Statistics and Probability Commons^™

The Quantum Dialectic, Logan Kelley

Pitzer Senior Theses

A philosophic account of quantum physics. The thesis is divided into two parts. Part I is dedicated to laying the groundwork of quantum physics, and explaining some of the primary difficulties. Subjects of interest will include the principle of locality, the quantum uncertainty principle, and Einstein's criterion for reality. Quantum dilemmas discussed include the double-slit experiment, observations of spin and polarization, EPR, and Bell's theorem. The first part will argue that mathematical-physical descriptions of the world fall short of explaining the experimental observations of quantum phenomenon. The problem, as will be argued, is framework of the physical descriptive schema. Part …

Markov Bases For Noncommutative Harmonic Analysis Of Partially Ranked Data, Ann Johnston

HMC Senior Theses

Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool …

Martingale Couplings And Bounds On Tails Of Probability Distributions, Kyle Luh

HMC Senior Theses

Wassily Hoeffding, in his 1963 paper, introduces a procedure to derive inequalities between distributions. This method relies on finding a martingale coupling between the two random variables. I have developed a construction that establishes such couplings in various urn models. I use this construction to prove the inequality between the hypergeometric and binomial random variables that appears in Hoeffding's paper. I have then used and extended my urn construction to create new inequalities.

How Other Drivers’ Vehicle Characteristics Influence Your Driving Speed, Russell Brockett

CMC Senior Theses

An analysis of the effect of passing vehicles’ characteristics and their impact on other drivers’ velocities was investigated.  Three experimental studies were proposed and likely outcomes were discussed.  Experiment 1 focused on the effect of passing vehicle type (SUV, sedan or truck) on driver speed.  Drivers were hypothesized as going faster when the same vehicle type as they were driving passed them versus when no vehicle or a different vehicle passed them.  Experiment 2 focused on the effect of passing SUV age on driver’s speed.  Evidence suggests passing older SUVs will increase the driver’s speed more than new SUVs.  Experiment …

Applying Localized Realized Volatility Modeling To Futures Indices, Luella Fu

CMC Senior Theses

This thesis extends the application of the localized realized volatility model created by Ying Chen, Wolfgang Karl Härdle, and Uta Pigorsch to other futures markets, particularly the CAC 40 and the NI 225. The research attempted to replicate results though ultimately, those results were invalidated by procedural difficulties.

Is A Basketball Free-Throw Sequence Nonrandom? A Group Exercise For Undergraduate Statistics Students, Stephen C. Adolph

All HMC Faculty Publications and Research

I describe a group exercise that I give to my undergraduate biostatistics class. The exercise involves analyzing a series of 200 consecutive basketball free-throw attempts to determine whether there is any evidence for sequential dependence in the probability of making a free-throw. The students are given the exercise before they have learned the appropriate statistical tests, so that they can come up with ideas on their own. Students spend a full class period working on the problem, with my guidance and hints. In the next class period, we discuss how each student group approached the problem. I then present several …

A Robust Measure Of Correlation Between Two Genes On A Microarray, Johanna S. Hardin, Aya Mitani '06, Leanne Hicks, Brian Vankoten

Pomona Faculty Publications and Research

Background

The underlying goal of microarray experiments is to identify gene expression patterns across different experimental conditions. Genes that are contained in a particular pathway or that respond similarly to experimental conditions could be co-expressed and show similar patterns of expression on a microarray. Using any of a variety of clustering methods or gene network analyses we can partition genes of interest into groups, clusters, or modules based on measures of similarity. Typically, Pearson correlation is used to measure distance (or similarity) before implementing a clustering algorithm. Pearson correlation is quite susceptible to outliers, however, an unfortunate characteristic when dealing …

Analyzing Dna Microarrays With Undergraduate Statisticians, Johanna S. Hardin, Laura Hoopes, Ryan Murphy '06

Pomona Faculty Publications and Research

With advances in technology, biologists have been saddled with high dimensional data that need modern statistical methodology for analysis. DNA microarrays are able to simultaneously measure thousands of genes (and the activity of those genes) in a single sample. Biologists use microarrays to trace connections between pathways or to identify all genes that respond to a signal. The statistical tools we usually teach our undergraduates are inadequate for analyzing thousands of measurements on tens of samples. The project materials include readings on microarrays as well as computer lab activities. The topics covered include image analysis, filtering and normalization techniques, and …

