The matrix we obtain from inverting the Hessian, \(\textbf {H}^{-1}\), turns out to be very useful. It was given the name variance-covariance matrix by Albert Einstein but at least some of us, perhaps those of us who were not raised speaking with large German polysyllabic words, find that a mouthful and call it the covariance matrix instead. Like the Hessian, it is also a symmetric matrix.
One property of the covariance matrix is that it can be used to compute the correlation coefficient between two parameters. If \(\textbf {V} = \textbf {H}^{-1}\), then the correlation coefficient between parameters \(j\) and \(k\) is given by \(V_{jk} / \sqrt {V_{jj}V_{kk}}\). If two parameters have completely independent effects on the fit, or in other words, are orthogonal, then the correlation coefficient will be 0. If two parameters have nearly the same effect, then the correlation coefficient will be close to 1. If the correlation coefficient were to be 1, then the Hessian would be singular. If two parameters have opposite effects on the fit, then the correlation coefficient between them will be negative. Correlation is impossible to avoid in crystallographic fitting. As an example, as will be discussed in Chapter 13, site occupancies will always have some correlation with ADP values, but as will be made clear when correlation becomes too severe, then fitting fails. It is at this point where alternate models need to be considered.
In GSAS-II, covariance values are found in the Covariance data tree entry, or in sequential fits, by double-clicking on the row for the fit. The values are plotted, as shown in Fig. 7.1.
One of the strengths of crystallography is not only that it gives us values for parameters, such as atomic positions, that are derived directly from our experimental observations, but also that it provides a measure for the precision of these values that is derived from the uncertainties on the observations. These uncertainty estimates are known as standard uncertainties. Previously, the term estimated standard deviation (e.s.d.) was used for these values, but use of this term is now discouraged.
The diagonal values in the covariance plot (Fig. 7.1) are all green, since \(V_{jj} / \sqrt {V_{jj}V_{jj}} = 1\), but for these diagonal terms in the matrix, the value \(\sqrt {V_{jj}}\) provides the standard uncertainty for parameter \(j\). Some programs increase this number by multiplying by the reduced \(\chi ^2\), but as was pointed out by Ted Prince, the relationship between the covariance matrix and standard uncertainties is only valid when there is no significant level of systematic error and if that is true then the reduced \(\chi ^2\) value is close to 1, and in that case this choice would make little difference. Further, if there is systematic error, the effect of that can be to bias a parameter by a factor much more, or much less, than the value of the reduced \(\chi ^2\), so Ted frowned on multiplying the standard uncertainty by the reduced \(\chi ^2\).
In GSAS-II’s covariance plot (in Fig. 7.1), placing the mouse over the diagonal elements shows the value and the s.u. value for a parameter as a “tooltip.” The values are also shown in the status line at the bottom of the plot. Other places where the s.u. values can be found are in the .lst file produced during each refinement, in a project-based CIF export and in the sequential results table.
Crystallography uses a compact special notation for display of numbers with their standard uncertainty. The uncertainty was originally expressed as a single digit, expressing the uncertainty on the last digit in a value, so 1.234(5) means a value of 1.234 with a standard uncertainty of 0.005. There is a bit of a loss of precision when that final digit would be (1) so it is common to extend this notation to two digits for uncertainty values up to (19) so 1.2345(14) means a value of 1.2345 with a standard uncertainty of 0.0014. For numbers where the s.u. value is greater than 10, use scientific notation, such as \(1234(5) \times 10^3\) for a value of 1,234,000 with an s.u. of 5000. This practice of displaying numbers with their uncertainties is referred to by some (particularly outside the U.S.) as bracket notation.
