The Probability Distribution of Symmetry Values

Here the question of interest is not the closest symmetric configuration, but rather the symmetry measure or the probability distribution of the symmetry measure values given the probability distributions of the point locations. 

The matrix ACov(X)A, being symmetric and positive definite, we find the 2n x 2n matrix V diagonalizing Cov(Y), i.e., 

Thus the random variables Z that compose Z are independent and, being linear combinations of Xjt they are of normal distribution. The symmetry measure, as defined in Section III, is equivalent, in the current notations, to S = Y Y. Having S orthonormal we have  

If Z were a random variable of standard normal distribution, we would have s being of a %2 distribution of order 2 (n - 1). In the general case Z( are normally distributed but not standard and Z cannot be standardized globally. We approximate the distribution of s as a normal distribution with  

Application of the method described here to thermal ellipsoids in X-ray analysis is in progress. 

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