数理解析研究所講究録
第 1889 巻 2014 年 52-64

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An approximate approach to E–optimal designs for weighted
polynomial regression by using Tchebycheff systems andorthogonal polynomials
Hiroto Sekido
Department of Applied Mathematics and Physics, Graduate School ofInformatics, Kyoto University, Kyoto 606-8501, Japan.

Introduction

1

In experimental designs, –optimal designs are defined as designs which minimize the maximum eigenvalue of the covariance matrix of the estimator. The
Tchebycheff systems sometimes play an important role in optimal designs.
We firstly give a brief introduction to E–optimal designs and Tchebycheff systems
as preliminaries. Then an approximate approach to -optimal designs for weighted
polynomial regression is proposed.
$E$

$E$

-optimal designs for linear regression and Tchebycheff
systems

2

$E$

2.1

Linear regression and its$E$-optimal designs

A weighted linear regression model is defined by$Y=\theta^{T}f(x)+\epsilon(x)$$=$$(\theta_{0}$$\theta_{1}$...

$\theta_{m-1})(\begin{array}{l}f_{0}(x)f_{l}(x)|f_{m-1}(x)\end{array})+\epsilon(x)$,where $f(x)=(f_{0}(x), f_{1}(x), \ldots, f_{m-1}(x))^{T}$ is a vector of known linearly indeis a vector of unknown papendent continuous functions,,rameters, anddenotes a random error term. We call the functions$f_{1}(x),$$f_{m-1}(x)$ basis functions. Here we assume that the error term satisfies$\theta=(\theta_{0}, \theta_{1}, \ldots, \theta_{m-1})$$f_{0}(x)$$\epsilon(x)$$\ldots,$$E[\epsilon(x)]=0, V[\epsilon(x)]=\frac{\sigma^{2}}{w(x)},$is a positive constant and $w(x)\geq 0$ is called a weight function of regreswheresion.
Consider making $N$ observations at experimental conditions
such as$\sigma^{2}$$x_{1},$$y_{i}=\theta^{T}f(x_{i})+\epsilon_{i}, i=1,2, \ldots, N$$x_{2},$$x_{N}\in \mathcal{X}$$\ldots,$

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to estimate all unknown parameters. Here the design spacedenotes the set of all possible points where observations can be made. Throughout
this paper, we assume that different errors are uncorrelated, namely,$\theta_{0},$$\theta_{m-1}$$\theta_{1},$$\mathcal{X}$$\ldots,$$E[\epsilon_{i}]=0, V[\epsilon_{i}]=\frac{\sigma^{2}}{w(x_{i})}, E[x_{i}x_{j}]=0, i,j=1,2, \ldots, N, i\neq j$even if they has the same experimental conditions. From Gauss-Markov’s
theorem, the best linear unbiased estimator of the parameter vector is$x_{i}=x_{j}$$\hat{\theta}$$\theta$$\hat{\theta}=(X^{T}WX)X^{T}Wy$

,

(1)

where
$X=(f(x_{1}), f(x_{2}), \ldots, f(x_{N}))^{T}\in M_{N,m}(\mathbb{R})$,$W=$ diag $(w(x_{1}), w(x_{2}), \ldots, w(x_{N}))\in M_{N}(\mathbb{R})$ ,$y=(y_{1}, y_{2}, \ldots, y_{N})^{T}$

It is well known that the its covariance matrix is
Cov

(2)

