## A new approach to the construction of optimal designs (1993)

Venue: | J. Statistical Planning and Inference |

Citations: | 32 - 10 self |

### BibTeX

@ARTICLE{Hardin93anew,

author = {R. H. Hardin and R. H. Hardin and N. J. A. Sloane and N. J. A. Sloane},

title = {A new approach to the construction of optimal designs},

journal = {J. Statistical Planning and Inference},

year = {1993},

volume = {37},

pages = {339--369}

}

### Years of Citing Articles

### OpenURL

### Abstract

By combining a modified version of Hooke and Jeeves ’ pattern search with exact or Monte Carlo moment calculations, it is possible to find I-, D- and A-optimal (or nearly optimal) designs for a wide range of response-surface problems. The algorithm routinely handles problems involving the minimization of functions of 1000 variables, and so for example can construct designs for a full quadratic response-surface depending on 12 continuous process variables. The algorithm handles continuous or discrete variables, linear equality or inequality constraints, and a response surface that is any low degree polynomial. The design may be required to include a specified set of points, so a sequence of designs can be obtained, each optimal given that the earlier runs have been made. The modeling region need not coincide with the measurement region. The algorithm has been implemented in a program called gosset, which has been used to compute extensive tables of designs. Many of these are more efficient than the best designs previously known.

### Citations

462 |
Empirical Model Building and Response Surfaces
- Box, Draper
- 1987
(Show Context)
Citation Context ...terion Extensive discussions of the relative merits of A-, D-, E-, G- and I-optimality and of the dangers of relying on any single numerical criterion have appeared in the literature (see for example =-=[13]-=-, [15], [48], [64], [87]), and our treatment will be brief. Suppose for concreteness that we wish to construct a design for a full quadratic responsesurface model y = β0 + k� βixi + i=1 k� i=1 � βiix ... |

250 |
Regular polytopes
- Coxeter
- 1948
(Show Context)
Citation Context ...imilarly, any one of our 14-point designs, supplemented by two center points, forms a rotatable (and D-optimal) 16-point design. The design based on the icosahedron has of course a long history [14], =-=[23]-=-, but the other designs appear to be new. Bose and Draper [8] give an infinite family of rotatable 16-point designs, but these lie on two concentric spheres, not one, and are neither D- nor I-optimal.... |

69 |
The equivalence of two extremum problems
- Kiefer, Wolfowitz
- 1960
(Show Context)
Citation Context ...Eq. (13) below). A number of interesting designs meet the bound. There is a similar bound for D-optimal designs. (b) It is known that for large numbers of runs D- and G-optimal designs are equivalent =-=[65]-=-. The results of Section 5 show that I-optimal designs are strictly different. For example, I-optimal designs make more measurements at the center of the region and fewer at the boundary. For quadrati... |

63 |
A Basis for the Selection of a Response Surface Design
- Box, Draper
- 1959
(Show Context)
Citation Context ...esign measure (the prime indicates matrix transposition). The prediction variance at an arbitrary point x is var �y(x) = σ2 −1 f(x)MX n f(x)′ . We define an I-optimal design (following Box and Draper =-=[11]-=-, [12] and others) to be one which minimizes the normalized average or integrated prediction variance I = n σ2 � var �y(x)dµ(x) , (3) R 4swhere R is the region of interest (or modeling region), and µ ... |

60 | Some New Three Level Designs for the Study of Quantitative Variables
- Box, Behnken
- 1960
(Show Context)
Citation Context ...n by pairs of complex numbers. It consists of the points [0, 0], [ω r , 0], [0, ω r ], 2 −1/2 [−ω r , −ω s ], where ω = e 2πi/3 , and r, s ∈ {0, 1, 2}. For 27 points the program finds the Box-Behnken =-=[10]-=- design, which consists of three center 21spoints together with the vertices of the 24-cell [19], [23], and is I-optimal (although not unique). Roquemore’s six-dimensional design 628A, although it is ... |

