National University of Kaohsiung Repository System:Item 310360000Q/10504

全文

(1)國立高雄大學統計學研究所碩士論文. A Study on Optimal Star Points Based on a Locally Optimal Criterion 以一個局部最適設計準則下的最佳星點研究. 研究生：洪義忠撰指導教授：陳瑞彬博士. 中華民國. 九十九年. 七月.

(2) 致謝辭碩士生涯一轉眼即將結束，回顧這兩年，最大的收穫就是看事情的角度。首先我要感謝. 陳瑞彬老師的指導，無論是課業、論文或是待人處事，您的言教與. 身教均帶給我許多學習的地方。感謝. 黃錦輝老師，平時待我就像第二個指導老. 師一般，常在關鍵時刻幫我一把，讓我覺得您亦師亦友。感謝了讓我的論文更加完善外，也感謝老師您平時的關懷。感謝. 羅夢娜老師，除黃文璋老師，讓我. 親身體驗何謂大師風範。亦感謝蘭屏姐，謝謝您的關心與協助，讓我完成不少的任務。另外就是統計所的同學及學長姐們，和大家相處的這段時間，彼此同甘共苦，這段充滿歡笑和淚水的時光，將是我永遠懷念的回憶。最後，我要感謝我的父母和乾爹、乾媽，你們的支持、付出與犠牲，是我內心的最大支柱，讓我得以成就我的求學之路，辛苦你們了，謝謝！願我下階段的喜悅，也能成為你們的喜悅。. 洪義忠於高雄大學統計學研究所中華民國 99 年 7 月.

(3) A Study on Optimal Star Points Based on a Locally Optimal Criterion. by Yi-Chung Hung Advisor Ray-Bing Chen. Institute of Statistics, National University of Kaohsiung Kaohsiung, Taiwan 811 R.O.C. July 2010.

(4) Contents. Z` zZ`. ii iii. 1 Introduction. 1. 2 Locally D-optimal Criterion. 4. 3 Selection Method. 6. 4 Locally D-optimal Star Points 4.1. 4.2. 4.3. 10. Full Factorial Designs in the First-Order Design . . . . . . . . . . . . .. 10. 4.1.1. 22 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 11. 4.1.2. 23 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 15. 4.1.3. 24 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 18. 4.1.4. 25 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 20. 4.1.5. 26 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 22. 4.1.6. 27 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 23. 4.1.7. 28 Full Factorial Design. . . . . . . . . . . . . . . . . . . . . . .. 25. Fractional Factorial Designs in the First-order Design . . . . . . . . . .. 27. 4.2.1. 25−1 Fractional Factorial Design . . . . . . . . . . . . . . . . . . V. 28. 4.2.2. 26−1 V I Fractional Factorial Design . . . . . . . . . . . . . . . . . .. 29. 4.2.3. 27−1 V II Fractional Factorial Design . . . . . . . . . . . . . . . . . .. 31. 4.2.4. 28−2 Fractional Factorial Design . . . . . . . . . . . . . . . . . . V. 33. Comparison with Spherical CCD . . . . . . . . . . . . . . . . . . . . .. 35. 5 Conclusion. 41. Reference. 42. A Appendix: The Proof of Theorem. 43. i.

(5) |×Í ItÊ'ãJìÝt·ÏF@~ ¼0>0: Wü }ÿ »ñ{..Ù.@~X .ß: ÷LD »ñ{..Ù.@~X ` 39S¡Z )W'ÎãëÍI5àW5½ÎTF.'Ï F&ÆkjE)W'3TFõ.'`´0t·ÏF 'k @.ÝÍó TFõ.'`&ÆqA ID-tÊ'ã J´092kÍt·ÏFÝH ky2`ãyêýÐó´ Ó&Æ¸à ÿa[jÕ°¼´0 ID-tÊ'ãJìÝt·ÏFt¡| ID-tÊ[ £ Ý£ãJbt·ÏFÝ)W'ÎfFÙÝ¦«³)W'b[£ n"C: ³)W' ID-tÊãJÿa[jÕ°. ii.

(6) A Study on Optimal Star Points Based on a Locally Optimal Criterion Advisor: Dr. Ray-Bing Chen Institute of Statistics National University of Kaohsiung. Student: Yi-Chung Hung Institute of Statistics National University of Kaohsiung. ABSTRACT In this thesis, we are interested in finding the conditionally optimal star points for the central composite designs. There are three segments in the central composite designs: center points, factorial design, and star points. Given the center points and factorial design, we search for the 2k additional points based on the locally D-optimal criterion, where k is the number of factors, instead of 2k star points. When k is more than two, a simulated annealing algorithm is used to find the corresponding locally D-optimal star points numerically. Finally our composite designs with the optimal star points are compared with the spherical central composite designs via the relative locally D-optimal efficiencies. Keyword: central composite design, locally D-optimal criterion, simulated annealing algorithm.. iii.

(7) 1. Introduction Response surface methodology (RSM) is connected with fitting a local response. surface of interest by a typically small set of observations, and its objective is to determine what levels of the independent variables, x1 , · · · , xk , maximize or minimize the response. In most RSM problems, there is no prior information about the true surface, i.e. the functional relationship, f , between the response, y, and independent variables, x1 , · · · , xk , is assumed to be “unknown”. Thus the key idea of RSM is to find a suitable approximation for the true but unknown functional relationship. Under some smooth conditions, a lower-order polynomial model over a limited experimental region, X , is employed to approximate the function, f . Since RSM is a two-stage procedure, at the first stage of RSM, the approximation model is the first-order polynomial model, i.e. y = β0 + β1 x1 + · · · + βk xk + ε, where β0 , β1 , · · · , βk are the unknown parameters and ε is a random variable with mean 0 and variance σ 2 . If the surface curvature exists, a polynomial of higher-order terms are incorporated into the first-order polynomial model. At the second stage of RSM, the second-order polynomial model is adopted here, i.e. y = β0 +. k X i=1. β i xi +. k X. βii x2i +. i=1. X. βij xi xj + ε,. 1≤i<j≤k. where βij are the unknown parameters for the second-order terms and the interaction terms. In this second-order polynomial model, there are 12 (k + 2)(k + 1) unknown parameters. Finally the optimal point is identified based on this second-order polynomial surface. The central composite design (CCD), introduced by Box and Wilson (1951) and Box and Hunter (1957), is popularly used in RSM and is very efficient for the sequential experiments. There are three segments in CCD: 2k factorial (or fractional factorial with resolution V or higher) design, (±1, ±1, · · · , ±1); 2k star points, (±α, 0, · · · , 0),· · · , (0, · · · , 0, ±α), and n0 center points, (0, 0, · · · , 0), where α is the distance from center point to star point, and n0 is the number of center points. The 2k factorial (or fractional factorial with resolution V or higher) design and n0 center points are used for fitting the first-order polynomial model, and then when the surrogate model becomes the 1.

(8) second-order polynomial model, the star points are added. There are two important parameters in CCD, the number of center points (n0 ) and the distance (α) from center point to star point, which need to be specified. As mentioned before, RSM is sequential procedure. In the first step, the first-order model is used. At this time, we might not have any information about the response surface. Thus fractional factorial design with center points might be a proper choice. When the second-order terms are included, star points are added for extra experiments. However, the reason why star points are on the axes is not so clear. The goal of this thesis is that we choose additional star points based on the information of the firstorder design, which contains the factorial design and n0 center points. This idea might result better response surface prediction. Chen and Lin (2009) have used this idea to study how to choose star points properly. Since a design ξ can be considered as a probability measure over the design space X , the CCD can be represented by    0 , c , ··· , c , s , ··· , s  1 n1 1 2k ξ = ,  n /n , 1/n , · · · , 1/n , 1/n , · · · , 1/n  0 where 0 = (0, · · · , 0) is the center point, c1 , · · · , cn1 are the design points of the 2k factorial design or fractional factorial design with resolution V or higher, and s1 , · · · , s2k are the additional star points. Here n is the total number of design points, and is equal to n0 + n1 + 2k. In Chen and Lin (2009), φp -optimal criteria were used for selecting s1 , · · · , s2k , for example, A-, D-, and Ds -optimal criteria. In this thesis, a locally optimal design criterion is employed instead of φp -optimal criteria. The locally D-optimal criterion is an idea from estimating an extremum point, b, of a multivariate quadratic regression model in the unit hyperball. Unlike the general estimation problem in the linear regression model, by a reparametrization, the estimation problem can be reduced to the problem of estimating some interesting parameters, usually they are relative to extremum point, b, of the nonlinear (in parameters) regression model. The details of the locally D-optimal criterion is introduced in Section 2. ∗ As shown in Cheng et al. (2000), and Melas et al. (2003), a design, ξD , is called a. locally D-optimal design for a given extremum point, b(0) , is defined as ¢¢ ¡ ¡ ∗ = arg min det Mb−1 (ξ) = arg max (det (Mb (ξ))) , ξD ξ. ξ. 2.