$[\hat{\theta}]=\sigma^{2}(X^{T}WX)^{-1}$

Note that the weighted least squares estimator of the parameter vector is also
given by (1).
The multisetof the experimental conditions is called a design,
and it is considered as a probability measure such that$\theta$$\{x_{1}, x_{2}, \ldots, x_{N}\}$$\mu$$\mu(\{x\})=\frac{\#\{i|x_{i}=x,i=1,2,\ldots,N\}}{N}.$Hereinafter we consider designs as probability measures. Since accuracy of estimators depend designs, it is important to choose a good design. There are various
criteria for good designs, and some of them minimize the covariance matrix of estimators in some sense. In this paper, we consider -optimal designs which minimize
the maximum eigenvalue of the covariance matrix. Using the Fisher information
matrix$E$$M_{f,w}( \mu)=\int_{\mathcal{X}}w(x)f(x)f(x)^{T}d\mu(x)$$= \int_{\mathcal{X}}(\sqrt{w(x)}f(x))(\sqrt{w(x)}f(x))^{T}d\mu(x)$the covariance matrix (2) is rephrased asCov$[ \hat{\theta}]=\frac{\sigma^{2}}{N}M_{f,w}(\mu)^{-1}$,54Hence E–optimal designs are defined as the probability measures which maximize
the minimum eigenvalue of the Fisher information matrix. Namely, -optimal
designs are defined as the optimal solutions of the optimization problem$E$maximize$\lambda_{\min}(M_{f,w}(\mu))$subject to(3)$\mu\in \mathcal{P}_{\mathcal{X}}$denotesdenotes the minimum eigenvalue of the matrix , andwhere
the set of all probability measures on the Borel sets of . If the sample size
$N$ is fixed, all probabilities corresponding to the design
must be multiple of–optimal designs$1/N$ , however we do not consider this constraint. In this case,are sometimes called approximate E–optimal designs instead of exact E–optimal
designs. Thus, in this paper, we consider approximate E–optimal designs definedby (3), whose probabilities may be rounded to be multiples of $1/N$ in order to. Then E–optimal designs no
consider the corresponding multiset
longer depend on the sample size $N$ , they depend only the vector of functions$A$$\lambda_{\min}(A)$$\mathcal{P}_{\mathcal{X}}$$\mathcal{X}$

$\mu$

$E$

$\{x_{1}, x_{2}, \ldots, x_{N}\}$$g(x)=(g_{0}(x)$$=$$g_{1}(x)$$(\sqrt{w(x)}f_{0}(x)$For convenience, if the design...

$g_{m-1}(x))^{T}$$\sqrt{w(x)}f_{1}(x)$

$\mu$

...

$\sqrt{w(x)}f_{m-1}(x))^{T}$(4)

is a discrete probability measure, then we write$\mu=(\begin{array}{llll}\mathcal{S}_{1} s_{2} \cdots s_{n}\rho_{1} \rho_{2} \cdots \rho_{n}\end{array}),$where$s_{1},$$s_{2},$$\ldots,$$s_{n}$

denote the support points of$\mu$, and $\rho_{k}=\mu(\{s_{k}\})$ ,$k=$$1,2,$ $\ldots,$ $n.$See [3, 6] for details of optimal designs and other optimality criteria.

2.2

Tchebycheff systems and their applications to $E$-optimal
designs

denote linearly independent continuous functions:
Let
defined on a closed finite interval $I=[a, b]$ . If there exists $\delta\in\{1, -1\}$ such that$u_{1},$ $u_{2},$$\ldots,$$u_{n}$$Iarrow \mathbb{R}$$a\leq t_{1}<t_{2}<\cdots<t_{n}\leq b$$\Rightarrow\delta\det(u_{i}(t_{j}))_{i,j=1}^{n}=\delta\det(\begin{array}{llll}u_{l}(t_{1}) u_{1}(t_{2}) \cdots u_{1}(t_{n})u_{2}(t_{1}) u_{2}(t_{2}) \cdots u_{2}(t_{n})\vdots \vdots \ddots \vdots u_{n}(t_{1}) u_{n}(t_{2}) \cdots u_{n}(t_{n})\end{array})>0,$

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of functions is called a Tchebycheff system onthen the set
Similarly, if there exists $\delta\in\{1, -1\}$ such that$\{u_{1}, u_{2}, \ldots, u_{n}\}$$I.$$a\leq t_{1}<t_{2}<\cdots<t_{n}\leq b\Rightarrow\delta\det(u_{i}(t_{j}))_{i,j=1}^{n}\geq 0,$

then the setof functions is called a weak Tchebycheff system on
is a weak
I. It is well known [4, Theorem II 10.2] that the set
given byTchebycheff system if and only if there exists a unique function$\{u_{1}, u_{2}, \ldots, u_{n}\}$$\{u_{1}, u_{2}, \ldots, u_{n}\}$$\kappa(t)$$\kappa(t)=\gamma^{T}u(t)$$\gamma=$$(\gamma_{1}$,...

$\gamma_{2}$$u(t)=(u_{1}(t)$$\gamma_{n})^{T}\in \mathbb{R}^{n},$$u_{2}(t)$...