44 |
A Comparison of Algorithms for Constructing Exact D-optimal Designs
- Cook, Nachtsheim
- 1980
(Show Context)
Citation Context ...individual ingredient of our algorithm is new. But although computers have been used by many people to construct experimental designs (surveys of such work can be found for example in [13, Chap. 15], =-=[21]-=-, [27], [70], [78], [79], [87], [94], [100]), we believe that no algorithm comparable to ours is presently available. Implementation. We have implemented the algorithm in the C language in a program c... |

35 |
Multi-factor experimental designs for exploring response surfaces
- Box, Hunter
- 1957
(Show Context)
Citation Context ...es are maximized if the B surface points can be arranged to form a spherical 4-design. The proof for I-efficiency is given in [54], while for D-efficiency this follows from the work of Box and Hunter =-=[14]-=- and Kiefer [63] (see also [38], [80]). If the surface points do form a spherical 4-design, the I- and D-values of the design can be calculated analytically, and we obtain the following bounds. The I-... |

26 |
Determinantenabschätzungen für binäre Matrizen
- Ehlich
- 1964
(Show Context)
Citation Context ... = 1.6863 · 10 13 . 5.3. Linear designs in the cube There is a substantial body of work dealing with the construction of designs for linear regression in the k-dimensional cube (see for example [17], =-=[34]-=-, [35], [36], [37], [39], [43]–[45], [46], [47], [73], [74], [77], [88], [89], [91], [99], and the references therein). The problem is to construct a design for the model β0 + β1x1 + · · · + βkxk, usi... |

24 |
Statistics for Experimenters: Design
- Hunter
- 2005
(Show Context)
Citation Context ...hm is new. But although computers have been used by many people to construct experimental designs (surveys of such work can be found for example in [13, Chap. 15], [21], [27], [70], [78], [79], [87], =-=[94]-=-, [100]), we believe that no algorithm comparable to ours is presently available. Implementation. We have implemented the algorithm in the C language in a program called gosset. A built-in parser perm... |

22 |
The Application of the Annealing Algorithm to the Construction of Exact Optimal Designs for Linear-Regression Models
- Haines
- 1987
(Show Context)
Citation Context ...s section by briefly mentioning some earlier papers that look for I-optimal designs. Studden [95] gives a method for constructing approximations to I-optimal designs for 1dimensional problems. Haines =-=[49]-=-, Meyer and Nachtsheim [72], Crary [24] and Snow and Crary [93] all use simulated annealing to search for I-optimal designs. This approach seems to be restricted to low dimensions and small numbers of... |

18 |
Operating Manual for Gosset: A General-Purpose Program for Constructing Experimental Designs
- Hardin, Sloane
- 2003
(Show Context)
Citation Context ...gorithm in the C language in a program called gosset. A built-in parser permits a very flexible input. A brief description is given in Section 4, and further details can be found in the users’ manual =-=[50]-=-. The program is named after the amateur mathematician Thorold Gosset (1869–1962), who was one of the first to study polytopes in six, seven and eight dimensions [23, p. 164], and his contemporary, th... |

14 |
Time- and space-saving computer methods, related to Mitchell's DETMAX, for finding D-optimum designs
- Galil, Kiefer
- 1980
(Show Context)
Citation Context ...ity has been rigorously established.) For three dimensions there are a number of papers dealing with D-optimal 3-level designs (see Atkinson [1], Mitchell and Bayne [75], [76]; Galil and Kiefer [41], =-=[44]-=-; Welch [98]; Meyer and Nachtsheim [72]). We have used gosset to search for both I- and D-optimal designs, both with continuous and with 3-level coordinates, and the designs with from 10 to 20 runs ar... |

