William L. Welbourn, Jr., M.Sc., PKP, GKIHS, ISLP
The problems we face cannot be solved
at the same level of thinking we were at when we created them Albert
Einstein
"Who we are today is a reflection of our actions from the
past"- Dr. David Chen, Professor California State University, Fullerton
"Greatness is 1% final product, and 99% passion in the pursuit of
completing it"- William Welbourn, Jr.
"Mathematics is the queen of the sciences and number theory is the queen
of mathematics"- Carl Friedrich Gauss
"The excitement that a gambler feels when making a bet is equal to the
amount he might win times the probability of winning it"- Blaise Pascal
MATHEMATICS/STATISTICS/GENETICS
BACKGROUND
COMPUTATIONAL ALGORITHM
DEVELOPED 3/22/2008
Some information about myself:
- Affiliations
- Member,
- Member, Mathematical Association of
- Member,
- Member,
- International Scholar Laureate Program (ISLP) of GKIHS
- Candidate,
Doctor of Philosophy Program in Statistics (Emphasis: Statistical Genetics),
August 2007 December 2011 (Projected)
- Utah State
University
- Teaching Assistant, Math 1050: College Algebra,
August 2007 May 2008
- Passed PhD Qualifying Exam, May 2008
- Teaching Assistant, Stat 2000: Statistical
Methods, August 2008 May 2009
- SAS/R Lab Instructor, Stat 5100/5200: Linear
Regression and Time Series/Experimental Designs, August 2009 December
2009
- Recitation Leader, Stat 3000 (x3 Sections):
Statistics for Scientists, January 2010 May 2010
- Instructor, Stat 3000: Statistics for
Scientists, Summer 2010 and Summer 2011
- Research Assistant, January 2009 May 2009:
Database Management of a genetic data warehouse (287 participants;
660,918 SNPs; ~ 8 gigabytes)
- Research Assistant, June 2009 July 2009: Data
Management of a genetic data warehouse (11 participants; 1,145,510 SNPs; ~ 3 gigabytes)
- Research Assistant, February 2010 : Analyzing
GWAS data set (2,035 participants; 769,672 SNPs;
~ 6 gigabytes of text files)
- Elected to the
- Research Writing Award, Department of
Mathematics and Statistics,
- Nominated to the International Scholar Laureate
Program of GKIHS, December 2009
- Passed PhD Comprehensive Exam, March 2010
- Technology Expert Award, Department of
Mathematics and Statistics,
- Successfully Defended Doctoral Dissertation
Proposal and Advanced to Candidacy,
- Master
of Science in Biostatistics, May 2006, Honors Distinction
- University of
Southern California
- Elected to the
- Bachelor
of Arts in Mathematics with an emphasis in Probability and Statistics,
January 2001
- California
State University, Fullerton
- CSUF GPA 3.71/4.00 with an Upper Division
Mathematics GPA of 4.0/4.0
- Summited
- Summited Half Dome (EL. 8,842 FT.),
- Computer
System, Built
- Motherboard: Foxconn BloodRage,
Intel X58 Chipset
- CPU: Intel
Core i7 920 2.66GHZ 8MB L3 Cache 4.8GT/s QPI Quad Core, Supporting
Hyper-Threading (i.e., handles 8+ threads simultaneously)
- Memory: OCZ 3GB
- Hard Drives: Seagate SATA II 4 x 250GB RAID 0;
120GB Corsair
- OS: Windows XP Home/Windows Vista Home
Premium/Windows 7 RC - Desktop (Windows Vista, Business Edition - Laptop)
32bit
- Graphics: EVGA NVIDIA GeForce
GTX 470 Fermi 1.2GB (x2) in
- Monitor: ASUS 24" LCD
- Sound: SoundBlaster X-Fi
Fatal1ty Pro Series
- Whetstone Benchmarks show a 180% increase in
integer/floating point arithmetic over my prior Intel Core 2 Duo E6750
System
- Programming:
- CUDA for C (CUDA
Toolkit 3.2);
- C,
Emphasis on Multi-Threaded Applications Computational timing
comparison of a single thread of R to multiple threads of C
- R (Version
2.13.0)
- Bioconductor Project, Analysis of Genomic Data
- Database Management, Interfacing R with the SQLite RDBMS Example code,
working with a genetic database, warehousing ~1.2 million SNPs
- Graphical User Interface (GUI) for OSA Linear Regression
Methodology; Logistic
Regression Methodology
- SAS (9.1.3)
- Mathematica (4.0)
- LaTeX
- LEd Editor
- Compiler
- Beamer Class for professional looking
presentations Example
Past Academic News
Undergraduate
·
· January 2, 2001- Conferral Date for the Bachelor of Arts Degree in Mathematics.
Graduate (Master of Science)
·
March 2004 -
Admitted to the Master's Program in Biostatistics at the
·
May 2004 - Began
my Graduate Studies at
· May 2005 - Began my Master's Thesis Research - Topic: Ordered Subset Analysis (OSA).
· April 2006 - Successfully defended my Master's Thesis and completed the requirements for the Degree, Master of Science in Biostatistics.
·
·
Graduate (Doctoral)
- March
2007 - Admitted to the Doctor of Philosophy Program in Mathematics,
Statistics Option, at Utah State University
and the
·
August 2007 -
Began my Graduate Studies at
· February 2008 - Revising MS Thesis and subsequent papers for publication.
· May 2008 - Overall Course Score of 100.5% in Mathematical Statistics II (Stat 6720).
· May 2008 - Passed the Mathematical Statistics PhD Qualifying Exam.