$u_{n}(t))^{T}$

which satisfies the following two properties:(a) $|\kappa(t)|\geq C$ for all $t\in I,$(b) There exist$n$points$\kappa(s_{k})=(-1)^{k}C,$$s_{1},$$s_{2},$$k=1,2,$$\ldots,$$\ldots,$$s_{n}$

such that $a\leq s_{1}<s_{2}<\cdots<s_{n}\leq b$ and$n.$

Here is a fixed positive constant, usually which is set as 1. In this paper, wecall the function
a Tchebycheff function, and we call the points
Tchebycheff points.
If the set $\{g_{0}(x), g_{1}(x), \ldots, g_{m-1}(x)\}$ is a weak Tchebycheff system on , then wecan construct a design, called a Tchebycheff design, by the following procedure.
Here $g_{k}(x)=\sqrt{w(x)}f_{k}(x)$ , where $w(x)$ denotes a weight function of regression,bedenotes a basis function. Let the Tchebycheff function$C$$\kappa(t)$$s_{1},$$s_{2},$$\ldots,$$s_{n}$$\mathcal{X}$$f_{k}(x)$$\kappa(x)$$\kappa(x)=\gamma^{T}g(x) , \gamma\in \mathbb{R}^{m},$and let the Tchebycheff points of the Tchebycheff function. Then the Tchebycheff design
is defined by$\mathcal{X}$$\kappa(x)$be$s_{1},$$s_{2},$$\ldots,$$s_{m}\in$$\mu^{*}$$\mu^{*}=(\begin{array}{llll}s_{1} s_{2} \cdots s_{m}\rho_{1} \rho_{2} \cdots \rho_{m}\end{array}),$$(\rho_{1}$$\rho_{2}$...

$\rho_{m})^{T}=\frac{F^{-1}\gamma}{\gamma^{T}\gamma},$$F=((-1)^{j+1}f_{i-1}(s_{j}))_{i,j=1}^{m}$(5)$f_{m-1}(x)$ has a
If the linear regression with the basis functions $f_{0}(x),$ $f_{1}(x),$unique -optimal design, then the Tchebycheff design is the -optimal design [5,pp. 94-97].
See [4] for more details of Tchebycheff systems.
$\ldots,$

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3

An approximate approach to -optimal designs for weighted
polynomial regression
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3.1

The proposed algorithm

Consider weighted polynomial regression, that is, consider the case of $f(x)=$$(1, x, \ldots, x^{m-1})^{T}$ . Thus $g(x)$ defined by (4) is $g(x)=\sqrt{w(x)}(1, x, \ldots, x^{m-1})$ here.
In this section, we propose an approximate approach to E–optimal designs forpolynomial regression with almost general weight function.
At first we introduce to the E–optimal designs calculated exactly. The following
theorem is shown by Dette [2].
Theorem 3.1. Let the design space be $\mathcal{X}=[-1,1]$ , and let the weight function ofregression be$w(x)=(1-x)^{\alpha}(1+x)^{\beta},$$\alpha,$$\beta\in\{0,1\}$.

(6)

Then the set $\{g_{0}(x), g_{1}(x), \ldots, g_{m-1}(x)\}$ is a weak Tchebycheff system, and thefunction(7)$(1-x)^{\alpha/2}(1+x)^{\beta/2}J_{m-1}^{\alpha-1/2,\beta-1/2}(x)$is the Tchebycheff function. Here

$J_{n}^{a,b}(x)$denotes a Jacobi orthogonal polynomial$J_{n}^{(a,b)}(x)= \frac{(-1)^{n}}{2^{n}n!(1-x)^{a}(1+x)^{b}}\frac{d^{n}}{dx^{n}}((1-x)^{n+a}(1+x)^{n+b})$$= \frac{1}{2^{n}}\sum_{k=0}^{n}(\begin{array}{l}n+an-k\end{array})(\begin{array}{l}n+bk\end{array})(x-1)^{k}(x+1)^{k}.$And the Tchebycheff function, calculated by (5), is thenomial regression with weight function (6).

