The method of least squares

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 88 | 1 | 2 |

Section 1.3.4.4.7.1. The method of least squares

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.4.7.1. The method of least squares

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Hughes (1941) was the first to use the already well established multivariate least-squares method (Whittaker & Robinson, 1944) to refine initial estimates of the parameters describing a model structure. The method gained general acceptance through the programming efforts of Friedlander et al. (1955), Sparks et al. (1956), Busing & Levy (1961), and others.

The Fourier relations between $[\rho\llap{$-\!$}]$ and F (Section 1.3.4.2.2.6) are used to derive the `observational equations' connecting the structure parameters $[\{u_{p}\}_{p = 1, \ldots, n}]$ to the observations $[\{|F_{{\bf h}}|^{\rm obs}, (\sigma_{{\bf h}}^{2})^{\rm obs}\}_{{\bf h} \in {\scr H}}]$ comprising the amplitudes and their experimental variances for a set $[{\scr H}]$ of unique reflections.

The normal equations giving the corrections δu to the parameters are then $[({\bf A}^{T}{\bf WA})\delta {\bf u} = - {\bf A}^{T}{\bf W}\Delta,]$ where $[\eqalign{A_{{\bf h}p} &= {\partial | F_{{\bf h}}^{\rm calc}| \over \partial u_{p}}\cr \Delta_{{\bf h}} &= |F_{{\bf h}}^{\rm calc}| - |F_{{\bf h}}|^{\rm obs}\cr {\bf W} &= \hbox{diag } (W_{{\bf h}}) \quad \hbox{with} \quad W_{{\bf h}} = {1 \over (\sigma_{{\bf h}}^{2})^{\rm obs}}.}]$ To calculate the elements of A, write: $[F = |F| \exp (i\varphi) = \alpha + i\beta\hbox{;}]$ hence $[\eqalign{ {\partial |F| \over \partial u} &= {\partial \alpha \over \partial u} \cos \varphi + {\partial \beta \over \partial u} \sin \varphi\cr &= {\scr Re} \left[{\partial F \over \partial u} \overline{\exp (i\varphi)}\right] = {\scr Re} \left[{\overline{\partial F} \over \partial u} \exp (i\varphi)\right].}]$

In the simple case of atoms with real-valued form factors and isotropic thermal agitation in space group P1, $[F_{{\bf h}}^{\rm calc} = {\textstyle\sum\limits_{j \in J}}\; g_{j} ({\bf h}) \exp (2\pi i{\bf h}\cdot {\bf x}_{j}),]$ where $[g_{j} ({\bf h}) = Z_{j}\;f_{j} ({\bf h}) \exp [-{\textstyle{1 \over 4}} B_{j} (d_{{\bf h}}^{*})^{2}],]$ $[Z_{j}]$ being a fractional occupancy.

Positional derivatives with respect to $[{\bf x}_{j}]$ are given by $[\eqalign{ {\partial F_{{\bf h}}^{\rm calc} \over \partial {\bf x}_{j}} &= (2\pi i{\bf h}) g_{j} ({\bf h}) \exp (2\pi i{\bf h}\cdot {\bf x}_{j})\cr {\partial |F_{{\bf h}}^{\rm calc}| \over \partial {\bf x}_{j}} &= {\scr Re} [(- 2\pi i{\bf h}) g_{j} ({\bf h}) \exp (-2 \pi i{\bf h}\cdot {\bf x}_{j}) \exp (i\varphi_{{\bf h}}^{\rm calc})]}]$ so that the corresponding $[3 \times 1]$ subvector of the right-hand side of the normal equations reads: $[\eqalign{&- \sum\limits_{{\bf h}\in {\scr H}} W_{{\bf h}} {\partial |F_{{\bf h}}^{\rm calc}| \over \partial {\bf x}_{j}} (|F_{{\bf h}}^{\rm calc}| - |F_{{\bf h}}|^{\rm obs})\cr &\quad= - {\scr Re} \left [\sum\limits_{{\bf h}\in {\scr H}} g_{j} ({\bf h}) (-2\pi i{\bf h}) W_{{\bf h}} (|F_{{\bf h}}^{\rm calc}| - |F_{{\bf h}}|^{\rm obs})\right.\cr &\qquad \times \left.\exp (i\varphi_{{\bf h}}^{\rm calc}) \exp (-2\pi i{\bf h}\cdot {\bf x}_{j})\right ].}]$

The setting up and solution of the normal equations lends itself well to computer programming and has the advantage of providing a thorough analysis of the accuracy of its results (Cruickshank, 1965b, 1970; Rollett, 1970). It is, however, an expensive task, of complexity $[\propto n \times |{\scr H}|^{2}]$ , which is unaffordable for macromolecules.

References

Busing, W. R. & Levy, H. A. (1961). Least squares refinement programs for the IBM 704. In Computing methods and the phase problem in X-ray crystal analysis, edited by R. Pepinsky, J. M. Robertson & J. C. Speakman, pp. 146–149. Oxford: Pergamon Press.Google Scholar

Cruickshank, D. W. J. (1965b). Errors in least-squares methods. In Computing methods in crystallography, edited by J. S. Rollett, pp. 112–116. Oxford: Pergamon Press.Google Scholar

Cruickshank, D. W. J. (1970). Least-squares refinement of atomic parameters. In Crystallographic computing, edited by F. R. Ahmed, pp. 187–197. Copenhagen: Munksgaard.Google Scholar

Friedlander, P. H., Love, W. & Sayre, D. (1955). Least-squares refinement at high speed. Acta Cryst. 8, 732.Google Scholar

Hughes, E. W. (1941). The crystal structure of melamine. J. Am. Chem. Soc. 63, 1737–1752.Google Scholar

Rollett, J. S. (1970). Least-squares procedures in crystal structure analysis. In Crystallographic computing, edited by F. R. Ahmed, pp. 167–181. Copenhagen: Munksgaard.Google Scholar

Sparks, R. A., Prosen, R. J., Kruse, F. H. & Trueblood, K. N. (1956). Crystallographic calculations on the high-speed digital computer SWAC. Acta Cryst. 9, 350–358.Google Scholar

Whittaker, E. T. & Robinson, G. (1944). The calculus of observations. London: Blackie.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 88