(9) where Mb (ξ) is the partition information matrix of ξ with respect to b(0) . In order to compare which designs are better under the locally D-optimal criterion, we use the relative locally D-efficiency, and the relative locally D-efficiency of ξ1 and ξ2 is defined as ¡ ¢ det Mb−1 (ξ1 ) ¡ ¢, det Mb−1 (ξ2 ). (1). where Mb (ξ1 ) and Mb (ξ2 ) are the partition information matrix of ξ1 and ξ2 , respectively, with respect to b(0) . This thesis is organized as follows. In Section 2, we introduce the details of the locally D-optimal criterion. In Section 3, we transfer our target problem as an optimization problem based on this locally D-optimal criterion, and a simulated annealing algorithm is used to find the 2k star points numerically when k is larger. In Section 4, based on the different first-order designs with n0 center points, the 2k additional star points are found respectively for k = 2, 3, · · · , 8. Then we compare our composite designs with the spherical CCD’s according to the relative locally D-efficiency. Finally we give a conclusion in Section 5.. 3.

(10) 2. Locally D-optimal Criterion According to Melas et al. (2003), we introduce the locally D-optimal criterion as. follows. First, we need to reparametrize the quadratic model in terms of the given extremum point. In one dimensional case, i.e. k = 1, the quadratic regression model is y = β0 + β1 x + β2 x2 + ε,. (2). where y is the result of an observation at point x ∈ [−1, 1] and ε is a random variable β1 with mean 0 and variance σ 2 . Set the extremum point b = − 2β , c = β0 − 2. β12 , 4β2. and. a = β2 , and then the model (2) can be transformed as y = a(x − b)2 + c + ε. For this reparametric model, the extremum point becomes simply one of the parameters, b, but the model becomes a nonlinear model. Consider the case of k, k ≥ 2. Suppose the experimental results at the design points x = (x1 , · · · , xk )T , x ∈ X , can be described as y = η(x, A, β, γ) + ε, where η(x) = η(x, A, β, γ) = xT Ax + β T x + γ, A is a positive definite k × k matrix, β is a k × 1 vector, γ is a real number, and ε is a random variable with mean 0 and variance σ 2 . So an estimate of the extremum point based on η(x) is b = (b1 , · · · , bk )T. ³. = arg min η(x, A, β, γ) x 1 = − A−1 β. 2. ´ or arg max η(x, A, β, γ) x. Hence we rewrite the regression function in the following form η¯(x, Θ) = (x − b)T A(x − b) + c, where Θ = (b1 , · · · , bk , a11 , · · · , akk , a12 , · · · , a1k , a23 , · · · , a2k , · · · , a(k−1)k , c)T and 1 c = γ − β T A−1 β. 4 4.

(11) Thus this reparametric model is still a nonlinear model with respect to b. Let a design ξ be a probability measure    x , ··· , x  (1) (n) ξ = .  µ , ··· , µ  1 n Here we assume Cov(y(x(i) ), y(x(j) )) = 0, ∀ i 6= j, i.e. Cov(εi , εj ) = 0, ∀ i = 6 j. Cheng et al. (2000) showed that the asymptotic variance matrix of the least squares estimate of Θ is σ 2 M −1 (ξ), and the information matrix, M (ξ), has the following form,     2A 0 2A 0  M (ξ)  , M (ξ) =  0 I 0 I where I is the identity matrix, M (ξ) = M (ξ, b) =. n X. f (x(l) )f T (x(l) )µl , and. l=1. f (x) = f (x, b) = ((b1 − x1 ), · · · , (bk − xk ), (b1 − x1 )2 , · · · , (bk − xk )2 , 2(b1 − x1 )(b2 − x2 ), · · · , 2(b1 − x1 )(bk − xk ), 2(b2 − x2 )(b3 − x3 ), · · · , 2(b2 − x2 )(bk − xk ), · · · , 2(bk−1 − xk−1 )(bk − xk ), 1)T . The matrix M (ξ) is rewritten as the block form, i.e.   M1 M2T , M (ξ) =  M2 M3 where M1 is a k × k matrix. As shown in Cheng et al. (2000), Mb = Mb (ξ) = M1 − M2T M3−1 M2 is defined as the partition information matrix for the extremum point b. Thus a design ξ ¢ ¡ is called a locally D-optimal design for a given extremum point, b(0) , if det Mb−1 (ξ) is minimum or det(Mb (ξ)) is maximum. Cheng et al. (2000) also showed that the locally D-optimal design only depends on the given extremum point b(0) and does not depend on the true values of A and c. 5.

(12) 3. Selection Method. A common choice of α, the locations of the star points in CCD, is to set α as √ k. This central composite design with α = k is called a spherical CCD, because √ all design points are on the surface of the sphere with radius k except for the center. √. points. In this thesis, our goal is to find the 2k additional points, s1 , · · · , s2k , according to the locally D-optimal criterion, based on a given first-order design with n0 center √ points. The design space considered here is the ball with radius k. We first use the polar coordinate system to represent the 2k additional points. Consider the case of two factors, i.e. k = 2, a additional point can be represented as (x1 , x2 ) = (r cos θ, r sin θ), where r =. p x21 + x22. is the radial distance from the design origin, and θ is the. counterclockwise angle from the x1 axis. Following the same structure of CCD, based on the symmetric and orthogonal properties of the four star points, we uniformly add the four points from each quadrant of a circle with radius r, and we define the four additional points as (r cos θ, r sin θ), (−r sin θ, r cos θ), (−r cos θ, −r sin θ), and (r sin θ, −r cos θ), √ where 0 < r ≤ 2, and − π4 ≤ θ < π4 . At this moment, the four additional points can be thought to rotate the axial points, whose distances from the origin are r, by an √ angle θ. Figure 1 shows our approach for k = 2 and r = 2.. Figure 1: The locations of four additional star points with k = 2 and r =. √. 2.. Because the rotation of two orthogonal vectors with an angle θ can be described by multiplying a Givens rotation matrix, the coordinates of the four additional points 6.

(13) are also represented as.  ± G12 (θ) .  where G12 (θ) = .  r 0 0 r. ,. (3).  cos θ. sin θ.  is a 2 × 2 Givens rotation matrix for axes x1 and. − sin θ cos θ x2 and angle θ. For the higher dimensionality, following the same idea of 2-dimension,. the 2k additional points, s1 , · · · , s2k , can be described by multiplying Givens rotation matrices, Gij (θij ), where 1 ≤ i < j ≤ k. The definition of Gij (θij ) is given as follows. Definition 1. A k-dimension Givens rotation matrix is defined by an orthogonal matrix form i. . Gij (θij ) =.      i       j     . j. 1 ··· .. . . . .. 0 .. .. ···. 0 ··· .. .. cos θij .. .. 0 .. .. · · · sin θij .. 1 .. 0 · · · − sin θij · · · cos θij .. .. .. . . . 0 ···. 0. ···. 0.  ··· 0  ..  .    ··· 0   ..  .    ··· 0   . . ..  . .   ··· 1. .. k×k. Gij (θij ) represents a plane rotation, is the space spanned by the unit vectors ei and ej , where 1 ≤ i < j ≤ k, with an angle θij , and is the identity matrix with two subsections: 1. Gij (θij )(i, j) = − Gij (θij )(j, i) = sin θij , 2. Gij (θij )(i, i) =. Gij (θij )(j, j) = cos θij .. Without loss of generality, the 2k symmetric additional points can be represented as ±. Y. Gij (θij )Dk (r),. 1≤i<j≤k. where Dk (r) is a k × k diagonal matrix with the diagonal element, r. In the locally D-optimal criterion, the believed extremum point, b(0) , is first prespecified. We also use the polar coordinate system to represent the given extremum point b(0) . For k = 2, b(0) can be represented as (0). (0). (b1 , b2 ) = (R cos φ1 , R sin φ1 ), 7. (4).

