Is a Transformed Low Discrepancy Design Also Low Discrepancy?

Abstract

Experimental designs intended to match arbitrary target distributions are typically constructed via a variable transformation of a uniform experimental design. The inverse distribution function is one such transformation. The discrepancy is a measure of how well the empirical distribution of any design matches its target distribution. This chapter addresses the question of whether a variable transformation of a low discrepancy uniform design yields a low discrepancy design for the desired target distribution. The answer depends on the two kernel functions used to define the respective discrepancies. If these kernels satisfy certain conditions, then the answer is yes. However, these conditions may be undesirable for practical reasons. In such a case, the transformation of a low discrepancy uniform design may yield a design with a large discrepancy. We illustrate how this may occur. We also suggest some remedies. One remedy is to ensure that the original uniform design has optimal one-dimensional projections, but this remedy works best if the design is dense, or in other words, the ratio of sample size divided by the dimension of the random variable is relatively large. Another remedy is to use the transformed design as the input to a coordinate-exchange algorithm that optimizes the desired discrepancy, and this works for both dense or sparse designs. The effectiveness of these two remedies is illustrated via simulation.

Appendix

We derive the formula in (5.18) for the discrepancy with respect to the standard normal distribution, $\varPhi$, using the kernel defined in (5.9). We first consider the case $d=1$. We integrate the kernel once:

$$\begin{aligned} {\int _{-\infty }^\infty K(t,x) \, \mathrm {d}\varPhi (t)}\\ =&\int _{-\infty }^{\infty } \left( 1+\frac{1}{2}|x|+\frac{1}{2}|t|-\frac{1}{2}|x-t|\right) \phi (t) \, \mathrm {d}t\\ =&1+ \frac{1}{\sqrt{2\pi }} + \frac{1}{2}|x| -\frac{1}{2}\left[ \int _{-\infty }^{x}(x - t)\phi (t) \, \mathrm {d}t +\int _{x}^{\infty } (t - x)\phi (t) \, \mathrm {d}t\right] \\ =&1+ \frac{1}{\sqrt{2\pi }} + \frac{1}{2}|x| - x [\varPhi (x)-1/2] - \phi (x) . \end{aligned}$$

Then we integrate once more:

$$\begin{aligned} {\int _{-\infty }^\infty \int _{-\infty }^\infty K(t,x) \, \mathrm {d}\varPhi (t) \mathrm {d}\varPhi (x)}\\&= \int _{-\infty }^{\infty } \left( 1+ \frac{1}{\sqrt{2\pi }} + \frac{1}{2}|x| - x [\varPhi (x)-1/2] - \phi (x) \right) \phi (x) \, \mathrm {d}x\\&= 1+ \sqrt{\frac{2}{\pi }} + \int _{-\infty }^{\infty } \{ - x \varPhi (x)\phi (x) + [\phi (x)]^2 \} \, \mathrm {d}x\\&= 1+\sqrt{\frac{2}{\pi }}-\frac{1}{\sqrt{4\pi }}+\int _{-\infty }^{\infty }\frac{1}{2\pi }\mathrm {e}^{-x^2}\mathrm {d}x =1+\sqrt{\frac{2}{\pi }}. \end{aligned}$$

Generalizing this to the d-dimensional case yields

$$\begin{aligned}\begin{gathered} \int _{\mathbb {R}^d\times \mathbb {R}^d} K(\varvec{x},\varvec{t}) \, \mathrm {d}\varPhi (\varvec{x})\mathrm {d}\varPhi (\varvec{t}) = \left( 1+\sqrt{\frac{2}{\pi }}\right) ^d, \\ \int _{\mathbb {R}^d}K(\varvec{x},\varvec{x}_n) \, \mathrm {d}\varPhi (\varvec{x}) = \prod \limits _{j=1}^d \left[ 1+\frac{1}{\sqrt{2\pi }}+\frac{1}{2}|x_j|-x_j[\varPhi (x_j)-1/2]-\phi (x_j)\right] . \end{gathered}\end{aligned}$$

Thus, the discrepancy for the normal distribution is

$$\begin{aligned}&D^2(\mathscr {X}, \varPhi , K)\\= & {} \left( 1+\sqrt{\frac{2}{\pi }}\right) ^d - \frac{2}{N}\sum \limits _{\varvec{x}\in P} \prod \limits _{j=1}^d\left[ 1+\frac{1}{\sqrt{2\pi }}+\frac{1}{2}|x_j|-x_j[\varPhi _1(x_j)-1/2]-\phi (x_j)\right] \\&+\frac{1}{N^2}\sum _{\varvec{x},\varvec{t}\in P}\prod _{j=1}^d \left[ 1+\frac{1}{2}|x_j|+\frac{1}{2}|t_j|-\frac{1}{2}|x_j-t_j|\right] . \end{aligned}$$