-optimal design for poly-$E$

We generalize this theorem by focusing that the Jacobi polynomialis theorthogonal polynomial with respect to the weight function $\eta(x)=(1-x)^{a}(1+x)^{b},$$J_{n}^{a,b}(x)$namely,$l_{1}^{1}J_{m}^{(a,b)}(x)J_{n}^{(a,b)}(x) \eta(x)dx=\frac{2^{a+b+1}\Gamma(n+a+1)\Gamma(n+b+1)}{(2n+a+b+1)\Gamma(n+a+b+1)n!}\delta_{m,n},$where
denotes the Kronecker delta. See [1, 7] for more details of orthgonal polynomials. Our method is applicable to almost general weight functions,however our method construct E–optimal designs approximately, not exactly, viaapproximate Tchebycheff functions. Next we define the approximate Tchebychefffunctions, and its Tchebycheff points.
$\delta_{m,n}$

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Definition 3.2. Suppose that the design space $\mathcal{X}=[-1,1]$ . For a general weightfunction $w(x)$ of regression such that if $-1<x<1$ then $w(x)>0$ , the function
obtained the following steps is called an approximate Tchebycheff function.
$\kappa^{\uparrow}(x)$(a) Compute the $(m-1)$ -th degree of the orthogonal polynomial with respect tothe weight function$\eta(x)=\frac{w(x)}{\sqrt{1-x^{2}}}$

Let the orthogonal polynomial be$\nu_{m}(x)=$const.

$v_{m-1}(x)_{f}$

.

(8)

then

$\cross\det(\begin{array}{lllll}c_{0} c_{1} c_{2} \cdots c_{m-1}c_{1} c_{2} c_{3} \cdots c_{m}\vdots \vdots \vdots \vdots c_{m-2} c_{m-1} c_{m} \cdots c_{2m-3}1 x x^{2} \cdots x^{m-1}\end{array}),$$c_{k}=l_{1}^{1}x^{k}d\eta(x)$.

The orthogonal polynomial $\nu_{m-1}(x)$ can be calculated by the above formula,or the $Gram-$ Schmidt orthogonalization for example.
(b) Obtain $\kappa^{\uparrow}(x)=\mu,(x)\sqrt{w(x)}.$Then the Tchebycheff points$s_{2},.,$of the approximate Tchebycheff function
are defined as local maximum points and local minimum points.
$s_{m}\dagger\dagger..\dagger$$s_{1},$$\kappa\dagger(x)$Note that if the assumption $w(x)>0$ for $-1<x<1$ is almost violated, approximate Tchebycheff functions and Tchebycheff points sometimes are not welldefined. However, in most cases, approximate Tchebycheff functions are similar
to Tchebycheff functions. We show some examples of approximate Tchebychefffunctions by Figures 1-2.
Figures 1-4 indicate that an approximate
Tchebycheff function
has local maximums and local minimums whose absolute values are almost the same but not exactly the same. Figures 5-8 show thecase where the assumption $w(x)>0$ for $-1<x<1$ is almost violated, because$w(-1/2)=0.001$ is close to . In this case the Tchebycheff points cannot be
defined as local maximum points and local minimum points for $m=4,10,20.$
It is also noted that if the weight function of regression is given by (6), thenthe approximate Tchebycheff function is equal to (7). Thus, in this case, theapproximate Tchebycheff function is the exact Tchebycheff function.
By using the approximate Tchebycheff function, the proposed algorithm for calculating the approximation of -optimal designs, as an approximate Tchebycheff
designs, for weighted polynomial regression is described as follows.
$\kappa\dagger(x)$

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The approximate
Figure 1:
The approximate Figure 2:
Tchebycheff function for $m=$ Tchebycheff function for $m=$$w(x)=(1-x)^{1/2}(2+x)^{1/2}$
10, $w(x)=(1-x)^{1/2}(2+x)^{1/2}$$3,$

The approximate
Figure 3:
The approximate Figure 4:
Tchebycheff function for $m=$ Tchebycheff function for $m=$$w(x)=e^{x}$
10, $w(x)=e^{x}$
$3,$

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Figure 5:
The approximate Figure 6:
The approximate
Tchebycheff function for $m=$ Tchebycheff function for $m=$$w(x)=(x+0.5)^{2}+0.001$
10, $w(x)=(x+0.5)^{2}+0.001$$4,$