(14) r³. ´2 ³ ´2 (0) (0) where R = b1 + b2 is the radial distance from the design origin with √ (0) 0 < R ≤ 2 and φ1 is the counterclockwise angle from the b1 axis with 0 ≤ φ1 < 2π. For k ≥ 3, b(0) can be represented as (0). (0). (0). (0). b(0) = (b1 , b2 , b3 , · · · , bk ), where (0). b1 = R sin φk−1 · · · sin φ2 cos φ1 , (0). b2 = R sin φk−1 · · · sin φ2 sin φ1 , (0). b3 = R sin φk−1 · · · sin φ3 cos φ2 , .. . (0). bk = R cos φk−1 , √ with 0 < R ≤ k, 0 ≤ φm < 2π, m = 1, 2, · · · , k − 2, and 0 ≤ φk−1 < π. Hence ¡ ¢ det Mb−1 (ξ) is a function of r, θij , R, and φj , j = 1, 2, · · · , k−1. Because the extremum ¡ ¢ point b(0) is given, i.e. R and φj are given, det Mb−1 (ξ) can be reduced to a function of r, θij , i.e. d(r, θ12 , · · · , θ1k , θ23 , · · · , θ2k , · · · , θ(k−1)k ). Our goal is to find the ”best” additional star points according to locally D-optimal criterion. It can be transformed into an optimization problem, because these additional ¡ ¢ star points are found by minimizing det Mb−1 (ξ) , i.e. min d(r, θ12 , · · · , θ1k , θ23 , · · · , θ2k , · · · , θ(k−1)k ), ¡ ¢ where d(r, θ12 , · · · , θ1k , θ23 , · · · , θ2k , · · · , θ(k−1)k ) = det Mb−1 (ξ) . When the number of factors, k, increases, the objective function d(r, θ12 , · · · , θ1k , θ23 , · · · , θ2k , · · · , θ(k−1)k ) is very complex, and is hard to write down the close form. At this moment, we introduce the Best Angles and Radius sampler (BAR sampler) algorithm, proposed by Chen and Lin (2009), to find the minimum of objective function d(r, θ12 , · · · , θ1k , θ23 , · · · , θ2k , · · · , θ(k−1)k ) numerically. For simplicity, we set θ = (θ12 , · · · , θ1k , θ23 , · · · , θ2k , · · · , θ(k−1)k ) = (θ1 , θ2 , · · · , θp ), where p =. k(k−1) , 2. and then the objective function is represented as d(r, θ) = d(r, θ1 , θ2 , · · · , θp ).. (5). In fact, the BAR sampler algorithm is a kind of a simulated annealing algorithm to optimize the objective function d(r, θ). We define a density, πT (t) (r, θ), for our BAR sampler algorithm as 8.

(15) πT (t) (r, θ) ∝ exp(−d(r, θ)/T (t)), where T (t) is the temperature at time t and is a decreasing function from the initial temperature, T (0) > 0, to 0+ . The BAR sampler algorithm is represented in Table 1.. (1) Select the initial radius, r(0) , 0 < r(0) ≤ (0). − π4 ≤ θi. √. (0). k and initial angles, θi , i = 1, · · · , p ,. < π4 .. (2) Run Nt iterations of the Gibbs sampler to sample r and θ from πT (t) (r, θ). At each iteration of Gibbs sampler, (2.1) Sample r from πT (t) (r| θ) and (2.2) Draw θi , i = 1, · · · , p , from πT (t) (θi | r, θ −i ). (3) Set t to t + 1, go to step 2 until t is large enough. Table 1: The Best Angles and Radius Sampler Algorithm. In Gibbs sampling algorithm, θ −i is the collection of all angles except for the ith angle, i.e. θ −i = (θ1 , · · · , θi−1 , θi+1 , · · · , θp ). Because the simulated annealing algorithm can be (0). affected by the initial situations, r(0) and θi , we would like to select several different initial situations to repeat the BAR sampler several times. Then we check the tendency of the objective function d(r, θ) to ensure that d(r, θ) is close to an extremum value.. 9.

(16) 4. Locally D-optimal Star Points In this section, two types of the factorial designs in the first-order designs are. considered. The first one is the full factorial designs and the second type is fractional factorial designs when k is large. When the the number of factors is low, for example, k = 2, we try to find the close form of the objective function and the locally D-optimal star points theoretically. When the number of factors is increasing, say k ≥ 3, the close form is too complex to search the exactly optimal points. At this time, we use the BAR sampler to search the locally D-optimal star points numerically. The number of center points we consider in this section is from one to four, i.e. n0 = 1, 2, 3, 4. One thing needs to mention is that the experimental region is a k-dimensional ball with √ radius k for each k.. 4.1. Full Factorial Designs in the First-Order Design. Due to the locally D-optimal criterion, we need to pre-specify the possible extremum point, b(0) , first. When the full factorial designs are used in the first-order designs, we have the following theorem to show that we only need to consider the extremum point (0). (0). (0). (0). such that b(0) = (b1 , b2 , · · · , bk ) with bi ≥ 0, i = 1, 2, · · · , k. The reason is that our objective function is invariant to orthogonal transformation. Theorem 1. Suppose the first-order design contains a 2k full factorial design and n0 center points. Let b(0) be the given extremum point and s∗1 , · · · , s∗2k be the star points of f (0) f (0) (0) (0) = (bf the locally D-optimal design, ξ ∗ , for b(0) . There exist a vector bf ,b ,···,b ) 1. D. 2. k. f (0) with bi ≥ 0, i = 1, 2, · · · , k, and the corresponding sf m , m = 1, 2, · · · , 2k, such that ³ ´T ¡ ¢ T (0) and (f sm )T = G (s∗m )T , where G = G(ϑ ), ϑ = (ϑ12 , · · · , ϑ1k , bf = G b(0) ϑ23 , · · · , ϑ2k , · · · , ϑ(k−1)k ), is the rotation matrix and can be represented as the prodQ uct of the Givens rotation matrices, 1≤i<j≤k Gij (ϑij ), with ϑij ∈ ( π2 )n, n ∈ Z. Then (0) . se , · · · , sf are the additional star points of the locally D-optimal design for bf 1. 2k. The proof of this theorem is shown in Appendix. In this subsection, we search the locally D-optimal star points when the full factorial design is used in the first-order design for k = 2, 3, · · · , 8. For k = 2, we find the close form of the objective function, and then numerical results are also shown. For k ≥ 3, only numerical results are shown. 10.

(17) 4.1.1. 22 Full Factorial Design. For the case of k = 2, the four star points are represented as (3) with radius r and angle θ12 , and the given extremum point b(0) is represented as (4) with radius R and angle φ1 . When k = 2, given n0 center points and the 22 full factorial design as the first-order design , the objective function d(r, θ12 ) is (n0 + 8)2 P1 + P2 + P3 + P4 + P5 + P6 ( ), 4 P7 where ¡ ¢ P1 = r4 (n0 +4)r4 − 16r2 +4(n0 + 4) , ¡ ¢ P2 = R2 (n0 +4)r10 +2(n0 − 4)r8 +2(5n0 + 8)r6 + 4(5n0 + 8)r4 +16(n0 − 4)r2 + 32(n0 +4) , ¡ ¢ P3 = R4 (3n0 +16)r8 +4(3n0 +8)r6 +24n0 r4 +16(3n0 +8)r2 +16(3n0 +16) , ¡ ¢ P4 = r4 (cos 4θ12 ) 2(n0 + 8)R2 (r2 + 2) + (n0 + 4)r4 − 16r2 + 4(n0 + 4) , P5 = −4(n0 + 8) (cos 4φ1 ) R4 (r2 + 2)2 , P6 = (n0 + 8) (cos 4(θ12 − φ1 )) R4 r4 (r2 + 2)2 , ¡ ¢ P7 = (1+cos 4θ12 )r4 (r2 +2)2 (n0 + 4)r4 −16r2 +4(n0 + 4) .. To identify the optimal additional points, we take the derivative of this objective function d(r, θ12 ) with respect to r and θ12 separately. It is easy to show that d(r, θ12 ) √ is a decreasing function for r. Thus the minimum of d(r, θ12 ) is obtained at r = 2. That is the four locally D-optimal star points are all on the boundary of the 2-ball √ √ √ with radius 2. Given r = 2, we find the minimum of d( 2, θ12 ) for θ12 by taking √ √ the derivative of d( 2, θ12 ) with respect to θ12 . Finally the minimum of d( 2, θ12 ) is attended at θ12. 1 = − arcsin 2. µ. (n0 + 8)R2 sin 4φ1 √ H. ¶ ,. (6). where H = 4(n0 + 8)2 R4 cos2 4φ1 − 8R2 (n0 + 8)((3n0 + 8)R2 + 2n0 ) cos 4φ1 +(37n20 + 208n0 + 320)R4 + 16n0 (3n0 + 8)R2 + 16n20 . To demonstrate the difference between our optimal additional points and original √ star points, according to (6), we show some plots of φ1 and θ12 by fixing R = 2, 11.

