Space-filling designs with a Dirichlet distribution for mixture experiments

Abstract

Uniform designs are widely used for experiments with mixtures. The uniformity of the design points is usually evaluated with a discrepancy criterion. In this paper, we propose a new criterion to measure the deviation between the design point distribution and a Dirichlet distribution. The support of the Dirichlet distribution, is defined by the set of d-dimensional vectors whose entries are real numbers in the interval [0,1] such that the sum of the coordinates is equal to 1. This support is suitable for mixture experiments. Depending on its parameters, the Dirichlet distribution allows symmetric or asymmetric, uniform or more concentrated point distribution. The difference between the empirical and the target distributions is evaluated with the Kullback–Leibler divergence. We use two methods to estimate the divergence: the plug-in estimate and the nearest-neighbor estimate. The resulting two criteria are used to build space-filling designs for mixture experiments. In the particular case of the flat Dirichlet distribution, both criteria lead to uniform designs. They are compared to existing uniformity criteria. The advantage of the new criteria is that they allow other distributions than uniformity and they are fast to compute.

Appendices

Appendices

1.1 Appendix A. The proof of theorem 1

We apply the Jensen’s inequality to the expected value ${I}_{f}\left(g\right)=E\left[\text{log}(g({\varvec{X}})\right]$.

Let denote the function $\varphi \left({\varvec{x}}\right)=\text{log}\left(g\left({\varvec{x}}\right)\right),$

$$ \varphi \left( {\varvec{x}} \right) = \log \left( {\frac{1}{{B\left( {\varvec{\alpha}} \right)}}\mathop \prod \limits_{k = 1}^{d} \left( {x_{k} } \right)^{{\alpha_{k} - 1}} } \right) = \mathop \sum \limits_{k = 1}^{d} \left( {\alpha_{k} - 1} \right){\text{log}}(x_{k} ) - \log \left( {B\left( \alpha \right)} \right). $$

$\varphi$ is a concave function since the logarithmic function is concave and $\left({\alpha }_{k}-1\right)$ is positive for ${\alpha }_{k}\ge 1$. The Jensen’s inequality implies that, $E\left[{\varphi }({\varvec{X}})\right]\le \varphi \left(E[{\varvec{X}}]\right)$. Let $E\left[{\varvec{X}}\right]=\left({\mu }_{1},\dots ,{\mu }_{d}\right)$, then

$$ \varphi \left( {E\left[ {\varvec{X}} \right]} \right) = \mathop \sum \limits_{k = 1}^{d} \left( {\alpha_{k} - 1} \right){\text{log}}(\mu_{k} ) - \log \left( {B\left( \alpha \right)} \right) < \infty $$

since $0<{\mu }_{k}<1$ ($supp\left({X}_{k}\right)=\left[{0,1}\right]$ and we exclude the special case of a constant random variable equals to 0).

1.2 Appendix B. The proof of theorem 2

The prof of Theorem 2 is a direct application of a result demonstrated by Joe (1989). We just have to verify the assumptions.

The choice of a Gaussian kernel satisfies the conditions.

• $K(-z)=K(z)$

• The kernel is of the form $K\left(z\right)=K\left({z}_{1},\dots ,{z}_{d}\right)=\prod_{j=1}^{d}{K}_{0}({z}_{j})$ where ${K}_{0}$ is a symmetric univariate density satisfying $\int {u}^{2}{K}_{0}\left(u\right)du=1.$

The $d$ components of ${\varvec{X}}$ have approximately the same scale in [0,1], the logarithmic function is thrice differentiable. Moreover, we suppose that $\int f\left({\varvec{x}}\right)log\left(f\left({\varvec{x}}\right)\right)d{\varvec{x}}$ and $\int f\left({\varvec{x}}\right) log^{2} \left(f\left({\varvec{x}}\right)\right)d{\varvec{x}}$ exists. Hence all conditions are satisfied to apply the results demonstrated by Joe (1989).

