Brooks Emerick

A minimium cardinality set covering problem is a well-known integer programming problem. Given a random matrix of ones and zeros with fixed dimensions and density, it is possible to determine the the smallest set of columns from that matrix that covers all rows. However, once this minmal set of columns is found, is there any optimal solutions of the same cardinality? The answer is sure, in some cases. In fact, there may be a large number of alternative solutions of the same cardinality. Is there a way to predict this number? We seek an answer to this question using statistics and machine learning. The plot above shows a boxplot of the number of optimal solutions for each minimal cardinality for 10x20 matrices with 20% density.

This project considered a system of linear ODEs to model the degradation of a peptide substrate reporter for protein kinase B (VI-B) in five different cell cultures from data provided by the Chemistry department.  After solving these equations one by one, we find the best fit parameters that match the data using least squares tools in Matlab with an iterative approach. Drawing from the histogram plots of parameter distributions, we conclude that the most popular values which yield the smallest residuals are best fit parameters. The figure shows the best fit of our model to the given data. The paper can be found here.