Question 6 of 8: Network Post-Analysis — Confidence Ellipses

Question 6: Network Post-Analysis — Confidence Ellipses (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The point covariance matrix from a least-squares network adjustment, from which standard and 95% confidence regions (ellipses in 2-D, ellipsoids in 3-D) are derived.

Find. (a) definitions of the standard error ellipse/ellipsoid and their confidence level; (b) the meaning of out-of-context vs in-context (simultaneous) regions; (c) how to scale the standard ellipse up to 95%.

(a) Standard error ellipse and ellipsoid. After adjustment, each point has a \(2\times2\) (or \(3\times3\)) covariance sub-matrix. The standard error ellipse is the locus obtained from the eigen-decomposition of that covariance matrix: its semi-axes are \(a=\sigma\sqrt{\lambda_1}\) and \(b=\sigma\sqrt{\lambda_2}\) (the square-roots of the eigenvalues, i.e. the largest and smallest standard deviations of position), and its orientation \(\theta\) is the direction of the major eigenvector. It shows the magnitude and direction of the point’s positional uncertainty at the “1-sigma” level. The standard error ellipsoid is the 3-D analogue (three semi-axes from a \(3\times3\) covariance matrix). Crucially, the standard ellipse does not correspond to 68% (that is the 1-D figure): in 2-D it encloses \(P=1-e^{-1/2}=\mathbf{39.4\%}\) of the probability, and the standard ellipsoid in 3-D encloses only about \(\mathbf{19.9\%}\).

(b) Out-of-context vs in-context. An out-of-context (or “relative-to-a-single-point”) ellipse is built from the covariance of one point considered in isolation — it answers “where is this point, ignoring the rest of the network.” An in-context (or simultaneous) ellipse accounts for the fact that we are making a confidence statement about many points (or several coordinates) at once: to keep the joint confidence at the stated level, each region must be enlarged. Statistically, the out-of-context ellipse is scaled with the degrees of freedom of the point alone (\(k=2\) for an ellipse, \(k=3\) for an ellipsoid), whereas the in-context region uses the dimension of the whole parameter vector, \(k=u\) (all network coordinates): \(c=\sqrt{\chi^2_{u,\,1-\alpha}}\) with a known variance factor, or \(\sqrt{u\,F_{u,\nu,\,1-\alpha}}\) when it is estimated from the redundancy \(\nu\). Because \(u\gg2\), e.g. \(u=20\) gives \(\sqrt{\chi^2_{20,0.95}}=5.60\) against 2.45 for a single point, and the simultaneous ellipse is therefore always larger than the out-of-context one for the same nominal confidence. Using out-of-context ellipses when you actually need a network-wide statement understates the true uncertainty.

(c) 95% ellipse from the standard ellipse. Keep the same shape and orientation and multiply both semi-axes by a single scale factor \(c\). For a 2-D region the positional statistic follows a \(\chi^2\) distribution with 2 degrees of freedom (variance factor assumed known), so

$$c_{95\%}=\sqrt{\chi^2_{2,\,0.95}}=\sqrt{5.991}=2.45.$$

Thus \(a_{95\%}=2.45\,a,\quad b_{95\%}=2.45\,b\). (For comparison, the 99% factor is \(\sqrt{\chi^2_{2,0.99}}=3.03\); and for the 3-D ellipsoid the 95% factor is \(\sqrt{\chi^2_{3,0.95}}=2.80\).) If the variance factor is estimated rather than known, \(\chi^2\) is replaced by \(2F_{2,\,\nu}\) with \(\nu\) the adjustment redundancy.