It should be noted that not all compositions sum to unity due to rounding error.
The incorporation of individual diet heterogeneity and/or nestedness into a mixing model presents a non-trivial computational challenge due to the highly constrained covariance structure of diet compositions (i.e. compositions must sum to unity) and the resultant non-normal variance associated with compositional data.
The neutron scattering-length density, ρn, of a molecule is given by eq II : IIwhere bn, i values are the coherent neutron scattering lengths of the individual nuclei in the molecule, a property that is sensitive to the nuclear composition (the sum is again over all nuclei in the molecule), and V is again the volume occupied by one molecule.
Despite the differences in compositional variation, complexity and origin, the raw measurements recorded in a multicomponent chemical system can be very often described with a simple model consisting of the composition-weighted sum of the signals of their pure compounds.
For a given m and composition ((p_i)) summing to m, we have the mixed terms begin{aligned}&a_{p_1}(x cdots a_{p_k}(x)(2izeta )^{-m}&quad times left[ pm left( {begin{array}{c}k+1 kend{array}}right) A_M x,zeta )mp left( {begin{array}{c}k+2 kend{array}}right) A_M x,zeta )^2pm left( {begin{array}{c}k+3 kend{array}}right) A_M x,zeta )^3mp cdots right].
Nutrient intake for each patient was calculated from the weight of each food multiplied by its nutrient composition, and summed.
Lipid species were annotated either by sum composition or by molecular species composition as previously described [ 20, 27].
Reduced or base glass compositions re-summed to 100%% were used to compare data between sites.
A total of 1,527 unique lipids were measured across all laboratories and consensus location estimates and associated uncertainties were determined for 339 of these lipids measured at the sum composition level by five or more participating laboratories.
Lipid species were annotated using sum composition nomenclature as previously described (Klose et al., 2012).
