Least-squares estimation of diet composition from n-alkanes in herbage and faeces using matrix mathematics

Abstract

It is possible to estimate diet composition from an analysis of n-alkanes in the faeces of ruminant animals. For instance, to estimate the proportion of two species in a diet, two equations are constructed using the known concentrations of two different n-alkanes in the herbage and in the animal's faeces. These two equations are solved for the two unknown quantities of the diet components. Two problems exist with this method. First, it is often the case that we have estimated concentrations of more than two different n-alkanes. This can lead to a problem in deciding which two n-alkanes to use to construct the simultaneous equations. The choice of this pair of n-alkanes is arbitrary in its selection and wasteful of other useful information. The second problem is that sometimes the solution to the simultaneous equations yields nonsensical answers, such as a negative proportion of one species in the diet. In addition to making it difficult to estimate dietary proportions, estimating digestibility becomes impossible. In this paper, we present a technique which provides an estimate of the dietary proportions. This estimate uses information on all the n-alkanes available, and it has a very desirable property of being a least squares estimate. We also present a method for determining the least squares estimate subject to the constraint that all proportions must be non-negative. We provide examples for estimating the proportions of grass and clover in the diet of sheep and the digestibility of those diets.