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Analysis of three-dimensional grids: The cube and the octahedron
Applied Mathematics and Computation 179 (2006) 553–558 www.elsevier.com/locate/amc
Analysis of three-dimensional grids: The cube and the octahedron G.L. Silver Los Alamos National Laboratory, P.O. Box 1663, MS E517, Los Alamos, NM 87544, United States
Abstract The analysis of three-dimensional data is customarily performed by arranging eight measurements in a cubical array and representing them by the trilinear equation. The octahedral design offers the prospect of reduced laboratory costs because it requires only six measurements. Operational interpolating equations for six data in octahedral array can be more accurate than the trilinear equation for eight data in cubical array. The operational equations apply to the six- or seven-point octahedron. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Interpolation; Response surfaces; Cubic design; Octahedral design; Operational equations
1. Introduction The eight-point cube is a design that is used for the analysis of a phenomenon that depends on three independent parameters. Eight measurements are obtained, each one representing a different combination of the high and low levels of the parameters. The data are arranged at the vertices of a cube. They are then represented by the trilinear equation. That equation does not estimate quadratic coefficients but it estimates a coefficient that represents the simultaneous effect of all three parameters. In recent years, new equations for treating eight or nine data in cubical array have appeared. They assume polynomial, exponential, or trigonometric forms [1]. The equations estimate linear, cross-product, and curvature coefficients [2]. Many of the equations can accommodate a measurement at the center point of the cube, something that is not possible when applying the trilinear equation [3–5]. In their eight-point forms, the new equations use the same data that are used by the trilinear equation. Data that are suitable for the trilinear equation can be represented by operational equations at no additional cost. This convenience suggests an economic advantage because more experimental work may not be needed for curvature-term estimations. However, laboratory expenses remain the same as long as the eight-point cube is the standard design for experiments that depend on three independent parameters.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.105
554
G.L. Silver / Applied Mathematics and Computation 179 (2006) 553–558
2. Three equations for the octahedron The purpose of this paper is to suggest the octahedron as a potential alternative to the cube. The cube requires a minimum of eight experiments whereas the octahedron requires only six experiments. The 25% reduction in laboratory costs promised by octahedral designs may be appealing. This reduction is accompanied by a reduction in information about the system but there may be situations in which sacrificing information for reduced expenses can be justified. A skeleton illustration of the octahedral design appears in Fig. 1. It is formed by the intersection of three mutually perpendicular, equal lines. A measurement can be made at the end of each line in Fig. 1. The data are denoted A, B, C, E, F, G as in the figure. In the 1 .. 1 coordinate system, the data have coordinates (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively. These six measurements can be supplemented by an optional datum, D, at the center point of the design. It that case, the design represents a seven-point octahedron. Measurement D is located at (0, 0, 0). The analysis of four- and five-point diamond arrays has been illustrated in recent times [6,7]. A six-point octahedron is formed by three mutually perpendicular, four-point diamond arrays. They occur in the x, y-, x, z-, and y, z-planes. Three mutually perpendicular, five-point diamond arrays are formed if a measurement is obtained at the center point D of the octahedral array. The first equation that interpolates the three, four-point diamonds that generate an octahedron is based on Eq. (5) in Ref. [6]. The cross-product coefficients in the first equation for the octahedron do not utilize an estimated or measured datum D at the center point of the design. These three coefficients terms are listed as Eqs. (1)–(3) below (see Fig. 1). xyc ¼ ðC EÞðB F ÞðF þ B E CÞ=½ðB þ C F EÞðB C F þ EÞ;
ð1Þ
xzc ¼ ðC EÞðA GÞðG þ A E CÞ=½ðA þ C G EÞðA C G þ EÞ; yzc ¼ ðB F ÞðA GÞðG þ A F BÞ=½ðA þ B G F ÞðA B G þ F Þ.
