Gram-Charlier series

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... 

From the point of view of mathematics, the zone profiles encountered in chromatography belong to the class of distributions, which can be described in terms of their moments and cumulants [6] by the Gram-Charlier series... 

Figure 4.2 presents a computer-generated profile of this type and its best fit by the series 4.19. [9] On this occasion the approximation is better than that provided by 4.20. As could be expected, the Gram-Charlier series experiences problems in fitting the exponential tail of the profile. 

In addition to the EMG fimction, many other mathematical models have been suggested to accoimt for the profiles of experimental peaks and to determine characteristic shape-parameters, such as a number of theoretical plates, a skew and an excess. These parameters are related to the second, third and fourth moments (Eq. 6.77) of the peak, respectively. For example, the Gram-Charlier series (GC) [96,114,115] and the Edgeworth-Cramer series (EC) [115,116] have been... 

This expansion describes an arbitary distribution, P , for n>0 as a perturbation about a T-distribution, equivalent to the Gram-Charlier series resulting from a perturbation about a Gaussian distribution using Hermite polynomials [46]. 

Using a Gram-Charlier series with terms up to 4 order for f)((x) and a Gaussian distribution for fy(y) leads to the following set of equations for the stationary state ... 

For one particular set of data In F1g.3 the joint distribution of y and z, according to the method of non-6auss1an closure, Is compared with the distribution obtained by numerical simulation. On one hand It can be observed that the actual distribution of y and z Is entirely different from a joint normal one, hence It can not be approximated by a trunctated Gram-Charlier series in detail. Nevertheless It Is possible to see a strong correlation between both processes and some bimodality. But on the other hand negative values of the probability density function can also be noticed. 

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. ... 

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