The Geometric Model
March 5, 2007

The Geometric Model

Our first model involves two dimensional point data sets. The points will lie in the unit square. There will be two point sets, corresponding to the ELSs of two key words. Under the Null hypothesis, the points in each set are uniformly and independently distributed and the points in one set are independent of the points in the second set. The Alternative hypothesis states a dependence of the points in the first set with those of the second set. The Alternative hypothesis states that the points of one set are moved closer to that of the other set.

One model consistent with the Alternative hypothesis is a constructive probabilistic model. We generate two random point sets, each having N points, each uniformly distributed in the unit square. Each point from the second set has a closest point in the first set. The constructive probabilistic model states that a given fraction f of the points from the second set are going to be moved closer to their closest points in the first set. Those points chosen will be chosen at random and their number will satisfy the given fraction f. The movement closer will be in the geodesic direction. Suppose the distance between point p from the first set to point q of the second set is d. The new position of the point q will be on its geodesic to point p at a distance of kd, where k is a uniformly distributed variate on the interval [0,ad] and where a is a specified fraction in the interval [0,1].

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