ð2Þ ð3Þ
An expression for the center point of a four-point diamond array appears as Eq. (2) in Ref. [6]. In the case of three mutually perpendicular, four-point diamond arrays, the center point is response estimated by Eq. (4) below: 2
2
D ¼ ½ðF BÞ ðE þ CÞ ðE CÞ ðF þ BÞ=½6ðB C þ E F ÞðB þ C E F Þ 2
2
2
2
þ ½ðG AÞ ðE þ CÞ ðE CÞ ðG þ AÞ=½6ðA C þ E GÞðA þ C E GÞ þ ½ðG AÞ ðF þ BÞ ðF BÞ ðG þ AÞ=½6ðA B þ F GÞðA þ B F GÞ.
ð4Þ
The first interpolating equation for the octahedron illustrated in Fig. 1 is based on Eqs. (1)–(4). It appears as Eq. (5). The letter R represents the interpolated value for any combination of the x-, y-, and z-coordinates in the 1 .. 1 coordinate system. The subscript 1, as in R1, means that Eq. (5) is the first operational equation for the six-point octahedral design. D is estimated by means of Eq. (4).
Fig. 1. Skeleton of the octahedral design. Only the three mutually-perpendicular axes are shown.
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R1 ¼ D þ ðE CÞx=2 þ ðF BÞy=2 þ ðG AÞz=2 þ ðxycÞxy þ ðxzcÞxz þ ðyzcÞyz þ ðC 2D þ EÞx2 =2 þ ðB 2D þ F Þy 2 =2 þ ðA 2D þ GÞz2 =2.
ð5Þ
The cross-product coefficients in Eq. (5) can also be estimated by means of Eq. (4a) in Ref. [7]. When they are estimated by the cited Eq. (4a), they take the forms of Eqs. (6)–(8) below. These coefficients can also be used in Eq. (5) above. xyc ¼ ½ðF BÞðE CÞðF þ B þ C þ E 4DÞ=½ðF BÞ2 þ ðE CÞ2 ;
ð6Þ
2
2
ð7Þ
2
2
ð8Þ
xzc ¼ ½ðG AÞðE CÞðG þ A þ C þ E 4DÞ=½ðG AÞ þ ðE CÞ ; yzc ¼ ½ðG AÞðF BÞðG þ A þ B þ F 4DÞ=½ðG AÞ þ ðF BÞ .
The second equation that interpolates the octahedron in Fig. 1 uses Eqs. (6)–(8). The new equation is denoted R2. Eq. (R2) takes the same form as Eq. (5). The cross-product coefficients can also be based on Eq. (4b) in Ref. [7]. They are listed as Eqs. (9)–(11) below. 2
2
2
2
ð9Þ
2
2
2
2
ð10Þ
2
2
2
2
ð11Þ
xyc ¼ ½ðE þ CÞðF BÞ þ ðF þ BÞðE CÞ 2DððF BÞ þ ðE CÞ Þ=½2ðF BÞðE CÞ; xzc ¼ ½ðE þ CÞðG AÞ þ ðG þ AÞðE CÞ 2DððG AÞ þ ðE CÞ Þ=½2ðG AÞðE CÞ; yzc ¼ ½ðF þ BÞðG AÞ þ ðG þ AÞðF BÞ 2DððG AÞ þ ðF BÞ Þ=½2ðG AÞðF BÞ.