13 |
Graphical Assessment of the Prediction Capability of Response Surface Designs
- Giovannitti-Jensen, Myers
- 1989
(Show Context)
Citation Context ...sive discussions of the relative merits of A-, D-, E-, G- and I-optimality and of the dangers of relying on any single numerical criterion have appeared in the literature (see for example [13], [15], =-=[48]-=-, [64], [87]), and our treatment will be brief. Suppose for concreteness that we wish to construct a design for a full quadratic responsesurface model y = β0 + k� βixi + i=1 k� i=1 � βiix 2 i + k−1 k�... |

12 |
The choice of a second order rotatable design
- BOX, DRAPER
- 1963
(Show Context)
Citation Context ...measure (the prime indicates matrix transposition). The prediction variance at an arbitrary point x is var �y(x) = σ2 −1 f(x)MX n f(x)′ . We define an I-optimal design (following Box and Draper [11], =-=[12]-=- and others) to be one which minimizes the normalized average or integrated prediction variance I = n σ2 � var �y(x)dµ(x) , (3) R 4swhere R is the region of interest (or modeling region), and µ is uni... |

11 |
Experiments with mixtures
- Cornell
- 1981
(Show Context)
Citation Context ... . . , 47. For k ≡ 3 (mod 4) these are Plackett-Burman-Rao designs, while the others are discussed in Section 5.3. We have also used gosset to construct other classes of designs: mixture designs (cf. =-=[22]-=-, [97]), block designs, and designs which are intended for use in circumstances when one of the measurements may be lost. 5.1. Quadratic designs in the ball Study of the extensive collection of quadra... |

11 |
Computer construction of d-optimal first-order designs
- Mitchell
- 1974
(Show Context)
Citation Context ...s a substantial body of work dealing with the construction of designs for linear regression in the k-dimensional cube (see for example [17], [34], [35], [36], [37], [39], [43]–[45], [46], [47], [73], =-=[74]-=-, [77], [88], [89], [91], [99], and the references therein). The problem is to construct a design for the model β0 + β1x1 + · · · + βkxk, using n ≥ k + 1 points in the unit 23sTable 6: Conjecturally I... |

11 |
Branch-and-Bound Search for Experimental Designs Based on 'VOptimality and Other Criteria
- Welch
- 1982
(Show Context)
Citation Context ... rigorously established.) For three dimensions there are a number of papers dealing with D-optimal 3-level designs (see Atkinson [1], Mitchell and Bayne [75], [76]; Galil and Kiefer [41], [44]; Welch =-=[98]-=-; Meyer and Nachtsheim [72]). We have used gosset to search for both I- and D-optimal designs, both with continuous and with 3-level coordinates, and the designs with from 10 to 20 runs are summarized... |

10 |
Optimal and efficient designs of experiments
- Atwood
- 1969
(Show Context)
Citation Context ...in (4), (6)–(9) as the I-, A-, etc. values of the design. As the number of runs n → ∞, these quantities approach limits I∞, A∞, etc., and the I-, A-, etc. efficiencies of the design are given by (cf. =-=[2]-=-, [70]). I∞ I , A∞ A , D∞ D E∞ G∞ , , E G We are most interested in minimizing the prediction variance, var � f(x), for x ∈ R, which suggests the use of I- or G-optimality. On the other hand our algor... |

10 |
D-optimum weighing designs
- Galil, Kiefer
- 1980
(Show Context)
Citation Context ...r designs in the cube There is a substantial body of work dealing with the construction of designs for linear regression in the k-dimensional cube (see for example [17], [34], [35], [36], [37], [39], =-=[43]-=-–[45], [46], [47], [73], [74], [77], [88], [89], [91], [99], and the references therein). The problem is to construct a design for the model β0 + β1x1 + · · · + βkxk, using n ≥ k + 1 points in the uni... |

10 |
D-optimality for Regression Designs
- JOHN, R
- 1975
(Show Context)
Citation Context ...lgorithm is new. But although computers have been used by many people to construct experimental designs (surveys of such work can be found for example in [13, Chap. 15], [21], [27], [70], [78], [79], =-=[87]-=-, [94], [100]), we believe that no algorithm comparable to ours is presently available. Implementation. We have implemented the algorithm in the C language in a program called gosset. A built-in parse... |