Figure 7:
The approximate Figure 8:
The approximate
Tchebycheff function for $m=$ Tchebycheff function for $m=$$20,$
$w(x)=(x+0.5)^{2}+0.001$
32, $w(x)=(x+0.5)^{2}+0.001$

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Algorithm 3.3. Suppose that the design space $\mathcal{X}=[-1,1]$ . For a general weightfunction $w(x)$ of regression such that if $-1<x<1$ then $w(x)>0$ , the design
is called the approximate Tchebycheff design.
$\mu^{\uparrow}$(a) Compute the approximate Tchebycheff function$\kappa^{\uparrow}by$(b) Compute the Tchebycheff pointsof the approximate Tcheby-cheff function$s_{1}^{\dagger},$$s_{2}^{\dagger},$$s_{m}^{\uparrow}$$\ldots,$

the Definition 3.2.

$\kappa^{\uparrow}.$(c) Compute the design$\mu^{\uparrow}$

given by$\mu^{*}=(\begin{array}{llll}s_{1}^{\dagger} s_{2}^{\dagger} \cdots s_{m}^{\uparrow}\rho_{1} \rho_{2} \cdots \rho_{m}\end{array}),$$(\rho_{1}$$\rho_{2}$...

$\rho_{m})^{T}=\frac{F^{-1}\gamma}{\gamma^{T}\gamma},$$F=((-1)^{j+1}f_{i-1}(s_{j}^{\dagger}))_{i,j=1}^{m}$

In the next subsection, we show some results of numerical examples in order toverify that approximate Tchebycheff designs are close to E–optimal designs.

3.2

Numerical examples

In this subsection, we give numerical examples corresponding to the approximate
Tchebycheff functions of Figures 1-4. These 4 examples indicate that approximate
Tchebycheff designs have tendency to be close enough to E–optimal designs.
(a) The case of $m=3,$$w(x)=(1-x)^{1/2}(2+x)^{1/2}$ :the corresponding Tchebycheff functions are shown in Figure 1. The approximate Tchebycheff designis$\mu^{\dagger}\approx(\begin{array}{lll}-l.000 -0.1252 0.92l50.1721 0.4896 0.3383\end{array})$

The optimal value of the optimization problem is$\lambda_{\min}(M_{f,w}(\mu^{\dagger}))\approx 7.693\cross 10^{-3}.$

The E–efficiency of the approximate Tchebycheff design$eff^{E}(\mu^{\uparrow})$satisfies$1-eff^{E}(\mu^{\dagger})\approx 8.720^{-5},$

where the -efficiency is defined$E$$eff^{E}(\mu^{\dagger})=\frac{M_{f,w}(,\mu\dagger)}{\sup_{\mu}M_{fw}(\mu)},$thus
denotes a relative error in some sense. Here $\sup M_{f,w}(\mu)$are calculated approximately by a random optimization.
$1-eff^{E}(\mu^{\dagger})$61(b) The case of $m=10,$ $w(x)=(1-x)^{1/2}(2+x)^{1/2}$ : the corresponding
Tchebycheff functions are shown in Figure 2. The approximate Tchebycheff
design is$\mu^{\dagger}\approx(\begin{array}{llll}x_{1} x_{2} \cdots x_{10}\rho_{1} \rho_{2} \cdots \rho_{10}\end{array}),$where$x_{1}=-1.000, \rho_{1}=0.03909,$$x_{2}=-0.9407, \rho_{2}=0.08305,$$x_{3}=-0.7710, \rho_{3}=0.09785,$$x_{4}=-0.5126, \rho_{4}=0.1201,$$x_{5}=-0.1969, \rho_{5}=0.1395,$$x_{6}=0.1396, \rho_{6}=0.1423,$$x_{7}=0.4592, \rho_{7}=0.1261,$$x_{8}=0.7269, \rho_{8}=0.1031,$$x_{9}=0.9118, \rho_{9}=0.08509,$$x_{10}=0.9949, \rho_{10}=0.06379.$

The optimal value of the optimization problem is$\lambda_{\min}(M_{f,w}(\mu^{\dagger}))\approx 1.714\cross 10^{-6}.$