The third equation that interpolates the octahedron in Fig. 1 uses Eqs. (9)–(11). It is denoted R3. Eq. (R3) also takes the same form as Eq. (5). Eqs. (R1), (R2), and (R3) are exact on trilinear data and on the squares of trilinear data. 3. The octahedron and the cube The trilinear equation represents the traditional method for interpolating the prismatic array. It requires eight data to estimate eight coefficients: the constant term, three linear-term coefficients, three two-member cross-product coefficients, and one three-member cross-product coefficient. Each equation for the octahedron estimates ten coefficients using only six data. The three equations do not estimate the coefficient of the threeway interaction but each operational equation estimates three quadratic-term coefficients. Operational equations for the six-point octahedron estimate two more coefficients than the trilinear equation for the eight-point cube. If three curvature coefficients are more significant than one three-way cross-product coefficient, the octahedral design may be a better choice than the trilinear equation. Eight-point equations for the cube use data obtained where the x-, y-, and z-coordinates are combinations of (±1, ±1, ±1), respectively. The six-point equations for the octahedron use data from the same kinds of experiments but they are evaluated at different combinations of the x-, y-, and z-coordinates: (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1), respectively. It is interesting to compare the trilinear equation for the cubic design to the three operational equations for the octahedral design. In the case of the cube, trial data are obtained by applying the functions listed in the first column of Table 1 to (5 + x/2 + y + 5z/2). In the 1 .. 1 coordinate system, this means that the cited functions are applied to the numbers (1, 2, 3, 4, 6, 7, 8, 9) at A .. I in Fig. 2, respectively. For example, the function M2 means that the trial data are derived from (5 + x/2 + y + 5z/2)2 so they are (1, 4, 9, 16, 36, 49, 64, 81) at A .. I, respectively. In this case, the true surface is R = (5 + x/2 + y + 5z/2)2. The first interpolating surface for the cited data is obtained from the trilinear equation. The sum of the squares of the deviations of the interpolating surface from the true surface is found by triple integration. These sums appear as the numerical entries in the second column of Table 1. The third column in Table 1 lists the sums of squares of deviations as obtained by Eq. (10) in Ref. [1]. That equation applies to the eight-point cube but it differs from the trilinear equation in that it also estimates quadratic-term coefficients. The six measurements needed by the three, six-point operational equations for the octahedron are M(5/2), M(4), M(9/2), M(11/2), M(6) and M(15/2) at vertices A, B, C, E, F, and G in Fig. 1, respectively. The sums of the squares of the deviations of the new interpolating surfaces are obtained from the three operational equations for the octahedron, R1, R2, and R3. Those sums appear in the fourth, fifth, and sixth columns of Table 1,
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Table 1 Sums-of-squares-of-deviations between functions representing trial surfaces and five surfaces that model the trial surfaces Functiona
Trilinear equation
Eq. (10) [1]
Eq. (R1)
Eq. (R2)
Eq. (R3)
M4/10 M3 M2 M(3/2) (1000)sin(10M°) (1000)cos(10M°) sinh(M/2) cosh (M/2) (M 1/M)3 ln(M!)
66,421 52,621 229 7.06 29,320 20,907 265 270 52,846 2.32
5092 1201 0 0.0651 408 551 36.7 36.0 1187 0.0696
1739 387 0 0.0130 115 157 9.28 9.15 387 0.0131
1672 374 0 0.0129 114 156 9.09 8.95 373 0.0130
1674 377 0 0.0130 114 156 9.02 8.88 376 0.0131
The equations R1, R2, and R3 are equations that model the six-point octahedron as described in the text. Center point response D is an estimation. a M = (5 + x/2 + y + 5z/2).
Fig. 2. The nine-point cube.
respectively. The revealing property of Table 1 is the observation that the smallest numerical entries appear in the fourth, fifth, and sixth columns of the table. Table 2 is similar to Table 1. This table also lists sums of squares of deviations. It is generated by adding a measurement at the center of the cube to the six previously available data. There is no popular nine-point analog of the trilinear equation, so the second column in the table represents Eqs. (1) and (9)–(13) in Ref. [3]. The cited equation applies to the nine-point cube and it estimates quadratic-term coefficients. The third, fourth, and fifth columns of Table 2 list the sums of squares of deviations of the interpolating surfaces as obtained from the three operational equations R1, R2, and R3 in which D represents a measurement instead of an esti-
Table 2 Sums-of-squares-of-deviations between functions representing trial surfaces and five surfaces that model the trial surfaces Functiona 4
M /10 M3 M2 M(3/2) (1000)sin(10M°) (1000)cos(10M°) sinh(M/2) cosh(M/2) (M 1/M)3 ln(M!)
