When approximate positions have been determined for most, if not all, of the atoms, it is time to begin the refinement of the structure. In this process the atomic parameters are varied systematically so as to give the best possible agreement of the observed structure factor amplitudes (the experimental data) with those calculated for the proposed trial structure. Common refinement techniques involve Fourier syntheses and processes involving least-squares or maximum likelihood methods. Although they have been shown formally to be nearly equivalent—differing chiefly in the weighting attached to the experimental observations—they differ considerably in manipulative details; we shall discuss them separately here. Many successive refinement cycles are usually needed before a structure converges to the stage at which the shifts from cycle to cycle in the parameters being refined are negligible with respect to their estimated errors. When least-squares refinement is used, the equations are, as pointed out below, nonlinear in the parameters being refined, which means that the shifts calculated for these parameters are only approximate, as long as the structure is significantly different from the “correct” one. With Fourier refinement methods, the adjustments in the parameters are at best only approximate anyway; final parameter adjustments are now almost always made by least squares, at least for structures not involving macromolecules. As indicated earlier (Chapters 8 and 9, especially Figure 9.8 and the accompanying discussion), Fourier methods are commonly used to locate a portion of the structure after some of the atoms have been found—that is, after at least a partial trial structure has been identified. Initially, only one or a few atoms may have been found, or maybe an appreciable fraction of the structure is now known. Once approximate positions for at least some of the atoms in the structure are known, the phase angles can be calculated. Then an approximate electron-density map calculated with observed structure amplitudes and computed phase angles will contain a blend of the true structure (from the structure amplitudes) with the trial structure (from the calculated phases).
Statistical Inference for Periodic Self-Exciting Threshold Integer-Valued Autoregressive Processes
