Uniformity Criteria

Abstract

Motivated by the overall mean model and the famous Koksma–Hlawka inequality, the main idea of uniform experimental design is to scatter the experimental design points uniformly over the experimental domain. In this regard, uniformity measures constitute the main concept of the uniform experimental design. Discrepancy is a measure which is defined as the deviation between the empirical and the theoretical uniform distribution. Therefore, discrepancy is a measure of uniformity which provides a way of construction of uniform designs. However, there are several discrepancies under different considerations. This chapter introduces the definitions and derives lower bounds for different discrepancies, which can be used to construct uniform designs.

Exercises

2.1

For a one-factor experiment ($s=1$), prove that the n-run design with minimum ${\mathcal {L}}_{\infty }$-star discrepancy is the following set of evenly spaced points:

$$ \mathcal {P}=\left\{ \frac{1}{2n}, \frac{3}{2n}, \ldots , \frac{2n-1}{2n} \right\} . $$

2.2

Prove that a minimum star discrepancy one-run design takes the form $\mathcal {P}=\{(z, \ldots ,z)\}$ for z satisfying $z^{s} + z -1=0$.

2.3

Let $\mathcal {P}=\{\varvec{x}_k=(x_{k1}, \ldots , x_{ks}), k=1, \ldots , n\}$ be a set of n points on the unit cube $C^s=[0,1]^s$. The wrap-around $L_2$-discrepancy can be calculated by

$$ (WD(\mathcal {P}))^2 = -\left( \frac{4}{3}\right) ^s +\frac{1}{n^2}\left( \frac{3}{2}\right) ^s \sum _{k=1}^n\sum _{l=1}^{n}\left( \frac{5}{6}\right) ^{d_H(k,l)}, $$

where $d_H(k,l)$ is the Hamming distance between $\varvec{x}_k$ and $\varvec{x}_l$.

Prove that $WD_2(D_1)=WD_2(D_2)$ if $D_1$ and $D_2$ are equivalent. Two U-type designs are called equivalent if one can be obtained from the other by (i) exchanging rows or/and (ii) exchanging columns.

2.4

Compare the uniformity of the following two designs under WD, CD, and MD:

$$ \varvec{X}_{1} = \left[ \begin{array}{ccc} 3/4 &{}\quad 3/4 &{}\quad 3/4 \\ 3/4 &{}\quad 3/4 &{}\quad 3/4\\ 3/4 &{}\quad 1/4 &{}\quad 3/4\\ 3/4 &{}\quad 3/4 &{}\quad 1/4 \\ 1/4 &{}\quad 3/4 &{}\quad 1/4 \end{array}\right] ,\ \ \ \varvec{X}_{2} = \left[ \begin{array}{ccc} 1/4 &{}\quad 1/4 &{}\quad 1/4\\ 1/4 &{}\quad 3/4 &{}\quad 3/4\\ 3/4 &{}\quad 1/4 &{}\quad 1/4\\ 3/4 &{}\quad 3/4 &{}\quad 1/4 \\ 1/4 &{}\quad 3/4 &{}\quad 1/4\end{array}\right] . $$

2.5

Write a MATLAB code for calculating $WD(\mathcal {P}), CD(\mathcal {P}),{MD}(\mathcal {P})$, where $\mathcal {P}$ is a design on the domain $[0,1]^s$ with n runs and s-factors. Apply your own code to the two designs in Example 2.3.1.

2.6

Prove that the CD, WD, and MD defined in Sect. 2.3 can be expressed as function of their Hamming distances.

2.7

Consider the following four designs.

$\mathcal {P}_{5-1}$		$\mathcal {P}_{5-2}$		$\mathcal {P}_{5-3}$		$\mathcal {P}_{5-4}$
1	2	1	2	1	5	1	1
2	4	2	5	2	1	2	5
3	1	3	3	3	4	3	4
4	3	4	1	4	2	4	3
5	5	5	4	5	3	5	2

Calculate the corresponding star discrepancy, CD, WD, MD, and LD for each design, and show your conclusion.

2.8

The inequality between arithmetic mean and geometric mean has been known and is useful for deriving some lower bounds. Prove this inequality stated as follows:

Let $a_1, \ldots , a_m$ be m nonnegative numbers, and then

$$ \bar{a}\equiv \frac{1}{m}\sum _{i=1}^ma_i\geqslant \left[ \prod _{i=1}^m a_j\right] ^{1/m}\equiv \bar{a}_g, $$

where $\bar{a}$ is the arithmetic mean and $\bar{a}_g$ is the geometric mean of $a_1, \ldots , a_m$. The above equality holds if and only if all the $a_i$’s are the same.

2.9

For easily understanding the so-called curse of dimensionality, let us study on the volume of the ball in $R^n$. A ball of radius r with the center at the origin in $R^n$ can be expressed as

$$ B_n(r)=\{\varvec{x}: \varvec{x}^T\varvec{x}=x_1^2+\cdots +x_n^2\leqslant r^2\}. $$

It is known that the volume of $B_3(r)$ is $\frac{4}{3} \pi r^3$. In general,

$$ \text{ Vol }(B_n(r))=\frac{\pi ^{n/2}}{\Gamma (\frac{n}{2} +1)}r^n. $$

Intuitively, one may think that the volume of $B_n(r)$ becomes larger and larger. For the unit ball, its volume increases in the first five dimensions, but decreases as n tends to infinity. Study the behavior of the volume of $B_n(r)$ as n increases.

2.10

What is the Cauchy–Schwarz inequality in the linear inner product space? Give a proof. From the literature, find more applications of the Cauchy–Schwarz inequality.

2.11

Prove Theorems 2.6.27 and 2.6.29.