Construction of Uniform Designs—Deterministic Methods

Abstract

In this chapter and the following chapters, the different construction methods for uniform design tables are given. Typically, there are three approaches of constructing uniform design tables: (i) quasi-Monte Carlo methods; (ii) combinatorial methods; (iii) numerical search.

Exercises

3.1

For the case of $n=8,s=2$, give the design space $\mathcal {D}(8;C^2), \mathcal {D}(8;, 8^2), \mathcal {U}(8; 8^2)$ and $\mathcal {U}(8; 2\times 4)$.

3.2

Transfer the design ${\varvec{U}}$ in Example 3.1.2 into ${\varvec{X}}_{lft}, {\varvec{X}}_{ctr}, {\varvec{X}}_{ext}$ and ${\varvec{X}}_{mis}$ in $C^4$, respectively. Calculate WD, CD, and MD for these four designs in $C^4$. Are these four designs equivalent?

3.3

Give necessary conditions for a uniform design table $U_n(n^s)$.

3.4

By the use of the Fibonacci sequence introduced in Sect. 3.3.1, construct $U(n;n^2)$ for $n=5, 8, 13$ and compare their CD-values with designs in Tables 3.1 and 3.2.

3.5

Suppose an experiment has three factors, temperature, pressure, and reaction time, and the ranges are [50, 100] $^{\circ }$C, [3, 5] atm and [10, 25] min, respectively. Then, the experimental domain $\mathcal {X}=[50,80]\times [3,6]\times [10,25]$. Suppose each factor has four levels. Two possible choices of design matrices ${\varvec{Z}}_1$ and ${\varvec{Z}}_2$ and their mapping to $[0,1]^2$ are as follows:

$$\begin{aligned}&{\varvec{Z}}_1=\begin{pmatrix} 60 &{} 6 &{} 25 \\ 70 &{} 6 &{} 15 \\ 60 &{} 3 &{} 15 \\ 80 &{} 5 &{} 20 \\ 50 &{} 5 &{} 10 \\ 70 &{} 3 &{} 25 \\ 80 &{} 4 &{} 10 \\ 50 &{} 4 &{} 20 \end{pmatrix} \Rightarrow {\varvec{X}}_1=\begin{pmatrix} 0.3333 &{} 1 &{} 1 \\ 0.6667 &{} 1 &{} 0.3333 \\ 0.3333 &{} 0 &{} 0.3333 \\ 1 &{} 0.6667 &{} 0.6667 \\ 0 &{} 0.6667 &{} 0 \\ 0.6667 &{} 0 &{} 1 \\ 1 &{} 0.3333 &{} 0 \\ 0 &{} 0.3333 &{} 0.6667 \\ \end{pmatrix},\\&{\varvec{Z}}_2=\begin{pmatrix} 61.25 &{} 5.625 &{} 23.125 \\ 68.75 &{} 5.625 &{} 15.625 \\ 61.25 &{} 3.375 &{} 15.625 \\ 76.25 &{} 4.875 &{} 19.375 \\ 53.75 &{} 4.875 &{} 11.875 \\ 68.75 &{} 3.375 &{} 23.125 \\ 76.25 &{} 4.125 &{} 11.875 \\ 53.75 &{} 4.125 &{} 19.375 \end{pmatrix} \Rightarrow {\varvec{X}}_2=\begin{pmatrix} 0.375 &{} 0.875 &{} 0.875\\ 0.625 &{} 0.875 &{} 0.375\\ 0.375 &{} 0.125 &{} 0.375\\ 0.875 &{} 0.625 &{} 0.625\\ 0.125 &{} 0.625 &{} 0.125\\ 0.625 &{} 0.125 &{} 0.875\\ 0.875 &{} 0.375 &{} 0.125\\ 0.125 &{} 0.375 &{} 0.625 \end{pmatrix}. \end{aligned}$$

Answer the following questions:

(1) Compare designs ${\varvec{Z}}_1$ and ${\varvec{Z}}_2$ and give your comments.

(2) For obtaining ${\varvec{X}}_1$ and ${\varvec{X}}_2$, one mapping includes 0 and 1, but another does not involve 0 and 1. Find the formulae that use for the two mappings.

3.6

Find the Euler function $\phi (n)$ for $n=8,9,10,14,15$ and the corresponding generating vectors $\mathcal {H}_n$.

3.7

Give three cases satisfying $\phi (n)>\phi (n+1).$ In these cases, the leave-one-out glpm is not worth to be recommended.

3.8

Give the cardinality of $\mathcal {G}_{8,2}$. Under the discrepancy MD, put designs of $\mathcal {G}_{8,2}$ into groups such that designs in the same group are equivalent and designs in different group have different MD-values.

3.9

For given (n, s), find the cardinality of $\mathcal {A}_{n,s}$ for $n=11, s=2,3,4,5;$ and $n=30$, $s=2,3,4,5$, where

$$\begin{aligned} \mathcal {A}_{n,s}= & {} \{a: a<n,\ \gcd (a^j,n)=1,\ j=1, \ldots , s-1; 1, a, a^2, \ldots , a^{s-1}(\widetilde{\text{ mod }}~n)\\ ~&\text{ are } \text{ distinct } \text{ each } \text{ other }\}. \end{aligned}$$

3.10

By use of the design $U_{47}(47^3)$, construct a nearly uniform design $U_8(8^3)$ under MD by the cutting method.

3.11

Let $n=47$ and $s=46$. Find the $L_1$-distance and MD of the corresponding glp set U. Moreover, consider the simple linear level permutations $U + i$ $(\widetilde{\text{ mod }}~ n)$ and give your conclusion.

3.12

A Latin square of order n is an $n\times n$ matrix filled with n different symbols, each symbol in each row/column appears once and only once.

(1) Give a Latin square for order 3 and order 4.

(2) Find the definition for the concept of orthogonal Latin squares.

(3) Find the way to use orthogonal Latin squares to the construction of an orthogonal designs $L_{n^2}(n^{n-1})$.

(4) Sudoku puzzle such that the final $9\times 9$ matrix to be a Latin square. Fill the following Sudoku puzzle:

9		4	2			8	3
	5	7	4		9
6	2			7	1		9	5
8	9	2	7	3			1	4
1			6		4	9
4	7				8		5
5				4		3	8	2
	8		5					9
			8	9	2		6	1