Monday, July 1, 2013

Jon Ubnoske's Internship, Summer 2013 - Part 1

I’m working with Dr. Dmitry Dukhovskoy and Dr. Steven Morey at the Florida State University Center for Ocean-Atmospheric Prediction Studies on developing oil spill model validation metrics. The goal is to quantify the goodness of fit of proposed models of the Deepwater Horizon oil spill to actual observations. In contrast to most traditional model validation techniques, our technique quantifies similarity in shape, and it does so in a way that corresponds to human intuition.

The key idea is to develop a proper notion of distance between shapes themselves rather than between points in space. We make this notion rigorous by defining the set of all possible oil spill models as a metric space:

For those unfamiliar with topology, you can think of a topological space as an abstraction of the human perception of space. By making the set of all oil spill models into a metric space, we are giving ourselves a way to actually measure how “close” or “far” two models are from one another, and we are guaranteed by the restrictions (1)-(3) on our distance function that this notion of distance is free of pathological cases. We are essentially creating a very abstract space (which is impossible to visualize) in which the points themselves are oil spills, and in which we can measure a natural notion of distance between these points. This provides a simple framework in which to rank the goodness of fit of various proposed models.

My current efforts have been focused on what is called the modified Hausdorff distance:

This is a generalization of a traditional notion of shape distance known as the Hausdorff distance. It improves on its predecessor by showing decreased sensitivity to outliers. Significant empirical evidence indicates that this function measures similarity in shape in a way that corresponds with human intuition (for example, I successfully applied it in a simple facial recognition test). Unfortunately, the function does not always pass requirement (3) of a metric space distance function. Much of my work this week has been dedicated to investigating precisely when this inequality fails. When the data is placed on a regular grid, as is the case with our oil spill models, my intuition tells me the inequality is always satisfied. Experimentation in MATLAB has confirmed this conjecture, with the inequality holding for up to 1 million randomly generated sets A, B, and C which are organized on a regular grid. The next step is to prove this mathematically. This will then provide a fully rigorous framework for validating oil spill models developed by our collaborators at the research organization SINTEF in Norway.

Posted by:
Jon Ubnoske, Florida State University

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