Browsing by Subject "Distributions"
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Item Archimedes, Gauss and Stochastic computation: A new (old) approach to Fast Algorithms for the evaluation of transcendental functions of generalized Polynomial Chaos Expansions(2011-05) Mckale, Kaleb D.; Long, Kevin; Howle, Victoria E.; Barnard, Roger W.; Monico, Christopher J.In this paper, we extend the work of Debusschere et al. (2004) by introducing a new approach to evaluating transcendental functions of generalized polynomial chaos expansions. We derive the elementary algebraic operations for the generalized PC expansions and show how these operations can be extended to polynomial and rational functions of PC expansions. We introduce and implement the Borchardt-Gauss Algorithm, an Arithmetic-Geometric Mean (AGM)-type method to derive the arctangent for the Jacobi-Chaos expansion. We compare numerically the BG Algorithm versus the Line Integral Method of Debusschere et al. and the Non-intrusive Spectral Projection (NISP) Method. We present the future direction of our research, including incorporating more efficient AGM-type methods proposed by Carlson (1972) and Brent (1976) to calculate the arctangent and other transcendental functions.Item Distributions of sums of binary variables in survey research(2012-05) Morgan, Dorothy Lam; Lin, Tse-min; Roberts, BrianIn survey research, researchers often add up a finite number of binary responses to form an index of some political attitude or behavior, such as political knowledge and political participation. Indices of this sort are called grouped binary variables in political science research; they comprise finite and countable binary items that take on only integer values ranging from zero to the total number of survey items. Commonly-used distributions for modeling these kinds of indices are the binomial, beta binomial, and extended beta binomial distributions. But whether these distributions are appropriate depends on the assumptions that the binary responses are identically and independently distributed Bernoulli random variables. If these assumptions are violated, the binomial, beta binomial, and extended beta binomial models are rendered questionable, and it may be more useful to turn to other distributions of sums of Bernoulli variables, called generalized binomial distributions. To facilitate the use of generalized binomial distributions in political science research, this report is a review of the various probability distributions of grouped binary variables. This report clarifies the nature of the distributions of sums of Bernoulli variables in survey research by considering whether the Bernoulli variables are independently and/or identically distributed, whether there is heterogeneity across survey items and/or across respondents, and the consequences of these considerations for the relative dispersion of each generalized binomial distribution.