Fitting Kent models to compositional data with small concentration

Abstract

Compositional data can be transformed to directional data by the square root transformation and then modelled by using the Kent distribution. The current approach for estimating the parameters in the Kent model for compositional data relies on a large concentration assumption which assumes that the majority of the transformed data is not distributed too close to the boundaries of the positive orthant. When the data is distributed close to the boundaries with large variance significant folding may result. To treat this case we propose new estimators of the parameters derived based on the actual folded Kent distribution which are obtained via the EM algorithm. We show that these new estimators significantly reduce the bias in the current estimators when both the sample size and amount of folding is moderately large. We also propose using a saddlepoint density approximation for the Kent distribution normalising constant in order to more accurately estimate the shape parameters when the concentration is small or only moderately large.

Appendix: Proof of Proposition 1

The value $\hat{t}$ is obtained by solving the saddlepoint equation

which implies

$$ \frac{(\kappa-2\hat{t})^2}{p(\kappa-2\hat{t})+\kappa^2}=\frac{1}{1+ \frac{p-1}{\kappa-2\hat{t}} -\sum_{m=2}^p \frac{1}{ ( \kappa -2\beta_m -2\hat{t} )}}. $$

(22)

From Kume and Wood (2005, p. 468), it follows that the lower and upper bounds for $\hat{t}$ are

$$ \frac{\kappa-2\beta_2}{2} -\frac{p}{4} -\frac{1}{4} \bigl(p^2+4p \kappa^2 \bigr)^{1/2}, \qquad\frac{\kappa-2\beta_2}{2} - \frac{1}{2}, $$

(23)

and therefore

$$ \kappa-2\beta_2 -2\hat{t} > 1. $$

Using this bound, the limit conditions in Theorem 3 in Scealy and Welsh (2011) and (22), it follows that

$$ \frac{(\kappa-2\hat{t})^2}{p(\kappa-2\hat{t})+\kappa^2}=\frac{1}{1+ O(\kappa^{-1})}=1+O\bigl(\kappa^{-1}\bigr), $$

which implies

The solution within the bounds (23) to the above quadratic equation is

(24)

and when β=0 the O(κ ⁻¹) terms are exactly zero for all κ giving

$$ \hat{t}=\frac{\kappa}{2}-\frac{p}{4}-\frac{1}{4} \bigl( p^2+4\kappa^2 \bigr)^{1/2}. $$

This is slightly different from $\hat{t}$ given in Kume and Wood (2005, Sect. 3.1) because they set their matrix A to be a matrix of zeros, which is not equivalent to our approach. Note that both approaches still lead to the same normalising constant approximation for the von Mises-Fisher case.

Under the limit conditions in Theorem 3 in Scealy and Welsh (2011) and using the fact that $\hat{t}=O(1)$, it follows that

$$ \hat{t}\kappa(\kappa-2\hat{t})^{-1}-\hat{t}=O\bigl( \kappa^{-1}\bigr), \qquad T=O\bigl(\kappa^{-1}\bigr), $$

and

$$ \Biggl( \biggl(1-\frac{2\hat{t}}{\kappa} \biggr) \prod_{m=2}^p \biggl( 1-\frac{2\hat{t}}{\kappa-2\beta_m} \biggr) \Biggr )^{-1/2}=1+ O\bigl( \kappa^{-1}\bigr). $$

It also follows that

$$ \bigl(K_2(\hat{t}) \bigr)^{-1/2}=2^{-1} \kappa^{1/2} \bigl(1+ O\bigl(\kappa^{-1}\bigr) \bigr), $$

so

$$ \hat{f}^*(1)=(2\pi)^{-1/2}2^{-1}\kappa^{1/2} \bigl(1+O\bigl(\kappa^{-1}\bigr) \bigr) $$

and therefore

Using similar asymptotic arguments to Scealy and Welsh (2011, pp. 371–372), with the dominated convergence theorem, it follows that

(note that Kent et al. (2006, p. 754) give a detailed proof for the complex Bingham quartic distribution normalising constant which is similar) and therefore

with in the limit.