(18) √ √ √ 0.75 2, 0.5 2, 0.25 2, and n0 = 1, 2, 3, 4, because when θ12 = 0, then original star points are also our optimal points. From Figure 2, θ12 6= 0 when φ1 6= ( π4 )n, n = 0, 1, 2, · · · , 7. We also find that the absolute of θ12 increases when n0 decreases or R increases. That is when we put more center points, then optimal star points are close to the original star points, however, when extremum point b(0) is near the boundary of the experimental region, we need to move our star points from axes to obtain the better parameter estimations. (a). (b) 0.25. 0.2. 0.2. 0.15. 0.15. 0.1. 0.1. 0.05. 0.05 θ12. θ12. 0.25. 0 −0.05. 0 −0.05. −0.1. −0.1. n0=1. n0=1. −0.15. n0=2. −0.15. n0=2. −0.2. n0=3. −0.2. n0=3. −0.25. n0=4 0. 1. 2. 3. φ1. 4. 5. 6. −0.25. 7. n0=4 0. 1. 2. (c). 3. φ1. 4. 5. 6. 7. (d). 0.2. 0.1 0.08. 0.15. 0.06 0.1 0.04 0.02 θ12. θ12. 0.05 0. −0.04. n0=1. −0.1 −0.15 −0.2. 0 −0.02. −0.05. −0.06. n0=2. n0=3. −0.08. n0=3. n0=4 0. 1. 2. 3. φ1. 4. 5. 6. n0=1. n0=2. −0.1. 7. n0=4 0. 1. 2. 3. φ1. 4. 5. 6. 7. √ Figure 2: For k = 2, (a) the plot of φ1 and θ12 by fixing R = 2, (b) the plot of φ1 √ √ and θ12 by fixing R = 0.75 2, (c) the plot of φ1 and θ12 by fixing R = 0.5 2, and (d) √ the plot of φ1 and θ12 by fixing R = 0.25 2. To show the performance of BAR sampler, we choose b(0) with R =. √. 2 and φ1 =. π 10. as an example. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 2, based on (5), we need to have one rotation angle, θ12 , and one radius r in our objective function.. 12.

(19) • n0 = 1 When n0 = 1, θ12 is -0.2152 from (6), and then the four locally D-optimal star points are.  ±.  1.3816 −0.3020 0.3020. 1.3816. ,. (7). which are different from the original star points of the spherical CCD clearly. Now we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations, and then the four locally D-optimal star points are   1.3814 −0.3031 , ± 0.3031 1.3814 which are very close to the points shown in (7). Figure 3 presents the tendency ¡ ¢ of det Mb−1 (ξ) . 550 500 450 400 350 300 250 200 150 100. 0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. ¡. 8000. ¢ Figure 3: For k = 2 and n0 = 1, the tendency of det Mb−1 (ξ) for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.). • n0 = 2 When n0 = 2, θ12 is -0.1753 from (6), and then the four locally D-optimal star points are. .  ±. 1.3925 −0.2467 0.2467. 1.3925. ,. (8). which are different from the original star points of the spherical CCD clearly. Now we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations, and then the four locally D-optimal star points are   1.3921 −0.2490 , ± 0.2490 1.3921 13.

(20) which are very close to the points shown in (8). Figure 4 presents the tendency ¡ ¢ of det Mb−1 (ξ) . 500 450 400 350 300 250 200 150 100. 0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. ¡. 8000. ¢ Figure 4: For k = 2 and n0 = 2, the tendency of det Mb−1 (ξ) for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.) • n0 = 3 When n0 = 3, θ12 is -0.1519 from (6), and then the four locally D-optimal star points are  ±.  1.3979 −0.2140 0.2140. 1.3979. ,. (9). which are different from the original star points of the spherical CCD clearly. Now we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations, and then the four locally D-optimal star points are   1.3978 −0.2148 , ± 0.2148 1.3978 which are very close to the points shown in (9). Figure 5 presents the tendency ¡ ¢ of det Mb−1 (ξ) . 900 800 700 600 500 400 300 200 100. 0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. 8000. ¢ Figure 5: For k = 2 and n0 = 3, the tendency of det Mb−1 (ξ) for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.). 14. ¡.

(21) • n0 = 4 When n0 = 4, θ12 is -0.1365 from (6), and then the four locally D-optimal star points are  ±.  1.4011 −0.1925 0.1925. 1.4011. ,. (10). which are different from the original star points of the spherical CCD clearly. Now we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations, and then the four locally D-optimal star points are   1.4013 −0.1905 , ± 0.1905 1.4013 which are very close to the points shown in (10). Figure 6 presents the tendency ¡ ¢ of det Mb−1 (ξ) . 800. 700. 600. 500. 400. 300. 200. 100. 0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. ¡. 8000. ¢ Figure 6: For k = 2 and n0 = 4, the tendency of det Mb−1 (ξ) for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.) 23 Full Factorial Design. 4.1.2. Since a locally D-optimal design depends on b(0) , we choose b(0) by fixing the values √ √ √ √ π 2π 3π of R and φj , j = 1, 2, as R = 0.25 3, 0.5 3, 0.75 3, and 3, and φj = 0, 10 , 10 , 10 , and. 4π . 10. Thus totally there are 4 × 5 × 5 = 100 cases here.. To save the space, we only demonstarte one case by setting R = φ2 =. π . 10. √. 3, φ1 = 0, and. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 3, based. on (5), we need to have three rotation angles, θ1 , θ2 and θ3 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the six locally D-optimal star points numerically.. 15.

(22) • n0 = 1 When n0 = 1, the six locally D-optimal star points are   1.6856 0.0026 0.3986     ±  −0.0032 1.7320 0.0019  ,   −0.3986 −0.0026 1.6856 and the radius, r, is. √. 3. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the six locally D-optimal star points are   1.6976 0.0011 0.3436     ±  −0.0006 1.7320 −0.0029  ,   −0.3436 0.0027 1.6976 and the radius, r, is. √. 3. These points are different from the original star points. of the spherical CCD clearly. • n0 = 3 When n0 = 3, the six locally D-optimal star points are   1.7053 −0.0038 0.3032     ±  0.0033 1.7320 0.0034  ,   −0.3032 −0.0027 1.7053 and the radius, r, is. √. 3. These points are different from the original star points. of the spherical CCD clearly. • n0 = 4 When n0 = 4, the six locally D-optimal star points are   1.7102 −0.0029 0.2742     ±  0.0040 1.7320 −0.0067  ,   −0.2741 0.0072 1.7102 and the radius, r, is. √. 3. These points are different from the original star points. of the spherical CCD clearly. 16.

(23) We try to summarize some finds in our all 500 cases. For r, from our numerical √ results, we find that r = 3. Then for the rotation angles, θ1 , θ2 , and θ3 , basically we want to see that at least one of θ1 , θ2 , and θ3 are not equal to zero, because it means that our optimal additional points are not on axes anymore. The summarizations are as follows. (I) φ2 = 0 In this case, the possible extremum point b(0) is on the third axis, i.e. b(0) = (0, 0, R). The following two figures, Figures 7 and 8, are the plots of | θi | with n0 and R separately. From these figures, we conclude that when φ2 = 0, the original star points are also our optimal star points. (b). (a). Figure 7: For k = 3, R =. √. (c). 3, and φ2 = 0, (a) | θ1 | v.s. n0 , (b) | θ2 | v.s. n0 , and. (c) | θ3 | v.s. n0 .. (a). (b). (c). Figure 8: For k = 3, n0 = 1, and φ2 = 0, (a) | θ1 | v.s. R, (b) | θ2 | v.s. R, and (c) √ | θ3 | v.s. R . The unit of R is 3.. 17.

(24) (II) φ2 6= 0 When φ2 6= 0, all or some of the rotation angles θ1 , θ2 , and θ3 are not equal to zeros. That is our optimal star points should not on axes. We also have the following two figures, Figures 9 and 10, to show the relations of | θi | and n0 , R. Basically more center points we put in the first-order design, | θi | are smaller. That is our optimal additional points are more close to axes. But for large value of R, | θi | is large. Then at this time, our additional points are far from axes. These two summarizations are similar as what we find in the case of k = 2. (a). (b). √ Figure 9: For k = 3, R = 0.75 3, φ1 =. π , 10. (c). and φ2 =. 2π , 10. (a) | θ1 | v.s. n0 , (b) | θ2 |. v.s. n0 , and (c) | θ3 | v.s. n0 . (b). (a). Figure 10: For k = 3, n0 = 2, φ1 =. 3π , 10. and φ2 = √ R, and (c) | θ3 | v.s. R . The unit of R is 3.. 4.1.3. (c). 4π , 10. (a) | θ1 | v.s. R, (b) | θ2 | v.s.. 24 Full Factorial Design. Since a locally D-optimal design depends on b(0) , we choose b(0) by fixing the values √ √ √ √ π 2π of R and φj , j = 1, 2, 3, as R = 0.25 4, 0.5 4, 0.75 4, and 4, and φj = 0, 10 , 10 , 3π , 10. and. 4π . 10. Thus totally there are 4 × 5 × 5 × 5 = 500 cases here.. 18.