Eqs. (1), (9)–(13) [3]
Eq. (R1)
Eq. (R2)
Eq. (R3)
4957 1221 0 0.0553 373 527 33.0 32.2 1208 0.0586
1721 385 0 0.0130 114 157 9.20 9.06 385 0.0132
1555 357 0 0.0130 111 155 8.63 8.48 355 0.0132
1372 340 0 0.0132 108 153 7.75 7.57 336 0.0135
The equations R1, R2, and R3 are equations that model the seven-point octahedron as described in the text. Center point response D is a measurement. a M = (5 + x/2 + y + 5z/2).
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mate. The seven measurements used by the three, seven-point operational equations for the octahedron are M(5/2), M(4), M(9/2), M(5), M(11/2), M(6) and M(15/2) at vertices A, B, C, D, E, F, and G in Fig. 1, respectively. 4. Discussion The trilinear equation interpolates the cubical space bounded by eight data collected at the vertices of the cube. The octahedron uses six data to interpolate the space bounded by the centers of the faces of the cube. The numerical entries in Tables 1 and 2 are obtained by triple integration over the cubical space spanned by the eight-point equations. For the eight-point equations, the experimental space represents measured and interpolated responses. When Eqs. (R1), (R2), and (R3) are applied to the same functions, the experimental space represents measured, interpolated, and extrapolated responses. If the region near the center point of the interpolated space is the primary interest, the octahedral design may be preferred. This remark is based on the observation that the center point datum can be more accurately estimated by the octahedral design than by the cubic design. For example, let the trial data be generated by the function R = (5 + x/2 + y + 5x/2)4. The trilinear equation estimates the center point response to be 1839, the eight-point operational equation estimates it to be 681, while Eqs. (R1), (R2), and (R3) all estimate it to be 646. The true response at the center point of both designs is 625. Table 1 is constructed using commonplace, monotonic-increasing expressions to generate trial data. Laboratory data are not always of this kind but the table illustrates that the six-point octahedron is capable of rendering interpolating surfaces that are more accurate than the like surfaces generated by the trilinear equation. A lower sum-of-squares-of-deviations within the cubical space is the criterion by which superiority is judged. The numerical entries in columns 4–6 are smaller than the entries in column 2 of Table 1. Table 2 is similar to Table 1 but it applies to seven data in octahedral array and nine data in cubical array. In each case, the extra datum is a measurement at the center point of both arrays. The entries in Table 2 also suggest that the octahedral array can be a better choice than the cubic array even when the experimental spaces are the same. The numerical entries in Table 2 are commonly smaller than the like entries in Table 1. In other words, a measurement at the center point of the octahedral array is preferable to an estimate of what that measurement might be. The sums-of-squares-of-deviations in Tables 1 and 2 can be compared to the like entries in Table 4 of Ref. [2] and Table 1 of Ref. [4]. The comparison suggests that eight- and nine-point equations containing thirdorder coefficients are likely to be superior to Eqs. (R1), (R2), and (R3). Seen in this light, the principal advantage of the octahedral design is lower costs because of the need for only six measurements. The expenses associated with the eight-point cubical design can be reduced 25% by choosing the six-point octahedral design. The later design implies potential loss of accuracy if all points within the eight-point space are equally important or if regions near the vertices of the cube are the primary interest. The loss of accuracy may be an acceptable exchange if laboratory expenses are high and if the region near the center point of the design is the principal interest. The three operational equations for the octahedron do not permit estimations of the three-way interaction coefficient. This loss is compensated by the estimation of three quadratic-term coefficients. One interpretation of the entries in Tables 1 and 2 is that the three-for-one exchange is potentially beneficial. Eqs. (R1), (R2), and (R3) are based on polynomial representations of four- and five-point diamonds. Diamond arrays can be represented by equations based on exponential and trigonometric forms [6,7]. Those representations are potentially useful alternatives to the methods illustrated herein. Linear-term coefficients for a polynomial-type interpolating equation for the eight-point design are listed as Eqs. (6)–(8) in Ref. [2]. Their forms are complicated but they can be simplified by rewriting them as Eqs. (12)– (14) below. See also Eqs. (2)–(4) in Ref. [4]. nxc ¼ xc y2x z2x x3;
ð12Þ
nyc ¼ yc x2y z2y y3;
ð13Þ
nzc ¼ zc x2z y2z z3.
