Iowa State REU Projects

Six Alliance Scholars will work on projects offered by the Department of Mathematics and six Alliance Scholars will work on projects offered by the Department of Statistics at Iowa State University. Students will each have a private bedroom and shared living quarters at Frederiksen Court. Lunch or dinner will be provided each day at Union Drive Marketplace.

Participants spend the entire eight weeks working on research projects as part of active research groups at ISU that consist of faculty, graduate students, and undergraduates. The projects are in a variety of mathematical and statistical areas, representing the diverse research interests of the ISU Mathematics and Statistics Departments, such as dynamical systems, probability, linear algebra, spatial statistics, bioinformatics, and applications in social sciences and genetics. At the beginning of the summer the mentors explain the necessary background to the students, but there is no formal workshop or class component to this program.

For questions about the ISU program, please contact Prof. Leslie Hogben, lhogben@iastate.edu.

For more information, please visit http://orion.math.iastate.edu/reu/homepage.html


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Random walks in a game-theoretic environment

Mentor: Dr. Alexander Roitershtein

The project focuses on the analytical study of a class of self-interacting random walks. The random walks that we consider are on an integer lattice and are nearest-neighbor, that is all the steps are of the length one. In contrast to the usual setting we assume that there are two walkers (players) rather than one. The steps of the random walk are determined by the joint decision made in a "Stochastic Game" between players. The players receive rewards for any strategy that they choose, and each player's goal is to maximize the expectation of his total reward.

The project is in the intersection of two fascinating mathematical areas, probability theory and game theory. No prior knowledge of any of them is required. A decent proficiency with the Calculus and any basic course in probability taken in the past would be an advantage. The project might (or might not) involve simulations in MATLAB.

In the last year our group submitted two papers based on the outcome of our REU09 project. We will aim to write a research paper also in this year.

Eventually Nonegative Matrices and Sign Patterns

Mentors: Dr. Minnie Catral and Dr. Leslie Hogben

The square real matrix A is eventually nonnegative if there is a positive integer k 0 such that for every k > k 0, Ak > 0. A sign pattern is a matrix whose entries are elements of {+, -,0}; it describes the set of real matrices whose entries have the signs in the pattern. This project will investigate sign patterns that allow eventual nonnegative matrices and related problems.

Students involved in this project will be part the ISU Combinatorial Matrix Theory Research Group; more information is available on that page. This group regularly publishes its results. The summer 2004, 2005, 2006 groups all published papers that have appeared in in Linear Algebra and Its Applications and Electronic Journal of Linear Algebra (see list of papers); the 2009 group has papers in preparation. Linear algebra is a prerequisite for this project, graph theory is an advantage, and a strong theoretical mathematics background (usually including abstract algebra or real analysis) is expected. The software we use is Mathematica and/or Sage, but you can learn that here.

Dynamical Systems and Markov Chains

Menotr: Dr. Wolfgang Kliemann

Many ‘real world’ phenomena can be modeled mathematically via ordinary differential equations; this includes many engineering systems, reaction kinetics in chemistry, population dynamics in biology, market behavior in economics, etc. More often than not, these systems will be subjected to perturbations (cars traveling on roads, variations of influx into a chemical reactor, environmental conditions, etc) that need to be taken into account to obtain a workable model of the phenomenon. If we have statistical information about the perturbation (such as spectra of road surfaces, weather predictions, or market volatility), a finite state Markov chain in discrete time is often the model of choice for the perturbation.

We are then faced with a (continuous time) differential equation (on a continuous state space) that is perturbed by a (discrete time) Markov chain (with finitely many, hence discrete states). How can one build a mathematical theory that studies these (hybrid) systems? What can be said about the behavior of such systems, and about our ability to control them (i.e. to make the system behave like we want it to behave)? How can we estimate from actual data the parameters of Markov chains that model specific perturbations well?

The project will address at least some of these questions; its focus will depend on the interest and backgrounds of participants. We will start looking at mathematical descriptions of such hybrid systems (including their simpler version, switched systems, that omit the probabilistic part) and at numerical methods that allow reliable simulations of system behavior. We will then study either control theoretic aspects, or more statistical questions of parameter estimation.

This project is suitable for students with a first course in differential equations and in linear algebra and some knowledge of analysis/topology; background in probability/statistics would certainly be a plus, but is not required. Students will study an application area of their interest (biology, chemistry, engineering, economics, …) including results and simulations for their specific topic.

