the estimate . According to Press et al., it is 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. Weighted and generalised least squares Now consider the more general case where some distinct elements of have known nonzero covariance (in other words, the errors in the observations are not independently distributed), and/or the standard deviations of the errors are not all equal. Suppose the covariance matrix of is , where V is an n-by-n nonsingular matrix which was equal to in the more specific case handled in the previous section, (where I is the identity matrix,) but here is allowed to have nonzero off-diagonal elements representing the covariance of pairs of individual observations, as well as not necessarily having all the diagonal elements equal. It is possible to find a nonsingular symmetric matrix P such that In effect, P is a square root of the covariance matrix V. The least-squares problem can then be transformed by left-multiplying each term by the inverse of P, forming the new problem formulation where and A joint confidence region for the parameters, i.e. for the elements of , is then bounded by the ellipsoid given by: Here F represents the percentage point of the F-distribution and the quantities p and n-p are the degrees of freedom which are the parameters of this distribution. Nonlinear problems Confidence regions can be defined for any probability distribution. The experimenter can choose the significance level and the shape of the region, and then the size of the region is determined by the probability distribution. A natural choice is to use as a boundary a set of points with constant (chi-squared) values. One approach is to use a linear approximation to the nonlinear model, which may be a close approximation in the vicinity of the solution, and then apply the analysis for a linear problem to find an approximate confidence region. This may be a reasonable approach if the confidence region is not very large and the second derivatives of the model are also not very large. Bootstrapping approaches can also be used. See also Circular error probable Linear regression Confidence band Credible region Notes References Estimation