9 |
Optimum experimental designs V, with applications to systematic and rotatable designs
- Kiefer
- 1961
(Show Context)
Citation Context ... work on these “classical” problems can be regarded as an attempt to provide optimal “exact” designs with small numbers of runs to complement the “asymptotic” designs of Kiefer et al. [38], [39]–[47],=-=[63]-=-–[68]. As we shall see in Section 5, some of these designs overlap with and improve on designs already available in the literature. Other designs we have found will be published elsewhere (see [51], [... |

9 |
The exact D-optimal first order saturated design with 17 observations
- Moyssiadis, Kounias
- 1982
(Show Context)
Citation Context ...bstantial body of work dealing with the construction of designs for linear regression in the k-dimensional cube (see for example [17], [34], [35], [36], [37], [39], [43]–[45], [46], [47], [73], [74], =-=[77]-=-, [88], [89], [91], [99], and the references therein). The problem is to construct a design for the model β0 + β1x1 + · · · + βkxk, using n ≥ k + 1 points in the unit 23sTable 6: Conjecturally I-optim... |

8 |
The maximum determinant of 21
- Chadjipantelis, Kounias, et al.
- 1987
(Show Context)
Citation Context ... X ′ X = 1.6863 · 10 13 . 5.3. Linear designs in the cube There is a substantial body of work dealing with the construction of designs for linear regression in the k-dimensional cube (see for example =-=[17]-=-, [34], [35], [36], [37], [39], [43]–[45], [46], [47], [73], [74], [77], [88], [89], [91], [99], and the references therein). The problem is to construct a design for the model β0 + β1x1 + · · · + βkx... |

8 |
Uniform shell designs
- Doehlert
- 1970
(Show Context)
Citation Context ...mal), while for 14 points the icosahedron plus two center points is neither I- nor D-optimal. The 15-point central composite design mentioned above is also neither I- nor D-optimal, nor is Doehlert’s =-=[28]-=-, [29], 13-point uniform shell design (not shown in the table), which consists of the center and vertices of a cuboctahedron, and has Iand D-efficiencies of only 82.6 and 97.0%. Table 4: Conjecturally... |

8 |
Saturated fractions of 2” and 3” factorial designs
- Rechtschaffner
- 1967
(Show Context)
Citation Context ...runs in a cube. Draper and Lin [32] give a table comparing the D-values of designs found by Box and Draper [16], Draper and Lin [31], Lucas [69], Mitchell and Bayne [75], Notz [81] and Rechtschaffner =-=[85]-=-. We have reproduced this as Table 7, supplemented with an additional column showing the D-values of the designs found by gosset. In dimensions k ≥ 4 the new designs are better. The entries give the r... |

7 |
Construction of spherical t-designs
- Bajnok
- 1992
(Show Context)
Citation Context ...ensions if and only if B ≥ 20; in five dimensions if and only if B ≥ 29; and so on. These conjectures are far more precise than our existing knowledge of spherical 4-designs (compare [26], [90], [4], =-=[5]-=-, [84]). They are only conjectures, because in general we have only numerical evidence that the designs meet the bound (13). However, in several of these cases we have been able to prove that the poin... |

7 |
The Cell Structures of Lattices
- Conway, Sloane
- 1991
(Show Context)
Citation Context ...here Temp Zinc Water center 300 20 4 radius .5 30 discrete KEV 70 90 100 which indicate that a, b, c are continuous variables ranging from 90 to 120; Temp, Zinc and Water belong to a ball centered at =-=[300, 20, 50]-=- of radius .5, and KEV is a quantitative discrete variable with values 70, 90, 100. Any number of such lines may appear. To allow a more flexible input format, the program includes its own parser. (Pr... |