·
Research Interests
- CUDA paper for maxT MHT procedure 12/2009 Abstract
- CUDA paper for clustering GPUs 3/2010 Abstract
- Dissertation Abstract/TOC 10/2009 (MAIN OSA C CODE)
- Utilize Ordered Subset Analysis (OSA) in a Genome-wide Association Study (GWAS), to locate and suggest genes which express in the same amino acid transcription/translation process (i.e., locate genes which are correlated).
- Assign appropriate correlation magnitudes to the pairwise (and higher order) gene associations from OSA use.
- Working on the development of a permutation based multiple hypothesis testing (MHT) procedure, analogous to maxT/minP. However, novel algorithm will account for OSA determined gene-gene correlations.
- Adapting OSA to a binary phenotype. This would enable OSA to be used in place of traditional logistic regression modeling, in suggesting gene-environment interaction.
- Integration of permutation testing, particularly the use of the tilted hypergeometric distribution (GHG) for testing non traditional null hypotheses related to the odds ratio.
- Suggesting the use of Exact (permutation) Tests (as opposed to asymptotic tests) for GWAS MHT. This is a crucial notion, since within MHT, statistical significance is achieved way out in the tail of the null distribution. Hence, the assumption of an asymptotic sampling distribution under the null, could result in serious p-value calculation errors.
- Major Master's Thesis Extension Document (Potential Book/The basis for my PhD Dissertation paper currently stands at 194 pages) - Description, as of 1/4/2009:
- Chapter 1 - Overview of what Ordered Subset Analysis (OSA) is all about, and a few examples of where it could be applied
- Chapter 2 - Some of the underlying theory of OSA, including how one would draw inference from the empirical p-value.
- Chapter 3 - Computational efficiency issues. Topics include a theorem to reduce p-value computations, algorithms to search for the subset which provides maximum statistical evidence for gene-environment interaction, several algorithms for the implementation of power analyses, and maximizing computer power (i.e., parallel processing, utilizing multi-core CPU technology) in the implementation of the algorithms.
- Chapter 4 Numerical Justification to referenced assumptions within chapters 2 and 3. Benchmarking Theorem 8 of the paper.
- Chapter 5 Simulation and Implementation of OSA into an R GUI
- I am in the process of implementing the OSA methodology into the C programming environment (spent all of December 2008 doing this).
- I plan to look into
extending the OSA methodology to other (than linear regression)
- With regard to computational aspects, I plan to integrate database management (via the SQL querying language) into the framework. This should reduce computational timing substantially, over the use of "flat (text) file" storage techniques.
- Integrate the OSA statistical methodology into a novel R package.
My primary interest in statistics lies within distribution theory. My Master's Thesis, an Ordered
Subset Analysis investigation (hence the "osa" in osastatistician.com), entailed computational
burdensome algorithms, and during the course of the project I was able to prove
an interesting result (Paper #1 above) regarding the Central Student's-t
Distributions. Namely, given any positive real value, x, and two positive
integers, n1 and n2, such that n1<n2, then the value for the cumulative
distribution function (at the given value of x) for the Student's-t
distribution with n1 degrees of freedom is less than that for n2 degrees of
freedom. This result was an extremely important efficiency improvement
for this project, as a mere 4.7% of total data throughput was seen. The
Thesis essentially entails three unique and fresh distributions for
Gene-Environment Interaction Association Studies - The OSA distribution, the
CEO Null Distribution, and the
I continue to work on extending the Thesis. As mentioned in the body
of the paper, a source of error for the OSA investigation lies in the
assumption of the number of random orderings chosen for each of the CEO and
My most recent research endeavors:
(1) The Order Statistics for the Standard Normal Distribution. This project is designed to add credibility to the Order Statistic Simulation Methodology of my MS Thesis Appendices. I am continuously finding interesting results unfold... Keep coming back to this page for updates to this paper.
(2) An elegant result regarding the Central Student's-t Distributions. This result enhances the computational efficiency for the OSA.
Over the course of the months of September and October 2006, I kept a brief Journal of my thoughts regarding the OSA expansion. So many thoughts had filled my mind in such a short period of time, I documented my thoughts into Journal Form. These are notions which I hope to investigate in the near future.
An ongoing project which I began back in 2000, entails stochastic processes and Dynamical Systems. The motivation for this project was stemmed from Dr. Mario Martelli, Professor Emeritus of CSUF. Essentially, the project entails a group of sailors confined to an island with a fixed supply of food, a stack of bananas. The sailors, one-by-one approach the stack of bananas and take "their fair share." The complexity of the problem originates in the "stochastic thief action" of a monkey on the island - As the sailors take their bananas, there is a certain probability that the monkey will take some banana(s) from the stack. This project has the potential to be applied to medicine, where the sailors are human cells, the bananas are fuel for the human cells, and the monkey is a virus or bacterium.
During the Summer months of 1999, I investigated how the statistical tables in textbooks are derived. I used numerical integration techniques, such as the Simpson Rule and the Composite Boole's Rule, to generate the Standard Normal, Chi-Square, Student's-t, and Snedecor-F statistical tables. This research is documented in "notebook format," and a journal of the works is kept in my files. Since the writing of this technical document, I have further investigated the Chi-Square, Student's-t, and Snedecor-F distributions in much more detail. What has resulted in this research is the ascertainment of reduction formulas for the cumulative distribution functions for these distributions.
Links to Other Pages
Hobbies and interests
Probability Challenge Problems
A Special Outcome From My Math 375 Group Project
Math 375 Project Pictures
Page Last Updated June 1, 2011
.gif){kind=small}
.gif){kind=small}