory was a “method of sieves” construction. A second Ph.D. student (P. Shorter) developed a mathematical model by which a fetal head could be reconstructed from multiple ultrasound images obtained from the moving transducer; the technique automates the calculation of the biparietal diameter of the fetal head, the basic ultrasonic calculation for assessing abnormaltites, even if none of the images were at that level. Because the space of orientations (SO(3)) is a manifold, the resulting theory (of the algorithm) was established in such a context. The results were the establishment of stochastic relaxation and simulated annealing on a manifold via diffusion processes. A third student (W. Sun) modeled, probabilistically and statistically, the hypothalamic GnRH pulse generator. The pulsing was described by a point process whose “intensity function” was itself a process governed by time-delayed nonlinear feedback from the actual resulting pulsatile processes. A fourth student (R. Yang) developed a method for estimating the rates of secretion and elimination, from the concentrations of a pulsatile-secreting hormone. The statistical theory was the establishment of the MLE asymptotics for a general nonlinear mixed-effects model (i.e., where the parameters and the random effects interact). A fifth student (R. Rand ) worked on the problem of terrain mapping, from hyperspectral images (500 bands between 0.45 and 2.9 microns). The methodology developed was that of a hierarchy of Markov Random Fields, in which both (local) spatial and spectral information is being used. My sixth student (S. Chattopadhyay) developed methods by which to detect the pulse times, from a time profile of hormone concentrations for a pulsatile secreted hormone, such as luteinizing-hormone (LH). The method is based upon the selective smoothing procedures (nonlinear parabolic PDE’s) which have been used in image processing to detect boundaries of objects; boundaries are points of rapid change, just as the onset of a pulse is a point of rapid change. The necessary theory was established and it was then incorporated into an alternating Metropolis-diffusion scheme for the joint estimation of the pulses and the secretory/kinetic parameters. The pulse detection method is now widely applied in physiological modeling. My seventh student (X. Wang) filled in ”two missing pieces” in the application of a widely used method (especially in medicine): Approximate Entropy (ApEn). ApEn is a method by which a measure of the degree of regularity (and irregularity) is assigned to a given sequence of n values. The two missing pieces which have limited its application, are the need of a justified procedure for calculating the standard error of ApEn, and the removal of the small-sample bias which occurs in ApEn. Both of these issues were resolved in the thesis. My eighth student (Y. Liu) developed models of hormonal secretion, elimination and systemic noise, in the continuum, for which the observed data can be viewed as a sampling of this continuous process. Methods by which knowledge of the underlying ranges of kinetic and secretory values is used to establish sampling rates, based on that knowledge, that lead to optimal recovery of the underlying structure.
UNDERGRADUATE RESEARCH STUDENTS:
From Sept. 1989 - Dec.1990, I supervised 2 undergraduate math majors (3 credit hours) independent research (Math 493), on the topic of Computer Vision.
Matthew Harrison, as part of the Undergraduate Research Initiative in Mathematics, began working with me during the summer of 1996. He continued to do so, via 3 credit hours of directed research, for each semester through 1998. The research concerned the mathematical modeling of the female reproductive hormone system, in particular, the menstrual cycle. He received his PhD in Applied Mathematics at Brown University, and is currently an assistant professor there.
Senior Thesis Advisor for Renee Sunila (1997) and Colin VanAntwerp (1997), undergraduate Applied Mathematics majors, in Actuarial Science theses.
Lu Xia (2nd Year), Research, Fall, 2008 - Modeling the release of Insulin granules from Pancreatic Islet cells; Spring, 2009 - Reconstructing Unobserved Brain Signals for the Reproductive Hormone Axis.
Sida Peng (2nd Year), Research, Spring, 2009 - Nonlinear time series models and the pricing of financial derivatives.
Hang Chen (3rd, 4th Year), Research, Spring/Fall 2009, Spring/Summer 2010 - A comparison of the Approximate Entropy Statistic to the BDS statistic for Assessing Nonlinear Granger Causality; currently, a graduate student in Statistics at Stanford;
Zhen Wei (3rd Year, transfer student from Oxford Univ.), Research, Fall, 2009 - Mod- eling the Release of Insulin Granules from Pancreatic Islet Cells; Spring, 2010 - Estimating the dynamics of nonlinear time series.
Weilong Hu (2nd year exchange student from Auckland Univ) Fall, 2009. Worked on the same project as Hang Chen, above.
Sida Peng (3rd Year; Summer 2010 worked with Dr. Randall Moorman in Cardiology).
Zachary Terner (3rd and 4th years; 2012-2014); Modeling the Brain Arousal system.
Jiayin Zhao (3rd and 4th years, 2012-2014); Estimating a time-varying baseline in hormonal data;
Yue Xiang (2nd and 3rd year, 2012-2014) Fourier Analysis of circadian and ultradian rhythms.
Chun Li (3rd and 4th years, 2012-2014) Detecting circadian rhythms in mouse motor activity ; Alice Liu (3rd/4th-year student, Statistics)
Ph.D. THESIS COMMITTEES (other than my own students):
Statistics: Robert McDevitt, 1996; Krag Gregory, 1999; Kate Prescott, 2000; Y. Tomita, 2003; Youngsook Jeon, 2009, Wei Ma (2013); Xiaoxiao Tang (2013); Xiaoming Li (2015); Jingwei Wu (2015).
Mathematics: Raina Robeva, 1996; Patty Hunter, 1996; Christian Hellings, 2000; Aaron Haack, 2001; Carl Toews, 2002; Randall Helmstutler, 2004; Ryan Higginbottom, 2005; Michael Smith, 2006; Katie Quertermous, 2010; Joseph Johnson, 2012.
Applied Mathematics: Mehmet Camurdan, 1999.
Physics: Zongyi Gong 2013.
Electrical Engineering: John Anderson, 1992; Brian Sadler, 1992; Amod Dandawate, 1993; Sanyogita Shamsunder,1993; Ed Hall, 1994; Steve Halford, 1996; Erchin Ser- pedin, 1998; Tao Li, 2000; Terry Foreman, 2000.
Biomedical engineering: Greg Harrington, 1999;
Systems Engineering: Michael Porter, 2006.
Economics: Donald Sill, 1992; Greg Hopper, 1992.
Psychology: Elliot Tucker-Drob, 2009.