7 |
Minimal point second order design
- Notz
- 1982
(Show Context)
Citation Context ...h the minimal number of runs in a cube. Draper and Lin [32] give a table comparing the D-values of designs found by Box and Draper [16], Draper and Lin [31], Lucas [69], Mitchell and Bayne [75], Notz =-=[81]-=- and Rechtschaffner [85]. We have reproduced this as Table 7, supplemented with an additional column showing the D-values of the designs found by gosset. In dimensions k ≥ 4 the new designs are better... |

7 |
Hybrid Designs for Quadratic Response Surfaces
- ROQUEMORE
- 1976
(Show Context)
Citation Context ...d in [8], [30], [56]. Table 3 compares the I- and D-efficiencies of some designs found by our algorithm with those of earlier designs. In this table, a and b indicate Roquemore’s designs 310 and 311B =-=[86]-=-, c indicates the 12 vertices of the icosahedron, possibly supplemented with copies of the center point, and d is the central composite design with points 0, 0, 0; (±1, 0, 0); ±3 −1/2 , ±3 −1/2 , ±3 −... |

6 |
Second order rotatable designs in three dimensions
- Bose, Draper
- 1959
(Show Context)
Citation Context ... center points, forms a rotatable (and D-optimal) 16-point design. The design based on the icosahedron has of course a long history [14], [23], but the other designs appear to be new. Bose and Draper =-=[8]-=- give an infinite family of rotatable 16-point designs, but these lie on two concentric spheres, not one, and are neither D- nor I-optimal. The same is true of most of the rotatable designs constructe... |

6 |
and F.Pukelsheim, Another look at Rotatability
- Draper
- 1990
(Show Context)
Citation Context ... Neumaier and Seidel [80] that a design has D-value given by (14) if and only if the B surface points form a spherical 4-design. (Incidentally, in reference to the discussion in Draper and Pukelsheim =-=[33]-=-, this suggests that the difference between the D-value of a design and the minimum of (14) over all choices of γ = C/n with 1 ≤ C ≤ n is a good measure of the departure of the design from rotatabilit... |

6 |
Optimum multivariate designs, The
- Farrell, Kiefer, et al.
- 1967
(Show Context)
Citation Context ...into gosset. Our work on these “classical” problems can be regarded as an attempt to provide optimal “exact” designs with small numbers of runs to complement the “asymptotic” designs of Kiefer et al. =-=[38]-=-, [39]–[47],[63]–[68]. As we shall see in Section 5, some of these designs overlap with and improve on designs already available in the literature. Other designs we have found will be published elsewh... |

6 |
Optimum Composite Designs
- Lucas
(Show Context)
Citation Context ...empted to find D-optimal designs with the minimal number of runs in a cube. Draper and Lin [32] give a table comparing the D-values of designs found by Box and Draper [16], Draper and Lin [31], Lucas =-=[69]-=-, Mitchell and Bayne [75], Notz [81] and Rechtschaffner [85]. We have reproduced this as Table 7, supplemented with an additional column showing the D-values of the designs found by gosset. In dimensi... |

5 |
Simplex-Sum Designs: A Class of Second Order Rotatable Designs Derivable from Those of First Order
- BOX, BEHNKEN
- 1960
(Show Context)
Citation Context ...nt plus the well-known 27-point spherical 4-design associated with the Schläfli polytope (see [26], [20]). It is therefore both I- and D-optimal. This design was also found by gosset. Box and Behnken =-=[9]-=-, [10], Crosier [25], Doehlert [28], Doehlert and Klee [29] and others have given quadratic designs in higher-dimensional spheres, but these all contain many more points than ours. 5.2. Quadratic desi... |

5 |
third edition
- Conway, Sloane
- 1988
(Show Context)
Citation Context ...ω r , −ω s ], where ω = e 2πi/3 , and r, s ∈ {0, 1, 2}. For 27 points the program finds the Box-Behnken [10] design, which consists of three center 21spoints together with the vertices of the 24-cell =-=[19]-=-, [23], and is I-optimal (although not unique). Roquemore’s six-dimensional design 628A, although it is not identified as such in [86], consists of a center point plus the well-known 27-point spherica... |

