Gram-Charlier

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

A different expansion relies on using Gram-Charlier polynomials, which are the products of Hermite polynomials and a Gaussian function [41] These polynomials are particularly suitable for describing near-Gaussian functions. Even and odd terms of the expansion describe symmetric and asymmetric deformations of the Gaussian, respectively. To ensure that P0(AU) remains positive for all values of AU, we take... 

The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

Comparisons of Gram-Charlier with data and other distributions are in problems P5.02.15 and P5.02.16. In one of these, the third order GC fits better than the fourth order. More experience is needed, however, before a judgement can be made regarding the relative merits of GC and other distributions. At large variances the finite value of the ordinate at tr - 0 appears to be a fatal objection to both the Gaussian and GC distributions. 

In this problem the Gram-Charlier will be compared with the Gamma Gauss distributions for an impulse response curve with the equation... 

Plots of Equations (10-(4) reveal the Gram-Charlier to be closest to the original data, except near tr = 0. 

Construct the third and fourth order Gram-Charlier distributions corresponding to an Erlang with cr2(tr) = 0.5 or n = 2. 

The third order Gram-Charlier approximates the Erlang more closely than does the fourth order. 

Find second order segregated conversion with kC0t = 10. The reactor has a Gram-Charlier distribution with the same first three moments as a third order Erlang distribution. 

The three-dimensional Gram-Charlier expansion, first applied to thermal motion analysis by Johnson and Levy (1974), is a statistical expansion in terms of the zero and higher derivatives of a normal distribution (Kendal and Stuart 1958). If Dj is the operator d/du], the expansion is defined by... 

The Gram Charlier temperature factor is the Fourier transform of Eq. (2.30), which is given by... 

As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

Compared with the Gram-Charlier temperature factor of Eq. (2.31), the entire series now occurs in the exponent, so, in the cumulant formalism, terms are added to the exponent of the harmonic temperature factor P0(H) = exp — fijkhjhk. ... 

Schulz 1982), corresponds to a probability distribution which is the Taylor expansion of Eq. (2.32), and similar to the Gram-Charlier distribution of Eq. 

Unlike the Gram-Charlier and cumulant formalisms, the OPP model has a physical rather than a statistical basis. It assumes that each atom vibrates in a potential well K(u), determined by the interaction with the other atoms in the crystal, without any correlation between vibrations of adjacent atoms. 

Gram-Charlier temperature factors (chapter 2) were applied in this analysis, which made use of the XD programming package (Su and Coppens, to be published). In accordance with the results on H3P04, discussed in chapter 3, the values of n, were chosen as 6, 6, 7, and 7 for / = 1, 2, 3, and 4, respectively. However, the differences between the results from this choice and those of two alternative selections (4, 4, 4, and 4 and 6, 6, 6, and 6) are within one standard deviation. 

Mallinson PR, Koritsanszky T, Elkaim E, Li N, Coppens P (1988) The Gram-Charlier and multipole expansions in accurate X-ray diffraction studies can they be distinguished Acta Crystallogr A 44 336-343... 

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