(25) We show one case with R =. √. 4, φ1 = 0, φ2 = 0, and φ3 =. π 10. as an example. here. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 4, based on (5), we need to have six rotation angles, θ1 , θ2 , · · · , θ6 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the eight locally D-optimal star points numerically. • n0 = 1 When n0 = 1, the eight locally D-optimal star points are   −1.0786 −1.6842 0.0047 −0.0047      1.6843 −1.0785 0.0021 −0.0047   , ±   0.0011 0.0036 1.9524 0.4339    0.0013 −0.0075 −0.4339 1.9524 and the radius, r, is. √. 4. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the eight locally D-optimal star points are   1.9251 −0.5420 −0.0064 −0.0028      0.5420 1.9252 −0.0057 0.0004  , ±    0.0081 0.0034 1.9627 0.3842    0.0011 −0.0018 −0.3843 1.9627 and the radius, r, is. √. 4. These points are different from the original star points. of the spherical CCD clearly. • n0 = 3 When n0 = 3, the eight locally D-optimal star points are   0.7257 −1.8637 0.0007 0.0072      1.8637 0.7257 0.0073 −0.0054  ,  ±   −0.0066 −0.0006 1.9742 0.3202    0.0035 0.0089 −0.3202 1.9742 and the radius, r, is. √. 4. These points are different from the original star points. of the spherical CCD clearly. 19.

(26) • n0 = 4 When n0 = 4, the eight locally D-optimal star points are   1.9612 0.3921 0.0064 0.0031      −0.3921 1.9612 −0.0075 −0.0051  , ±    −0.0083 0.0067 1.9755 0.3120    −0.0028 0.0034 −0.3120 1.9755 and the radius, r, is. √. 4. These points are different from the original star points. of the spherical CCD clearly. 25 Full Factorial Design. 4.1.4. Since a locally D-optimal design depends on b(0) , we choose b(0) by fixing the values √ √ √ √ π of R and φj , j = 1, 2, · · · , 4, as R = 0.25 5, 0.5 5, 0.75 5, and 5, and φj = 0, 10 , 2π 3π , , 10 10. 4π . 10. Thus totally there are 4 × 54 = 2500 cases here. √ We show one case with R = 5, φ1 = 0, φ2 = 0, φ3 = 0, and φ4 = and. π 10. as an example. here. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 5, based on (5), we need to have ten rotation angles, θ1 , θ2 , · · · , θ10 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the ten locally D-optimal star points numerically. • n0 = 1 When n0 = 1, the ten locally D-optimal  0.1418 −0.2804 −2.2138    −2.1485 0.5821 −0.2115   ±  0.6029 2.1407 −0.2326    −0.0072 0.0041 0.0076  0.0112 −0.0115 −0.0108 and the radius, r, is. √. star points are  0.0110 −0.0098 −0.0026 2.1871 −0.4651. −0.0108.   0.0110    0.0065  ,   0.4652   2.1871. 5. These points are different from the original star points. of the spherical CCD clearly.. 20.

(27) • n0 = 2 When n0 = 2, the ten locally D-optimal  −0.8091 2.0117 0.5462    −1.0203 0.1288 −1.9855   ±  −1.8178 −0.9677 0.8714    −0.0044 −0.0045 0.0013  0.0053 −0.0041 −0.0042 and the radius, r, is. √. star points are  0.0009. 0.0069.   −0.0003 −0.0012    −0.0067 0.0030  ,   2.1979 0.4114   −0.4114 2.1979. 5. These points are different from the original star points. of the spherical CCD clearly. • n0 = 3 When n0 = 3, the ten locally D-optimal  −0.5068 −2.1221 −0.4895    −1.1637 0.6886 −1.7809   ±  1.8409 −0.1489 −1.2605    −0.0008 0.0111 −0.0052  −0.0020 −0.0151 0.0110 and the radius, r, is. √. star points are  0.0110 −0.0098 −0.0026 2.2082 −0.3515. −0.0108.   0.0110    0.0065  ,   0.3516   2.2082. 5. These points are different from the original star points. of the spherical CCD clearly. • n0 = 4 When n0 = 4, the ten locally D-optimal  0.5452 −0.9011 −1.9725    1.6356 −1.1647 0.9840   ±  −1.4240 −1.6827 0.3751    0.0008 0.0049 −0.0121  −0.0096 −0.0087 −0.0084 and the radius, r, is. √. star points are  −0.0075 −0.0098.   0.0063 0.0072    0.0078 −0.0102  ,   2.2115 0.3301   −0.3302 2.2115. 5. These points are different from the original star points. of the spherical CCD clearly.. 21.

(28) 26 Full Factorial Design. 4.1.5. Since a locally D-optimal design depends on b(0) , we choose b(0) by fixing the values √ √ √ √ π of R and φj , j = 1, 2, · · · , 5, as R = 0.25 6, 0.5 6, 0.75 6, and 6, and φj = 0, 10 , 2π 3π , , 10 10. 4π . 10. Thus totally there are 4 × 55 = 12500 cases here. √ We show one case with R = 6, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, and φ5 = and. π 10. as an. example here. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 6, based on (5), we need to have fifteen rotation angles, θ1 , θ2 , · · · , θ15 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the twelve locally D-optimal star points numerically. • n0 = 1 When n0 = 1, the twelve  0.3624 0.4399    0.1582 −2.3643    −1.0777 0.3692 ±   2.1639 0.2831    −0.0008 0.0026  −0.0063 0.0028 and the radius, r, is. √. locally D-optimal star points are  −0.5656. 2.3141. 0.0049. −0.0005. 0.3522. 0.5108. 0.0009. 0.0052. 2.0817. 0.6074. −0.0091. 0.0050. 1.1057. −0.1224 −0.0058. 0.0094. 0.0094. −0.0031. 0.4608. 2.4057. −0.0113 −0.0008 −0.4608.       ,      . 2.4057. 6. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the twelve  0.1516 −0.1873    −1.7754 0.1122    −0.3587 −2.4087 ±   −1.6421 0.3876    0.0053 −0.0032  0.0029 −0.0102 and the radius, r, is. √. locally D-optimal star points are  −2.2484 −0.9415. 0.0014. −0.0088. −0.7585. 0.0085. 0.0042. 1.5033. 0.2300. −0.1278 −0.0015 −0.0095. 0.5623. −1.6844 −0.0016. 0.0023. 0.0034. −0.0065. 0.3861. 2.4189. −0.0070 −0.0039 −0.3861.       ,      . 2.4188. 6. These points are different from the original star points. of the spherical CCD clearly. 22.

(29) • n0 = 3 When n0 = 3, the twelve locally D-optimal star points are   0.6273 2.2099 −0.8171 −0.2344 0.0073 0.0091      2.2321 −0.5024 0.5498 −0.6805 −0.0035 −0.0085       0.7589 −0.0802 −0.2965 2.3086 −0.0068 0.0079   , ±   −0.2202 0.9259 2.2231 0.3901 −0.0015 −0.0009       0.0037 −0.0083 0.0047 0.0050 2.4211 0.3718    0.0023 −0.0082 0.0061 −0.0096 −0.3719 2.4211 √ and the radius, r, is 6. These points are different from the original star points of the spherical CCD clearly. • n0 = 4 When n0 = 4, the twelve locally D-optimal star points are   −0.1639 0.2468 −1.5123 −1.9040 0.0051 0.0006      2.2762 −0.8702 −0.1072 −0.2236 −0.0016 −0.0090       −0.8419 −2.1003 0.6268 −0.6976 0.0066 0.0086   , ±   −0.2880 −0.8777 −1.8189 1.3557 0.0079 −0.0035       0.0066 0.0077 0.0065 0.0018 2.4247 0.3475    0.0101 0.0018 −0.0058 0.0038 −0.3475 2.4247 √ and the radius, r, is 6. These points are different from the original star points of the spherical CCD clearly. 27 Full Factorial Design. 4.1.6. Since a locally D-optimal design depends on b(0) , we choose b(0) by fixing the values √ √ √ √ π , of R and φj , j = 1, 2, · · · , 6, as R = 0.25 7, 0.5 7, 0.75 7, and 7, and φj = 0, 10 2π 3π , , 10 10. 4π . 10. Thus totally there are 4 × 56 = 62500 cases here. √ We show one case with R = 7, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, φ5 = 0, and. φ6 =. and. π 10. as an example here. The numbers of the center points, n0 , are 1, 2, 3, and 4.. For k = 7, based on (5), we need to have twenty-one rotation angles, θ1 , θ2 , · · · , θ21 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the fourteen locally D-optimal star points numerically. 23.