5 |
Computing D-optimumweighing designs: where statistics combinatorics and computationmeet
- Galil
- 1985
(Show Context)
Citation Context ...osset. Our work on these “classical” problems can be regarded as an attempt to provide optimal “exact” designs with small numbers of runs to complement the “asymptotic” designs of Kiefer et al. [38], =-=[39]-=-–[47],[63]–[68]. As we shall see in Section 5, some of these designs overlap with and improve on designs already available in the literature. Other designs we have found will be published elsewhere (s... |

5 |
Comparison of Design for Quadratic Regression on Cubes
- Galil, Kiefer
- 1977
(Show Context)
Citation Context ...ptimality has been rigorously established.) For three dimensions there are a number of papers dealing with D-optimal 3-level designs (see Atkinson [1], Mitchell and Bayne [75], [76]; Galil and Kiefer =-=[41]-=-, [44]; Welch [98]; Meyer and Nachtsheim [72]). We have used gosset to search for both I- and D-optimal designs, both with continuous and with 3-level coordinates, and the designs with from 10 to 20 r... |

5 |
Construction methods for D-optimum weighing designs when n ≡ 3 (mod 4
- Galil, Kiefer
- 1982
(Show Context)
Citation Context .... Our work on these “classical” problems can be regarded as an attempt to provide optimal “exact” designs with small numbers of runs to complement the “asymptotic” designs of Kiefer et al. [38], [39]–=-=[47]-=-,[63]–[68]. As we shall see in Section 5, some of these designs overlap with and improve on designs already available in the literature. Other designs we have found will be published elsewhere (see [5... |

5 | Computer-Generated Minimal (and Larger) Response-Surface Designs
- Hardin, Sloane
- 1992
(Show Context)
Citation Context ...7],[63]–[68]. As we shall see in Section 5, some of these designs overlap with and improve on designs already available in the literature. Other designs we have found will be published elsewhere (see =-=[51]-=-, [52], [53], [54], [92]). We have also used this collection of designs as data for theoretical investigations. Two results are worth mentioning here. (a) There is a simple lower bound on the average ... |

5 |
Collected Papers III: Design of Experiments
- Kiefer
- 1985
(Show Context)
Citation Context ...iscussions of the relative merits of A-, D-, E-, G- and I-optimality and of the dangers of relying on any single numerical criterion have appeared in the literature (see for example [13], [15], [48], =-=[64]-=-, [87]), and our treatment will be brief. Suppose for concreteness that we wish to construct a design for a full quadratic responsesurface model y = β0 + k� βixi + i=1 k� i=1 � βiix 2 i + k−1 k� i=1 j... |

5 |
Constructing Exact DOptimal Experimental Designs by Simulated Annealing
- Meyer, Nachtsheim
- 1988
(Show Context)
Citation Context ...ning some earlier papers that look for I-optimal designs. Studden [95] gives a method for constructing approximations to I-optimal designs for 1dimensional problems. Haines [49], Meyer and Nachtsheim =-=[72]-=-, Crary [24] and Snow and Crary [93] all use simulated annealing to search for I-optimal designs. This approach seems to be restricted to low dimensions and small numbers of design points, and further... |

5 |
D-Optimal Fractions of Three-level Factorial Designs
- Mitchell, Bayne
- 1978
(Show Context)
Citation Context ...mal, we do not believe this optimality has been rigorously established.) For three dimensions there are a number of papers dealing with D-optimal 3-level designs (see Atkinson [1], Mitchell and Bayne =-=[75]-=-, [76]; Galil and Kiefer [41], [44]; Welch [98]; Meyer and Nachtsheim [72]). We have used gosset to search for both I- and D-optimal designs, both with continuous and with 3-level coordinates, and the... |

