Higher Order Cumulant Truncation
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Jha, Shreya
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Abstract
Moment methods are extensively used to compute the evolution of probability distributions over time. These methods can involve nonlinear transformations where changes in lower order moments depend on higher order moments, resulting in an unclosed system. As a result, a method to express higher order moments in terms of lower order ones is required for system closure. A common closure method is the Gaussian, which approximates higher order moments using only in terms of the first and second cumulants, disregarding other relevant higher order information that could have been used. To address this, we propose higher order cumulant truncation methods that incorporate higher order information. Although only Gaussian distributions possess a finite number of non-zero cumulants, our approximation leverages the property of higher order cumulants decaying faster than lower order ones, suggesting its potential effectiveness. To obtain moments that correspond to feasible probability distributions, we project our higher order moment approximations onto the moments obtained from the set of feasible probability distributions using the set of polynomial sum-of-squares. We tested our methods through function approximations on various distributions, simulated particle collisions, moment closure applications on the Duffing oscillator, and moment closure applications on microbial consortium dynamics. We have identified situations where higher order truncation performs better than the Gaussian.
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Undergraduate Research Option Thesis