The Case of Independent, Identically Normally-distributed Errors
See also: Ordinary least squaresSuppose we have found a solution to the following overdetermined problem:
where Y is an n-dimensional column vector containing observed values, X is an n-by-p matrix which can represent a physical model and which is assumed to be known exactly, is a column vector containing the p parameters which are to be estimated, and is an n-dimensional column vector of errors which are assumed to be independently distributed with normal distributions with zero mean and each having the same unknown variance .
A joint 100(1 - ) % confidence region for the elements of is represented by the set of values of the vector b which satisfy the following inequality:
where the variable b represents any point in the confidence region, p is the number of parameters, i.e. number of elements of the vector and s2 is an unbiased estimate of equal to
Further, F is the quantile function of the F-distribution, with p and degrees of freedom, is the statistical significance level, and the symbol means the transpose of .
The above inequality defines an ellipsoidal region in the p-dimensional Cartesian parameter space Rp. The centre of the ellipsoid is at the solution . According to Press et al., it's easier to plot the ellipsoid after doing singular value decomposition. The lengths of the axes of the ellipsoid are proportional to the reciprocals of the values on the diagonals of the diagonal matrix, and the directions of these axes are given by the rows of the 3rd matrix of the decomposition.
Read more about this topic: Confidence Region
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