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Math Project Descriptions 2010

Random walks in a game-theoretic environment 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.

Eventually Nonegative Matrices and Sign Patterns  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, A^k > 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 investigated sign patterns that allow eventually positive or eventual;ly exponentially positve matrices. 

Zero forcing on hypercubes   Dr. Leslie Hogben

Initially a subset Z of the vertices of a graph G are colored black and the remaining vertices are colored white.  The color change rule is that if a black vertex v has exactly one white neighbor w, then change the color of w to black.   The set Z is a zero forcing set if after applying the color change rule until no more changes are possible, all the vertices of G are black.  The zero forcing number is the minimum size of a zero forcing set.  Zero forcing arose in the study of determining the minimum rank/maximum nullity among real symmetric matrices having nonzero off-diagonal positions described by the edges of a given graph (the zero forcing number is an upper bound for maximum nullitty.   This project was proposed by students and investigated zero forcing numbers of cut complexes of hypercubes.



ISU Stat REU      

Each statistics participant has an individual project that s/he investigates under the supervision of faculty and graduate student mentors.   Possible projects are listed below.

ISU Stat Project Descriptions 2010

Spatial Sampling Design with Ancillary Information  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   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    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 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  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  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.

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