Spatial Sampling Design with Ancillary Information

Mentor: Dr. Zhengyuan Zhu

In many applications one need to observe a random process over some space at a set of sample locations, and make inference about some functions of the process. Examples include surveys of soil materials, air pollution monitoring, ecological survey, Geological Survey for Oil and Gas Resources, etc. Since the number of locations one can sample is almost always constrained by available resources, it is of great importance to find efficient spatial sampling design which can lead to accurate and unbiased inference. In practice, ancillary information is often available which is related to the variable of interests. For example, in digital soil mapping, the soil properties of interests are related to some ancillary soil and environmental variables which can be obtained cheaply over large areas through remote sensing. We will study how to use ancillary information to improve the efficiency of spatial sampling, and develop innovative design approaches which can be implemented in practice. A digital soil mapping dataset will be used as a test bed to compare different approaches. This project will involve statistics and computation.

Development of Statistical and Computational Methods for Analysis of RNAseq Data

Mentor: Dr. Peng Liu

The next-generation sequencing technology allows digital measurements of gene expressions. The resulting RNAseq data provide much richer data about gene expressions than microarray technologies and calls for novel statistical analysis. This summer research project involves the development of statistical and computational methods for assessing RNAseq data. The project will involve descriptive and inferential statistics about RNAseq data and some computational activity.

Analysis of Data from the Family Transition Project

Mentor: Dr. Fred Lorenz

During the past two decades, the Family Transition Project has been following a panel of over 500 rural Iowa families. The objectives are to study family resilience to economic and family stress and to trace continuity in behaviors between one generation and the next. Lorenz's specific interests are in modeling change over time and in modeling the relationship between observer ratings and questionnaire reports of behavior.

Statistical Analysis of Microarray Data on Gene Expression

Mentor: Dr. Dan Nettleton

Microarray technologies allow researchers to simultaneously measure the expression of thousands of genes in multiple biological samples. By examining how genes change expression across different types of samples or samples collected under different conditions, researchers gain clues about how genes act together to carry out important biological processes. Genes can be organized into groups based on past research. Genes in a group may share a function or act together in the same biological process. Researchers often wish to learn whether known groups of genes change their behavior in response to new conditions. This summer research project involves the development of statistical and computational methods for assessing evidence of group expression change in response to stimuli. The project will involve mathematics, statistics, and computation. Although biological data will be used, no special background in biology is required.

The Effect of Nonconstant Variance on Spatial Prediction

Mentor: Dr. Mark Kaiser

Spatial prediction of phenomena such as weather variables, groundwater contamination, or mineral deposits (and many more) is often approached through the use of what statisticians call {\it geostatistical} methods. The standard form of predictors developed in a geostatistical analysis make a number of simplifying assumptions about the process being observed. In particular, it is often assumed that the process has a constant mean across all locations, a constant variance across all locations, and a covariance that depends only on distance (rather than, for example, direction). Of these, a good deal of work has been conducted on what happens when the assumptions of constant mean and covariance depending only on distance are violated. Much less is known about the effects of nonconstant variance. The goal of this project, most likely to be approached through Monte Carlo simulation, is to investigate the behavior of the standard methods when the true process being modeled has constant mean and covariance depending only on distance, but violates the assumption of constant variance. To undertake this project, the student will need to become familiar with basic geostatistical tools such as variograms and kriging predictors, and will be required to develop the skills needed to conduct a simulation study using the language R.

Estimating Water Quality Through Subsampling

Mentors: Dr. Mark Kaiser and Dr. Dan Nordman

Limnology is the study of lakes, focusing largely on the variables that govern the water quality in lakes, both natural and manmade (called reservoirs). Limnologists often sample a lake several (3 or 4) times during the summer and compute a ``seasonal mean'' as the average, and that value may be judged against water quality standards or regulations to determine the status of a lake.

There are questions about what a seasonal mean actually represents relative to what happens in a lake over the course of a year. This project will use data from a study in which 3 lakes were sampled every day from mid-May to mid-September. The data form a time series, as illustrated for one of the lakes in Figure 1 for the variables of phosphorus and chlorophyll (two of the primary indicators of water quality). Even visual examination of these plots suggests that the values of these variables over time should not be considered independent. In addition, what is the true mean, that limnologists attempt to estimate by sampling 3 or 4 particular days?

In this projet we will apply a statistical technique called subsampling to estimate the mean over the entire data record and provide a measure of uncertainty (standard error) for that estimate. The basic idea is to use portions of the data record (subsamples) that are short enough to give sufficient replication (number of subsamples) but long enough to preserve the dependence structure of the entire data record within each subsample. We can then examine a number of sampling strategies that might be used by limnologists through a Monte Carlo study. This project will introduce students to statistical dependence in time series, the techniques of subsampling, and hopefully an assessment of practical value for scientists and managers.