projects.2018.pdf - Optimization Projects At the beginning...

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Optimization Projects At the beginning of the semester, I indicated that I wanted for each of you to do an optimization project. It is now time for us to begin to try to define the individual projects. I know that some of you have projects that you would like to pursue. If you have a project in mind, please write up a one page description of the project. I will then evaluate what you have given me and determine if the project has sufficient depth and complexity without being overly ambitious. We can then talk about the project and come up with a final description of what you are going to do. For those of you who do not have a project in mind, I will now define four potential projects. Two of these projects come from statistical problems that I am interested in or have worked on. But they are self contained as optimization problems and knowledge of statistics or the statistical background of the problems is not required. A third project is a continuous graph theory problem. The final project can be classified as an inverse problem and this type of problem is quite typical of a wide range of interesting examples. Problem I. Local polynomial smoother - Nonlinear Regression Suppose that we would like to estimate a function f ( x ) given measurements { ( x i , y i ) } N i =1 . From this data how do we estimate the unknown function y = f ( x ) given the assumption that our measurements of y contains measurement errors (or noise)? This is commonly called the regression problem. Clearly, simple interpolation of the noisy data will give a poor solution to the problem. There are many partial solutions to the regression problem including fitting a straight line (or plane in higher dimensions) or perhaps fitting a quadratic function. But these solutions do not solve the more general problem where the unknown function f cannot be approximated by a plane or a quadratic surface. The solutions that I want you to implement are the local linear and local quadratic regression in dimensions one and two. A brief write up for this set of problems, and their solution from the book on Statistical Learning is attached. Please note that the solution to this portion of the problem is not iterative, but at each point x , the solution is given by solving a linear system.

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