The -efficiency of the approximate Tchebycheff design$E$$eff^{E}(\mu\dagger)$satisfies$1 -eff^{E}(\mu^{\dagger})\approx 3.334^{-5}$(c) The case of $m=3,$ $w(x)=e^{x}$ : the corresponding Tchebycheff functions areshown in Figure 3. The approximate Tchebycheff design is$\mu^{\dagger}\approx(\begin{array}{lll}-1.000 0.2405 1.0000.3204 0.5360 0.1436\end{array})$

The optimal value of the optimization problem is$\lambda_{\min}(M_{f,w}(\mu^{\dagger}))\approx 1.976\cross 10^{-1}$

The E–efficiency of the approximate Tchebycheff design$1-eff^{E}(\mu^{\dagger})\approx 4.082^{-8}$$eff^{E}(\mu^{\uparrow})$satisfies

62(d) The case of $m=10,$ $w(x)=e^{x}$ : the corresponding Tchebycheff functions
are shown in Figure 4. The approximate Tchebycheff design is$\mu^{\dagger}\approx(\begin{array}{llll}x_{1} x_{2} \cdots x_{10}\rho_{1} \rho_{2} \cdots \rho_{10}\end{array}),$where$x_{1}=-1.000, \rho_{1}=0.04351,$$x_{2}=-0.9326, \rho_{2}=0.09338,$$x_{3}=-0.7416, \rho_{3}=0.1119,$$x_{4}=-0.4566, \rho_{4}=0.1360,$$x_{5}=-0.1190, \rho_{5}=0.1494,$$x_{6}=0.2267, \rho_{6}=0.1404,$$x_{7}=0.5399, \rho_{7}=0.1164,$$x_{8}=0.7876, \rho_{8}=0.09315,$$x_{9}=0.9457, \rho_{9}=0.07880,$$x_{10}=1.000, \rho_{10}=0.03710.$

The optimal value of the optimization problem is$\lambda_{\min}(M_{f,w}(\mu^{\dagger}))\approx 1.660\cross 10^{-6}.$

The E–efficiency of the approximate Tchebycheff design$eff^{E}(\mu^{\dagger})$satisfies$1-eff^{E}(\mu^{\dagger})\approx 2.998^{-9}.$3.3

Properties and conjectures

In previous subsection, we ascertain that approximate Tchebycheff designs havetendency to be close to E–optimal designs. Here we show that exact Tchebycheff
designs gives E–optimal designs for weighted polynomial regression if regression
has a unique E–optimal design.

Theorem 3.4. Let $w(x)$ be the weight function of polynomial regression with thedesign space , and let $g_{k}(x)=x^{k}\sqrt{w(x)}$ . Then the set $\{g_{0}(x), g_{1}(x), \ldots, g_{m-1}(x)\}$is a weak Tchebycheff system on . Let polynomial regression with weight function $w(x)$ have a unique -optimal design. Then the Tchebycheff system is the-optimal design.
$\mathcal{X}$$\mathcal{X}$

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Proof. This theorem can be shown by using [4, pp. 9-20] and [4, Theorem II10.2].

$\square$

We have the following conjecture, which is derived from Figures 5-8 for examples.

Conjecture 3.5. Approximate Tchebycheff functions converge exact Tchebychefffunctions in some sense as $m$ goes to infinity.

At last, we note that if the design space isalmost same results by using$\mathcal{X}=[a, b]$, then we can discuss the$\eta(x)=\frac{w(x)}{\sqrt{(x-a)(b-x)}}$

instead of (8).

4Conclusions

In this paper, we first indicate a new definition of approximate Tchebycheff functions. By using this definition, we propose a new algorithm for constructing theapproximate Tchebycheff designs for weighted polynomial regression with almost
general weight functions. After that, we verify that the approximate Tchebycheff
designs are close to E–optimal designs by numerical examples.
As future works, it is necessary to discuss the definition of approximate Tchebycheff functions more strictly. We must clarify how much gaps of the absolute valuesof local maximums and local minimums of approximate Tchebycheff functions areadmitted.

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[4] Karlin, S. and Studden, W. J.: Tchebycheff Systems: With Applications inAnalysis and, Vol. 376, Interscience Publishers, New York, 1966.
$Stati_{\mathcal{S}}tics$[5] Melas, V. B.: Functional Approach to Optimal Experimental Design (Lecture
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64[6] Pukelsheim, F.: Optimal Design of Experiments, Wiley, New York, 1993.
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