(30) • n0 = 1 When n0 = 1, the fourteen locally D-optimal star points are   −0.0031 −0.0035 0.0193 0.0407 −2.6453 0.0086 0.0103      0.5452 −0.4518 −0.1086 −2.5466 −0.0400 0.0048 0.0119       2.2701 1.1807 0.6246 0.2498 0.0041 −0.0056 0.0002      ±  1.0505 −0.9765 −2.1684 0.4908 −0.0082 −0.0001 0.0120  ,      0.6676 −2.1091 1.3769 0.4581 0.0190 −0.0108 0.0028       0.0052 −0.0040 0.0083 0.0082 0.0103 2.6146 0.4045    −0.0091 0.0094 0.0076 0.0074 0.0091 −0.4046 2.6146 √ and the radius, r, is 7. These points are different from the original star points of the spherical CCD clearly. • n0 = 2 When n0 = 2, the fourteen locally D-optimal star points are   −0.2880 −0.5690 0.3767 −0.3176 −2.5200 −0.0038 0.0028      0.0669 1.3541 −2.0876 −0.7197 −0.5347 −0.0075 0.0089       −0.4026 0.2904 −0.5781 2.4984 −0.4209 0.0033 0.0059      ±  0.9311 2.0169 1.3760 0.1703 −0.3776 0.0088 0.0047  ,      −2.4256 0.8309 0.5219 −0.3314 0.2094 −0.0004 −0.0044       −0.0038 −0.0049 −0.0082 −0.0067 −0.0024 2.6187 0.3775    −0.0042 −0.0061 0.0076 −0.0027 0.0069 −0.3775 2.6187 √ and the radius, r, is 7. These points are different from the original star points of the spherical CCD clearly. • n0 = 3 When n0 = 3, the fourteen locally D-optimal star points are   0.0004 −0.0015 −0.0013 0.1463 2.6417 −0.0071 0.0098      −0.4337 0.0447 −0.5725 2.5421 −0.1410 −0.0074 −0.0016       −1.2164 2.1792 0.8768 −0.0481 0.0045 0.0095 0.0032      ±  −0.6485 −1.2411 2.2074 0.4071 −0.0221 0.0073 0.0090  ,      −2.2162 −0.8417 −1.0152 −0.5901 0.0320 0.0037 0.0054       0.0090 −0.0027 −0.0102 0.0075 0.0054 2.6248 0.3323    0.0068 0.0037 −0.0057 −0.0001 −0.0106 −0.3323 2.6248 24.

(31) and the radius, r, is. √. 7. These points are different from the original star points. of the spherical CCD clearly. • n0 = 4 When n0 = 4, the fourteen locally D-optimal star points are   −0.3179 −0.1713 0.3766 0.1114 2.5914 −0.0071 0.0098      1.3429 0.6922 −1.9489 −0.8002 0.5281 −0.0074 −0.0016       1.1791 −1.2834 −0.4307 1.9431 0.0389 0.0095 0.0032      ±  1.8795 0.7400 1.6852 −0.2792 0.0465 0.0073 0.0090  ,      0.4157 −2.0729 0.1863 −1.5791 −0.0453 0.0037 0.0054       −0.0079 0.0075 −0.0088 −0.0058 0.0070 2.6248 0.3323    −0.0057 0.0034 −0.0071 0.0016 −0.0103 −0.3323 2.6248 and the radius, r, is. √. 7. These points are different from the original star points. of the spherical CCD clearly. 4.1.7. 28 Full Factorial Design. Since a locally D-optimal design depends on b(0) , we choose b(0) by fixing the values √ √ √ √ π , of R and φj , j = 1, 2, · · · , 7, as R = 0.25 8, 0.5 8, 0.75 8, and 8, and φj = 0, 10 2π 3π , , 10 10. 4π . 10. Thus totally there are 4 × 57 = 312500 cases here. √ We show one case with R = 8, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, φ5 = 0, φ6 = 0, and. and φ7 =. π 10. as an example here. The numbers of the center points, n0 , are 1, 2, 3, and. 4. For k = 8, based on (5), we need to have twenty-eight rotation angles, θ1 , θ2 , · · · , θ28 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the sixteen locally D-optimal star points numerically.. 25.

(32) • n0 = 1 When n0 = 1, the sixteen locally D-optimal star points are  1.1995 0.8808 0.0373 −1.3777 −0.5529 −1.8921    1.1563 0.8180 0.1100 −1.3618 0.3876 1.9943    1.3185 −0.7656 −0.2194 0.7979 −2.1706 0.5284    1.3574 1.0112 0.9888 1.7806 0.9668 −0.2284 ±   0.1129 1.0743 −2.5338 0.6171 0.1776 0.0205    −1.2770 1.9473 0.7351 0.2437 −1.3661 0.3330    0.0087 −0.0077 −0.0004 −0.0112 0.0005 −0.0075  0.0038 −0.0054 −0.0115 −0.0010 −0.0050 0.0042 and the radius, r, is. √.  −0.0118. 0.0001. −0.0012 −0.0025 −0.0002 −0.0085 0.0041. 0.0073. 0.0056. −0.0073. 0.0107. 0.0070. 2.8078. 0.3406. −0.3406. 2.8078.          ,         . 8. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the sixteen locally D-optimal star points are   −0.0360 0.0220 −0.1349 0.0816 2.4172 1.4595 0.0116 −0.0124      −0.0215 −0.6300 −0.3645 0.3305 1.3831 −2.3340 0.0118 0.0047       1.1974 0.3041 −2.3252 0.9634 −0.2893 0.2351 0.0138 −0.0101       1.4587 1.4277 0.0731 −1.8673 0.3558 −0.4638 −0.0012 −0.0044   , ±   −0.6285 −1.5862 −1.2502 −1.8510 −0.1650 0.2693 0.0053 −0.0146       −2.0104 1.7191 −0.9348 −0.2075 0.0795 −0.2818 −0.0127 0.0129       −0.0114 0.0111 0.0104 −0.0055 −0.0127 0.0020 2.8123 0.3006    0.0137 −0.0128 −0.0116 −0.0077 0.0080 0.0129 −0.3005 2.8123 and the radius, r, is. √. 8. These points are different from the original star points. of the spherical CCD clearly.. 26.

(33) • n0 = 3 When n0 = 3, the sixteen locally D-optimal star points are   −0.0007 0.0009 −0.0002 0.0018 −0.1415 −2.8248 0.0124 −0.0157      −0.4159 0.2216 −0.1157 −0.0479 2.7825 −0.1393 0.0049 0.0138       −2.4337 1.3484 −0.1752 0.0168 −0.4769 0.0249 0.0002 −0.0009       0.7495 1.2588 −0.5757 2.3497 0.0282 0.0003 0.0001 0.0111  ,  ±   −1.0179 −2.0270 −1.2860 1.0958 −0.0252 0.0016 −0.0055 −0.0015       −0.5533 −0.6630 2.4434 1.1293 0.0910 −0.0039 0.0134 −0.0137       0.0009 −0.0024 −0.0123 −0.0036 −0.0062 0.0110 2.8119 0.3046    −0.0050 −0.0097 0.0154 −0.0025 −0.0136 −0.0163 −0.3045 2.8118 and the radius, r, is. √. 8. These points are different from the original star points. of the spherical CCD clearly. • n0 = 4 When n0 = 4, the sixteen locally D-optimal star points are   −0.0313 0.0386 0.1082 −0.7810 −2.7070 0.2180 0.0116 −0.0124      −0.1654 −0.2605 −0.3118 2.0839 −0.7526 −1.7025 0.0118 0.0047       2.3783 0.3584 −1.4309 −0.2535 −0.0321 −0.3199 0.0138 −0.0101       −0.0052 0.2733 −0.7050 1.6007 −0.3101 2.1840 −0.0012 −0.0044   , ±   1.5215 −0.5745 2.2057 0.6131 −0.0882 0.3256 0.0053 −0.0146       0.0071 −2.7198 −0.6938 −0.2127 0.0168 0.2747 −0.0127 0.0129       −0.0119 −0.0106 0.0012 −0.0058 0.0134 0.0099 2.8153 0.2715    0.0177 0.0129 0.0092 −0.0006 −0.0131 0.0055 −0.2714 2.8152 and the radius, r, is. √. 8. These points are different from the original star points. of the spherical CCD clearly.. 4.2. Fractional Factorial Designs in the First-order Design. Instead of the full factorial designs, the fractional factorial designs with resolution V or higher would be used in the central composite designs, especially when the number of factors, k, is large, for example, k ≥ 5. Here we have the conjecture that we only (0). consider the possible extremum point b(0) with bi 27. ≥ 0, i = 1, · · · , k, which is similar.

