Classification Using the Fourier Series Estimate of Multivariate Density Functions

1981 ◽  
Vol 11 (10) ◽  
pp. 726-730 ◽  
Keyword(s):  
Fourier Series ◽  
Density Functions ◽  
Multivariate Density

Some multivariate density functions of products of Gaussian variates

Biometrika ◽  
1965 ◽  
Vol 52 (3-4) ◽  
pp. 645-646 ◽  
Author(s):  
K. S. MILLER
Keyword(s):  
Density Functions ◽  
Multivariate Density

The probabilistic estimation of triplet and quartet invariants

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Fourier Series ◽  
Probability Distribution ◽  
Characteristic Function ◽  
Joint Probability ◽  
Higher Order ◽  
Probability Density Functions ◽  
Joint Probability Distribution ◽  
Density Functions ◽  
Series Representations ◽  
Joint Probability Density

This chapter describes how to estimate, by probabilistic approaches, triplet and quartet invariants from diffraction magnitudes. We will skip quintet (Fortier and Hauptman, 1977a,b,c; Hauptman and Fortier, 1977a,b; Van der Putten and Schenk, 1977; Giacovazzo, 1977b, 1980; Burla et al., 1977) and higher-order s.i.s, because their usefulness in modern phasing procedures is entirely marginal. For simplicity, we will also skip the mathematics necessary to obtain conclusive formulas (the general approach is described in Appendix 4.A), except for the triplet invariants, first representation, because of their prominent role. Triplet and quartet estimates will be discussed, particularly in relation to their impact on phasing procedures. For simplicity, some other specialized topics will also be skipped, even if theoretically relevant. For example: results obtained by Shmueli and Weiss (1986, 1992), who used Fourier series representations of joint probability density functions to estimate triplets; the effect of pseudotranslational symmetry on the triplet phase estimates, as described by Cascarano et al. (1985a,b, 1987b, 1988a,b); algebraic formulas obtained by Karle and Hauptman (1957), Vaughan (1958), Hauptman et al. (1969), Hauptman (1970), Fischer et al. (1970a,b), all related to (and encompassed by) the estimation of triplet phases via their second representation. Interested readers are referred to the original papers. Let us first consider the space group P1. According to Chapter 4, the simplest way to estimate the triplet s.i. . . . Φ = φh1 + φh2 + φh3 with h1 + h2 + h3 = 0 (5.1) . . . is to study the joint probability distribution . . . P(Eh1 , Eh2 , Eh3 ) ≡ P(Rh1 , Rh2 , Rh3 , φh1 , φh2 , φh3 ). (5.2) . . . According to Section 4.1 we must first calculate the characteristic function C and then, by Fourier inversion, recover the distribution (5.2). Because of the importance of the triplet invariant, we report the necessary calculations in Appendix 5.A.

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