5 |
Tools for Computer-Aided Design of Experiments
- Nachtsheim
- 1987
(Show Context)
Citation Context ... our algorithm is new. But although computers have been used by many people to construct experimental designs (surveys of such work can be found for example in [13, Chap. 15], [21], [27], [70], [78], =-=[79]-=-, [87], [94], [100]), we believe that no algorithm comparable to ours is presently available. Implementation. We have implemented the algorithm in the C language in a program called gosset. A built-in... |

5 |
Bounds for the number of nodes in Chebyshev type quadrative formulas
- Rabau, Bajnok
- 1991
(Show Context)
Citation Context ...ns if and only if B ≥ 20; in five dimensions if and only if B ≥ 29; and so on. These conjectures are far more precise than our existing knowledge of spherical 4-designs (compare [26], [90], [4], [5], =-=[84]-=-). They are only conjectures, because in general we have only numerical evidence that the designs meet the bound (13). However, in several of these cases we have been able to prove that the points fou... |

5 |
Corrigenda: A computer program for generating variance dispersion graphs
- Vining
- 1993
(Show Context)
Citation Context ...imal without being 100% efficient. To guard against the dangers of using any single number as a design criterion, we have plotted variance dispersion graphs (Giovannitti-Jensen and Myers [48], Vining =-=[96]-=-) for our designs; some examples will be found in [51]. These graphs show how the prediction variance var � f(x) varies over the region of interest, and also give the G-value of the design. We conclud... |

4 |
Multifactor Second Order Designs for Cuboidal Regions
- Atkinson
- 1973
(Show Context)
Citation Context ...spectively I- and D-optimal, we do not believe this optimality has been rigorously established.) For three dimensions there are a number of papers dealing with D-optimal 3-level designs (see Atkinson =-=[1]-=-, Mitchell and Bayne [75], [76]; Galil and Kiefer [41], [44]; Welch [98]; Meyer and Nachtsheim [72]). We have used gosset to search for both I- and D-optimal designs, both with continuous and with 3-l... |

4 |
Bayesian optimal design of experiments for sensor calibration
- Crary, Jeong
- 1995
(Show Context)
Citation Context ...rlier papers that look for I-optimal designs. Studden [95] gives a method for constructing approximations to I-optimal designs for 1dimensional problems. Haines [49], Meyer and Nachtsheim [72], Crary =-=[24]-=- and Snow and Crary [93] all use simulated annealing to search for I-optimal designs. This approach seems to be restricted to low dimensions and small numbers of design points, and furthermore is less... |

4 |
Some New Three-level Response Surface Designs
- Crosier
- 1991
(Show Context)
Citation Context ...own 27-point spherical 4-design associated with the Schläfli polytope (see [26], [20]). It is therefore both I- and D-optimal. This design was also found by gosset. Box and Behnken [9], [10], Crosier =-=[25]-=-, Doehlert [28], Doehlert and Klee [29] and others have given quadratic designs in higher-dimensional spheres, but these all contain many more points than ours. 5.2. Quadratic designs in the cube In t... |

4 |
Optimal Design of Experiments: An Overview
- Dodge, Fedorov, et al.
- 1988
(Show Context)
Citation Context ...dual ingredient of our algorithm is new. But although computers have been used by many people to construct experimental designs (surveys of such work can be found for example in [13, Chap. 15], [21], =-=[27]-=-, [70], [78], [79], [87], [94], [100]), we believe that no algorithm comparable to ours is presently available. Implementation. We have implemented the algorithm in the C language in a program called ... |

4 |
Optimum designs for polynomial extrapolation
- Hoel
- 1965
(Show Context)
Citation Context ... M. Landwehr for telling us about this problem. 5.5. Designs for extrapolation There have been a number of papers on designs for extrapolation, but these deal mostly with the asymptotic theory ([42], =-=[57]-=-, [58], [66], [67], [68]). In this section we give some examples of minimal or close to minimal designs. 27sThese designs have not yet found application, but are included to illustrate the use of goss... |