Yeast Through The Ages: A Statistical Analysis Of Genetic Changes In Aging Yeast, Alison Wise '05, Johanna S. Hardin, Laura Hoopes

Pomona Faculty Publications and Research

Microarray technology allows for the expression levels of thousands of genes in a cell to be measured simultaneously. The technology provides great potential in the fields of biology and medicine, as the analysis of data obtained from microarray experiments gives insight into the roles of specific genes and the associated changes across experimental conditions (e.g., aging, mutation, radiation therapy, drug dosage). The application of statistical tools to microarray data can help make sense of the experiment and thereby advance genetic, biological, and medical research. Likewise, microarrays provide an exciting means through which to explore statistical techniques.

The Ims New Researchers' Survival Guide, Naomi Altman, David Banks, Janis Hardwick, Kathryn Roeder, Peter Craigmile, Johanna S. Hardin, Mayetri Gupta

Pomona Faculty Publications and Research

Statistics is a wonderfully diverse profession and graduate students making career choices have many options — especially in light of the dearth of students moving into the statistical sciences today. The three main career paths at the PhD level are in academics, industry/business and government. Each of these job types offers its own mix of intellectual challenges, financial reward, pressure and security. How a new researcher selects (or is selected by) a specific occupation in the statistical sciences sometimes seems more a function of luck than of conscious decision making. This consideration was one of the first concerns addressed by …

Microarray Data From A Statistician’S Point Of View, Johanna S. Hardin

Pomona Faculty Publications and Research

No abstract provided.

Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d-dimensional torus ^d = ℝ^d/ℤ^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^−n/2, where C₁ = C(n, d) is …

No News Is Bad News: Characteristics Of Adolescents Who Provide Neither Parental Consent Nor Refusal For Participation In School-Based Survey Research, C. Anderson Johnson, Jennifer B. Unger, Peggy E. Gallaher, Paula Palmer, Lourdes Baezconde-Garbanati, Dennis R. Trinidad, Steven Cen

CGU Faculty Publications and Research

Schools offer a convenient setting for research on adolescents.However, obtaining active written parental consent is difficult. In a 6th-grade smoking study, students were recruited with two consent procedures: active consent (parents must provide written consent for their children to participate) and implied consent (children may participate unless their parents provide written refusal). Of 4,427 invited students, 3,358 (76%) provided active parental consent, 420 (9%) provided active parental refusal, and 649 (15%) provided implied consent (parental nonresponse). The implied consent procedure recruited more boys, African Americans, students with poor grades, and smokers. This dual-consent procedure is useful for collecting some limited …

Evaluation Of Multiple Models To Distinguish Closely Related Forms Of Disease Using Dna Microarray Data: An Application To Multiple Myeloma, Johanna S. Hardin, Michael Waddell, C. David Page, Fenghuang Zhan, Bart Barlogie, John Shaughnessy, John J. Crowley

Pomona Faculty Publications and Research

Motivation: Standard laboratory classification of the plasma cell dyscrasia monoclonal gammopathy of undetermined significance (MGUS) and the overt plasma cell neoplasm multiple myeloma (MM) is quite accurate, yet, for the most part, biologically uninformative. Most, if not all, cancers are caused by inherited or acquired genetic mutations that manifest themselves in altered gene expression patterns in the clonally related cancer cells. Microarray technology allows for qualitative and quantitative measurements of the expression levels of thousands of genes simultaneously, and it has now been used both to classify cancers that are morphologically indistinguishable and to predict response to therapy. It is …

On Choosing And Bounding Probability Metrics, Alison L. Gibbs, Francis E. Su

All HMC Faculty Publications and Research

When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.