(34) to Theorem 1. However we do not prove that this conjecture is correct theoretically. Several fractional factorial designs for k = 5, 6, 7, 8 are considered in the following numerical results which are obtained by BAR sampler. 4.2.1. 25−1 Fractional Factorial Design V. Based on the 25−1 design, since a locally D-optimal design depends on b(0) , we V √ √ choose b(0) by fixing the values of R and φj , j = 1, 2, · · · , 4, as R = 0.25 5, 0.5 5, √ √ π 2π 3π 0.75 5, and 5, and φj = 0, 10 , 10 , 10 , and 4π . Thus totally there are 4 × 54 = 2500 10 cases here. We show one case with R =. √. 5, φ1 = 0, φ2 = 0, φ3 = 0, and φ4 =. π 10. as an example. here. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 5, based on (5), we need to have ten rotation angles, θ1 , θ2 , · · · , θ10 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the ten locally D-optimal star points numerically. • n0 = 1 When n0 = 1, the ten locally D-optimal  −0.4716 1.5629 1.5281    −0.6932 1.3756 −1.6209   ±  −2.0729 −0.8156 0.1943    −0.0057 −0.0070 0.0071  0.0017 −0.0011 0.0010 and the radius, r, is. √. star points are  −0.0012. 0.0001.   0.0069 0.0038    −0.0085 −0.0011  ,   2.1618 0.5712   −0.5712 2.1619. 5. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the ten locally D-optimal  2.0749 0.6141 −0.5635    0.6130 −2.1487 −0.0843   ±  −0.5647 −0.0762 −2.1622    −0.0049 0.0047 0.0069  −0.0044 0.0028 −0.0044 28. star points are  0.0044. 0.0032.   0.0049    0.0066 −0.0040  ,   2.1868 0.4669   −0.4669 2.1868 0.0052.

(35) and the radius, r, is. √. 5. These points are different from the original star points. of the spherical CCD clearly. • n0 = 3 When n0 = 3, the ten locally D-optimal star points are   0.3603 −2.0144 −0.9014 −0.0059 0.0007      −0.3815 0.8428 −2.0357 −0.0042 0.0002      ±  2.1736 0.4817 −0.2079 0.0088 −0.0057  ,      −0.0072 −0.0052 −0.0053 2.1972 0.4152    0.0069 0.0028 0.0009 −0.4152 2.1972 √ and the radius, r, is 5. These points are different from the original star points of the spherical CCD clearly. • n0 = 4 When n0 = 4, the ten locally D-optimal star points are   1.8099 −0.8773 −0.9769 0.0033 −0.0082      −0.9421 0.2911 −2.0069 0.0065 0.0003      ±  0.9146 2.0360 −0.1340 −0.0103 0.0020  ,      0.0052 0.0088 0.0061 2.2043 0.3753    0.0051 −0.0067 −0.0043 −0.3752 2.2043 √ and the radius, r, is 5. These points are different from the original star points of the spherical CCD clearly. 4.2.2. 26−1 V I Fractional Factorial Design. (0) Based on the 26−1 V I design, since a locally D-optimal design depends on b , we √ √ choose b(0) by fixing the values of R and φj , j = 1, 2, · · · , 5, as R = 0.25 6, 0.5 6, √ √ π 2π 3π 0.75 6, and 6, and φj = 0, 10 , 10 , 10 , and 4π . Thus totally there are 4 × 55 = 12500 10. cases here. We show one case with R =. √. 6, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, and φ5 =. π 10. as an. example here. The numbers of the center points, n0 , are 1, 2, 3, and 4. For k = 6, based on (5), we need to have fifteen rotation angles, θ1 , θ2 , · · · , θ15 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the twelve locally D-optimal star points numerically. 29.

(36) • n0 = 1 When n0 = 1, the twelve locally D-optimal star points are   1.0927 −0.6063 −2.1060 0.0560 0.0057 −0.0075      −0.6013 0.1631 −0.4210 −2.3312 −0.0002 0.0038       −1.3916 −1.9959 −0.1409 0.2447 −0.0091 −0.0060   , ±   1.5836 −1.2736 1.1695 −0.7088 0.0010 −0.0072       −0.0069 −0.0078 0.0031 0.0014 2.3818 0.5718    0.0073 −0.0092 −0.0035 0.0020 −0.5718 2.3818 √ and the radius, r, is 6. These points are different from the original star points of the spherical CCD clearly. • n0 = 2 When n0 = 2, the twelve locally D-optimal star points are   −0.0103 0.0008 −0.3625 2.4225 0.0011 −0.0050      −1.0784 −0.7853 2.0323 0.2997 0.0077 −0.0091       2.1442 −0.8841 0.7776 0.1258 0.0060 0.0026  , ±    −0.4890 −2.1451 −1.0647 −0.1607 −0.0015 −0.0029       −0.0034 0.0024 −0.0077 −0.0012 2.4010 0.4847    −0.0063 −0.0050 0.0064 0.0061 −0.4846 2.4010 √ and the radius, r, is 6. These points are different from the original star points of the spherical CCD clearly. • n0 = 3 When n0 = 3, the twelve locally D-optimal star points are   0.6358 −0.0151 0.5542 −2.2996 0.0020 −0.0051      1.8680 −0.1645 1.3339 0.8390 0.0016 −0.0061       1.1465 1.6650 −1.3830 −0.0273 0.0012 −0.0023  ,  ±   −0.8898 1.7890 1.4145 0.0831 −0.0078 −0.0044       −0.0041 0.0057 0.0048 0.0011 2.4107 0.4341    0.0062 0.0034 0.0050 −0.0028 −0.4341 2.4107 √ and the radius, r, is 6. These points are different from the original star points of the spherical CCD clearly. 30.

(37) • n0 = 4 When n0 = 4, the twelve  −1.8824 0.1299    0.2763 −1.9522    −0.4904 −1.4731 ±   1.4628 0.0421    0.0028 −0.0066  0.0042 −0.0089 and the radius, r, is. √. locally D-optimal star points are  −1.2552 −0.9296 −0.0009. 0.0095.   −1.3928 −0.0092 −0.0052    −0.7161 1.7540 0.0036 −0.0085  ,  −1.9337 −0.3451 −0.0017 0.0026    0.0017 −0.0071 2.4159 0.4040   0.0051 0.0084 −0.4040 2.4159 0.4151. 6. These points are different from the original star points. of the spherical CCD clearly. 27−1 V II Fractional Factorial Design. 4.2.3. (0) Based on the 27−1 V II design, since a locally D-optimal design depends on b , we √ √ choose b(0) by fixing the values of R and φj , j = 1, 2, · · · , 6, as R = 0.25 7, 0.5 7, √ √ π 2π 3π , 10 , 10 , and 4π . Thus totally there are 4 × 56 = 62500 0.75 7, and 7, and φj = 0, 10 10. cases here. We show one case with R = φ6 =. π 10. √. 7, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, φ5 = 0, and. as an example here. The numbers of the center points, n0 , are 1, 2, 3, and 4.. For k = 7, based on (5), we need to have twenty-one rotation angles, θ1 , θ2 , · · · , θ21 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the fourteen locally D-optimal star points numerically. • n0 = 1 When n0 = 1, the fourteen locally D-optimal star points are   −0.7702 2.0196 1.1109 −0.6380 0.8288 0.0031 0.0022      −0.1569 −0.3116 0.0053 −2.3361 −1.1919 0.0030 0.0060       0.1020 0.9275 0.5880 0.9892 −2.1920 0.0038 −0.0033      ±  1.9317 −0.3991 1.7263 −0.2183 0.2857 0.0018 0.0075  ,      1.6249 1.3434 −1.5621 −0.3306 0.0758 0.0017 0.0006       −0.0024 −0.0036 −0.0033 0.0040 0.0029 2.5852 0.5627    −0.0043 0.0018 −0.0041 0.0071 −0.0022 −0.5627 2.5852 31.

(38) and the radius, r, is. √. 7. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the fourteen locally D-optimal star points are  0.1675 −0.1101 −0.1871 −1.0348 −2.4195 −0.0080    −0.2037 0.2945 0.7147 2.2879 −1.0613 −0.0041    2.2102 0.3115 −1.3041 0.5634 −0.0013 −0.0006   ±  0.2647 2.4499 0.8112 −0.5153 0.0645 0.0080    1.4055 −0.8954 2.0237 −0.3341 0.1244 −0.0094    0.0065 −0.0090 0.0068 0.0025 −0.0079 2.6001  0.0085 0.0081 0.0095 0.0083 0.0053 −0.4894 and the radius, r, is. √.  0.0072.   −0.0088    −0.0054    −0.0085  ,   −0.0101    0.4894   2.600. 7. These points are different from the original star points. of the spherical CCD clearly. • n0 = 3 When n0 = 3, the fourteen locally D-optimal star points are   −0.0767 0.0597 0.3680 −0.8893 −2.4626 −0.0019 0.0020      −0.4352 0.5650 1.0154 −2.1349 0.9500 0.0043 −0.0074       −0.3858 −2.4659 −0.4649 −0.7299 0.1463 0.0095 −0.0060      ±  −0.4491 −0.6646 2.3117 1.0062 −0.0201 −0.0059 −0.0088  ,      2.5405 −0.3933 0.5231 −0.3255 0.1071 0.0053 0.0020       −0.0049 0.0062 0.0057 0.0077 −0.0033 2.6108 0.4284    −0.0047 −0.0071 0.0080 −0.0047 0.0053 −0.4284 2.6108 and the radius, r, is. √. 7. These points are different from the original star points. of the spherical CCD clearly.. 32.