We have already noted that the existence hypothesis of $\int f\left({\varvec{x}}\right)log\left(f\left({\varvec{x}}\right)\right)d{\varvec{x}}$ is feasible since $f$ is close to $g$, and we proved the existence of this integral in Theorem 1. However, we have not demonstrated the existence of $\int f\left({\varvec{x}}\right) log^{2} \left(f\left({\varvec{x}}\right)\right)d{\varvec{x}}$ when $f=g$. This is demonstrated below in the case of $d=2$ to simplify notation. It remains true in the general case.

$$ I = \smallint f\left( {\varvec{x}} \right)log^{2} \left( {f\left( {\varvec{x}} \right)} \right)d{\varvec{x}} = \mathop \smallint \limits_{{S^{1} }}^{{}} f\left( {x_{1} ,x_{2} } \right)log^{2} \left( {f\left( {x_{1} ,x_{2} } \right)} \right)dx_{1} dx_{2} . $$

The line ${x}_{1}+{x}_{2}=1$ has the parametric representation,

$$ \left\{ {\begin{array}{*{20}c} {x_{1} \left( t \right) = - t} \\ {x_{2} \left( t \right) = 1 + t} \\ \end{array} } \right. $$

where $t\in \left[-{1,0}\right].$ We define the mapping function$M :\left[-{1,0}\right]\to \left[{0,1}\right]\times \left[{0,1}\right]$,$M\left(t\right)=(-t,1+t)$. Then,

$$ \begin{aligned} I & = \mathop \smallint \limits_{{ - 1}}^{0} f\left( { - t,1 + t} \right)log^{2} \left( {f\left( { - t,1 + t} \right)} \right)\left( {x^{\prime}_{1} \left( t \right),x^{\prime}_{2} \left( t \right)} \right)dt \\ & = \sqrt 2 \mathop \smallint \limits_{{ - 1}}^{0} f\left( { - t,1 + t} \right)log^{2} \left( {f\left( { - t,1 + t} \right)} \right)dt \\ \end{aligned} $$

If $f=g$,

$$ I = \frac{\sqrt 2 }{{B\left( \alpha \right)}}\mathop \smallint \limits_{ - 1}^{0} \left( { - t} \right)^{{\alpha_{1} - 1}} \left( {1 + t} \right)^{{\alpha_{2} - 1}} log^{2} \left( {\frac{1}{B\left( \alpha \right)}\left( { - t} \right)^{{\alpha_{1} - 1}} \left( {1 + t} \right)^{{\alpha_{2} - 1}} } \right)dt = I_{1} + I_{2} + I_{3} $$

With

$$ I_{1} = \frac{\sqrt 2 }{{B\left( \alpha \right)}}\mathop \smallint \limits_{ - 1}^{0} \left( { - t} \right)^{{\alpha_{1} - 1}} \left( {1 + t} \right)^{{\alpha_{2} - 1}} log^{2} \left( {\frac{1}{B\left( \alpha \right)}} \right)dt $$

$$ I_{2} = \left( {\alpha_{1} - 1} \right)\frac{\sqrt 2 }{{B\left( \alpha \right)}}\mathop \smallint \limits_{ - 1}^{0} \left( { - t} \right)^{{\alpha_{1} - 1}} \left( {1 + t} \right)^{{\alpha_{2} - 1}} log^{2} \left( { - t} \right)dt $$

$$ I_{3} = \left( {\alpha_{2} - 1} \right)\frac{\sqrt 2 }{{B\left( \alpha \right)}}\mathop \smallint \limits_{ - 1}^{0} \left( { - t} \right)^{{\alpha_{1} - 1}} \left( {1 + t} \right)^{{\alpha_{2} - 1}} log^{2} \left( {1 + t} \right)dt $$

${I}_{1}<+\infty$ since ${\alpha }_{i}\ge 1$.

${I}_{2}$ is an improper integral in 0, but ${\left(-t\right)}^{{\alpha }_{1}-1}{\left(1+t\right)}^{{\alpha }_{2}-1} log^{2} \left(-t\right)\sim {\left(-t\right)}^{{\alpha }_{1}-1} log^{2} \left(-t\right)$ when $t$ tends to 0 and ${\int }_{-1}^{0}{\left(-t\right)}^{{\alpha }_{1}-1} log^{2} \left(-t\right)dt$ is a convergent Bertrand’s integral.

${I}_{3}$ is an improper integral in -1, but ${\left(-t\right)}^{{\alpha }_{1}-1}{\left(1+t\right)}^{{\alpha }_{2}-1} log^{2} \left(1+t\right)\sim {\left(1+t\right)}^{{\alpha }_{2}-1} log^{2} \left(1+t\right)$ when $t$ tends to -1 and ${\int }_{-1}^{0}{\left(1+t\right)}^{{\alpha }_{2}-1} log^{2} \left(1+t\right)dt$ is a convergent Bertrand’s integral.