An Interactive Tutorial For Teaching Statistical Power, Christopher L. Aberson '99, Dale E. Berger, Michael R. Healy '04, Victoria L. Romero '07

CGU Faculty Publications and Research

This paper describes an interactive Web-based tutorial that supplements instruction on statistical power. This freely available tutorial provides several interactive exercises that guide students as they draw multiple samples from various populations and compare results for populations with differing parameters (for example, small standard deviation versus large standard deviation). The tutorial assignment includes diagnostic multiple-choice questions with feedback addressing misconceptions, and follow-up questions suitable for grading. The sampling exercises utilize an interactive Java applet that graphically demonstrates relationships between statistical power and effect size, null and alternative populations and sampling distributions, and Type I and II error rates. The applet …

Evaluating Java Applets For Teaching On The Internet, Michael R. Healy '04, Dale E. Berger, Victoria L. Romero '07, Christopher L. Aberson '99, Amanda Saw '11

CGU Faculty Publications and Research

The Web Interface for Statistics Education (http://wise.cgu.edu) is a website built around interactive tutorials designed to teach introductory and advanced statistical concepts. The tutorials use Java applets that dynamically illustrate the statistical concepts being taught. By using Java applets, we teach statistics in a manner not possible in a traditional classroom environment. In this paper, we provide examples of the applets, illustrate how students use them, and we report the outcome of a study that examined tutorial effectiveness as a learning tool.

Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su

All HMC Faculty Publications and Research

We analyze the drunkard's walk on the unit sphere with step size θ and show that the walk converges in order C/sin²(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.

What's Best?, Arthur T. Benjamin, Matthew T. Fluet '99

All HMC Faculty Publications and Research

No abstract provided in this article.

A Rational Solution To Cootie, Arthur T. Benjamin, Matthew T. Fluet '99

All HMC Faculty Publications and Research

No abstract provided in this article.

A Leveque-Type Lower Bound For Discrepancy, Francis E. Su

All HMC Faculty Publications and Research

A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on R^k / Z^k. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus.

Tea Tasting And Pascal Triangles, Paul Alper, Yongzhi Yang

Humanistic Mathematics Network Journal

No abstract provided.

Why The Player Never Wins In The Long Run At La Blackjack, Arthur T. Benjamin, Michael Lauzon '00, Christopher Moore '00

All HMC Faculty Publications and Research

No abstract provided in this article.

Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su

All HMC Faculty Publications and Research

Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp …

Methodological And Substantive Issues In Substance Abuse Prevention Research, C. Anderson Johnson, John W. Farquhar, Steve Sussman

CGU Faculty Publications and Research

This article summarizes current issues in drug abuse prevention research through integration of other articles in this journal and by heeding historical trends in prevention science. Recommendations are made for future research directions. For prevention to advance, iterative processes are needed involving both quasi-experimental and experimental designs and involving both small, simple units and large, complex, interactive units. Accuracy of measurement and replication are of paramount importance.

The Questionable Probability Theory Behind The Strange Story Of The Bell Curve's Bell Curve, Miriam Lipschütz-Yevick

Humanistic Mathematics Network Journal

No abstract provided.

Optimization In Chemical Kinetics, Arthur T. Benjamin, Gordon J. Hogenson '92

All HMC Faculty Publications and Research

No abstract provided in this article.

Students Become Data, Statistics Comes Alive., Linc. Fisch

Humanistic Mathematics Network Journal

No abstract provided.

Reliable Computation In The Presence Of Noise, Nicholas Pippenger

All HMC Faculty Publications and Research

This talk concerns computation by systems whose components exhibit noise (that is, errors committed at random according to certain probabilistic laws). If we aspire to construct a theory of computation in the presence of noise, we must possess at the outset a satisfactory theory of computation in the absence of noise.

A theory that has received considerable attention in this context is that of the computation of Boolean functions by networks (with perhaps the strongest competition coming from the theory of cellular automata; see [G] and [GR]). The theory of computation by networks associates with any two sets Q and …

Group Comparability: A Multiattribute Utility Measurement Approach To The Use Of Random Assignment With Small Numbers Of Aggregated Units, C. Anderson Johnson, John W. Graham, Brian R. Flay, William B. Hansen, Linda M. Collins

CGU Faculty Publications and Research

It is not always possible, especially in large-scale evaluation research, to ensure that random assignment will produce groups that are comparable on any number of potentially important factors. Typically, gaining comparability has been achieved only at the expense of random assignment. A method is presented that allows multivariate comparability while making only minimal restrictions on randomization. The procedure is demonstrated in the context of assigning 63 aggregated units (schools) to 28 experimental and control conditions. Good comparability of groups for all primary main effects and interactions was verified for 15 individual variables.