(39) • n0 = 4 When n0 = 4, the fourteen locally D-optimal star points are   −0.7621 −0.9338 −1.1294 0.6375 1.9661 −0.0060 −0.0044      −0.8485 −1.0400 −1.2986 0.6180 −1.7692 0.0033 0.0033       1.0430 −0.6565 −1.1941 −2.0129 0.0591 −0.0042 −0.0044      ±  1.3475 1.3743 −1.4408 1.1044 −0.0107 0.0036 0.0004  ,      −1.6721 1.6513 −0.7323 −0.9697 0.0299 −0.0034 −0.0011       −0.0029 −0.0017 −0.0021 −0.0058 0.0076 2.6137 0.4106    0.0010 −0.0006 −0.0020 −0.0028 0.0045 −0.4106 2.6137 and the radius, r, is. √. 7. These points are different from the original star points. of the spherical CCD clearly. 4.2.4. 28−2 Fractional Factorial Design V. Based on the 28−2 design, since a locally D-optimal design depends on b(0) , we V √ √ choose b(0) by fixing the values of R and φj , j = 1, 2, · · · , 7, as R = 0.25 8, 0.5 8, √ √ π 2π 3π 0.75 8, and 8, and φj = 0, 10 , 10 , 10 , and 4π . Thus totally there are 4 × 57 = 312500 10 cases here. We show one case with R = and φ7 =. π 10. √. 8, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, φ5 = 0, φ6 = 0,. as an example here. The numbers of the center points, n0 , are 1, 2, 3, and. 4. For k = 8, based on (5), we need to have twenty-eight rotation angles, θ1 , θ2 , · · · , θ28 , and one radius r in our objective function, and then we employ the BAR sampler with N t = 10, T (t) ∝ t−2 , and 400 times iterations to find the sixteen locally D-optimal star points numerically.. 33.

(40) • n0 = 1 When n0 = 1, the sixteen locally D-optimal star points are  −0.4179 −0.4886 0.1864 0.5445 0.0508 −2.6931    1.0047 0.6487 −0.8454 −1.1307 2.0747 −0.5216    0.3911 −2.1634 −0.4728 1.2000 1.0897 0.5622    −0.1634 1.6051 −0.6917 2.1558 0.5033 0.1316 ±   −2.5611 −0.1267 −0.6583 −0.5157 0.8024 0.2856    0.2777 −0.2603 −2.4728 −0.2677 −1.2682 −0.2451    0.0057 −0.0102 0.0115 −0.0074 0.0039 −0.0075  0.0140 0.0090 −0.0165 0.0047 −0.0026 −0.0112 and the radius, r, is. √.  −0.0057 −0.0083 −0.0011 −0.0108 −0.0042. 0.0025. 0.0164. −0.0074. 0.0028. 0.0129. 0.0126. −0.0142. 2.7535. 0.6465. −0.6464. 2.7534.          ,         . 8. These points are different from the original star points. of the spherical CCD clearly. • n0 = 2 When n0 = 2, the sixteen locally D-optimal star points are  −0.0039 0.0589 −0.0097 0.2043 0.7978 −2.7052    −2.0277 0.4522 −0.5805 −1.5654 0.9311 0.1713    −1.0453 0.5119 −0.2885 0.0675 −2.4611 −0.7071    1.0594 −1.0746 0.3303 −2.2414 −0.6614 −0.3905 ±   −1.2731 −2.2432 0.9775 0.6254 −0.0205 −0.0093    0.2288 −1.1590 −2.5524 0.2971 −0.0299 −0.0028    −0.0008 −0.0095 −0.0045 −0.0054 −0.0059 −0.0044  −0.0088 0.0119 0.0095 0.0045 0.0244 0.0009 and the radius, r, is. √.  −0.0006 −0.0069 0.0016. −0.0118. −0.0084. 0.0156. −0.0120. 0.0140. −0.0054. 0.0003. −0.0100. 0.0123. 2.7738. 0.5531. −0.5530. 2.7737. 8. These points are different from the original star points. of the spherical CCD clearly.. 34.          ,         .

(41) • n0 = 3 When n0 = 3, the sixteen locally D-optimal star points are  0.0447 −0.0068 0.0183 −1.8387 2.1370 −0.2239    −0.4135 0.0187 −0.0283 −0.6461 −0.8190 −2.5961    0.5810 −0.7941 0.3617 1.8885 1.4977 −1.0446    −2.6638 0.3541 0.4499 0.5339 0.5253 0.1233 ±   −0.5402 −1.1908 −2.5037 0.0863 0.1083 0.0491    −0.3187 −2.4136 1.1819 −0.5854 −0.4815 0.3180    −0.0086 −0.0105 0.0014 0.0061 0.0075 −0.0032  0.0176 0.0070 −0.0058 0.0008 0.0073 −0.0181 and the radius, r, is. √.  −0.0006 −0.0069 0.0016. −0.0118. −0.0084 −0.0136 −0.0120. 0.0140. −0.0054. 0.0003. −0.0100. 0.0123. 2.7846. 0.4957. −0.4956. 2.7845.          ,         . 8. These points are different from the original star points. of the spherical CCD clearly. • n0 = 4 When n0 = 4, the sixteen locally D-optimal star points are  −0.1154 0.0969 0.1144 −0.1490 −0.0345 2.8179    1.1404 −0.6068 1.0294 −1.9710 1.1759 −0.0641    2.1037 −1.2997 −0.0935 1.0461 −0.8659 0.1793    0.7290 0.4968 −2.5103 −0.9406 0.1742 0.0671 ±   0.8122 1.8104 0.7427 −0.7243 −1.7241 −0.1185    −1.0341 −1.5520 −0.2541 −1.2607 −1.6921 −0.0660    0.0090 −0.0041 −0.0079 −0.0045 −0.0103 0.0022  0.0141 −0.0039 0.0191 0.0075 0.0091 0.0071 and the radius, r, is. √.  −0.0006 −0.0069 0.0016. −0.0118. −0.0084 −0.0136 −0.0120. 0.0140. −0.0054. 0.0003. −0.0100. 0.0123. 2.7896. 0.4669. −0.4668. 2.7895. 8. These points are different from the original star points. of the spherical CCD clearly.. 4.3. Comparison with Spherical CCD. We employ the relative locally D-efficiency in (1) to compare our conditionally locally D-optimal central composite designs and the spherical CCD’s. The relative locally D-efficiency of our conditionally locally D-optimal central composite design. 35.          ,         .

(42) and spherical CCD is defined as ¡ ¢ ∗ det Mb−1 (ξD ) ¡ ¢, det Mb−1 (ξCCD ) ∗ where ξD is our conditionally locally D-optimal central composite design with respect. to b(0) and ξCCD is the spherical CCD . If the relative locally D-efficiency is less than 1, our conditionally locally D-optimal central composite design is better than the corresponding spherical CCD. That is the lower relative locally D-efficiency is, the better our conditionally locally D-optimal central composite design we get. Based on the different first-order design with n0 center points, the relative locally D-efficiencies are shown as follows. For each k, we have tried the cases of different b(0) by choosing values of R and angels φj , j = 1, · · · , k − 1. Given a first-order design (included the factorial design and n0 center points) and R, the best and the worst values of the relative locally D-efficiencies are shown in Tables 2 to 9. The results based on full factorial designs are in Tables 2 to 5, and the results in Tables 6 to 9 are the cases that we choose fractional factorial designs in the first-order designs.. R\k √ 0.25 k √ 0.5 k √ 0.75 k √. k. 2. 3. 4. 5. 6. 7. 8. 0.9920. 0.9847. 0.9777. 0.9698. 0.9595. 0.9546. 0.9516. (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). 0.9423. 0.9070. 0.8870. 0.8727. 0.8686. 0.8657. 0.8689. (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). 0.8935. 0.8436. 0.8241. 0.8137. 0.8152. 0.8230. 0.8357. (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). 0.8618. 0.8072. 0.7903. 0.7839. 0.7888. 0.8032. 0.8215. (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). (1.0000). Table 2. The relative locally D-efficiencies of our conditionally locally D-optimal central composite designs and the spherical CCD’s for n0 = 1 and 2k full factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest relative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s. 36.