ð14Þ
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References [1] [2] [3] [4]
G.L. Silver, Analysis of three-dimensional grids: the eight-point cube, Appl. Math. Comput. 153 (2004) 467–473. G.L. Silver, Analysis of three-dimensional grids: cubes and cubic coefficients, Appl. Math. Comput. 166 (2005) 196–203. G.L. Silver, Analysis of three-dimensional grids: five- and nine-point cubes, Appl. Math. Comput. 160 (2005) 133–140. G.L. Silver, Analysis of three-dimensional grids: cubic equations for nine-point prismatic arrays, Appl. Math. Comput. 170 (2005) 752– 760. [5] G.L. Silver, Analysis of three-dimensional grids: interpolation of data in prismatic array and the estimation of a missing datum, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.05.047. [6] G.L. Silver, Analysis of four-point grids: the diamond configuration, Appl. Math. Comput. 131 (2002) 215–221. [7] G.L. Silver, Analysis of five-point grids: the diamond configuration, Appl. Math. Comput. 144 (2003) 389–395.
Analysis of three-dimensional grids: The cube and the octahedron G.L. Silver Los Alamos National Laboratory, P.O. Box 1663, MS E517, Los Alamos, NM 87544, United States
Abstract The analysis of three-dimensional data is customarily performed by arranging eight measurements in a cubical array and representing them by the trilinear equation. The octahedral design offers the prospect of reduced laboratory costs because it requires only six measurements. Operational interpolating equations for six data in octahedral array can be more accurate than the trilinear equation for eight data in cubical array. The operational equations apply to the six- or seven-point octahedron. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Interpolation; Response surfaces; Cubic design; Octahedral design; Operational equations
1. Introduction The eight-point cube is a design that is used for the analysis of a phenomenon that depends on three independent parameters. Eight measurements are obtained, each one representing a different combination of the high and low levels of the parameters. The data are arranged at the vertices of a cube. They are then represented by the trilinear equation. That equation does not estimate quadratic coefficients but it estimates a coefficient that represents the simultaneous effect of all three parameters. In recent years, new equations for treating eight or nine data in cubical array have appeared. They assume polynomial, exponential, or trigonometric forms [1]. The equations estimate linear, cross-product, and curvature coefficients [2]. Many of the equations can accommodate a measurement at the center point of the cube, something that is not possible when applying the trilinear equation [3–5]. In their eight-point forms, the new equations use the same data that are used by the trilinear equation. Data that are suitable for the trilinear equation can be represented by operational equations at no additional cost. This convenience suggests an economic advantage because more experimental work may not be needed for curvature-term estimations. However, laboratory expenses remain the same as long as the eight-point cube is the standard design for experiments that depend on three independent parameters.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.105
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G.L. Silver / Applied Mathematics and Computation 179 (2006) 553–558
2. Three equations for the octahedron The purpose of this paper is to suggest the octahedron as a potential alternative to the cube. The cube requires a minimum of eight experiments whereas the octahedron requires only six experiments. The 25% reduction in laboratory costs promised by octahedral designs may be appealing. This reduction is accompanied by a reduction in information about the system but there may be situations in which sacrificing information for reduced expenses can be justified. A skeleton illustration of the octahedral design appears in Fig. 1. It is formed by the intersection of three mutually perpendicular, equal lines. A measurement can be made at the end of each line in Fig. 1. The data are denoted A, B, C, E, F, G as in the figure. In the 1 .. 1 coordinate system, the data have coordinates (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively. These six measurements can be supplemented by an optional datum, D, at the center point of the design. It that case, the design represents a seven-point octahedron. Measurement D is located at (0, 0, 0). The analysis of four- and five-point diamond arrays has been illustrated in recent times [6,7]. A six-point octahedron is formed by three mutually perpendicular, four-point diamond arrays. They occur in the x, y-, x, z-, and y, z-planes. Three mutually perpendicular, five-point diamond arrays are formed if a measurement is obtained at the center point D of the octahedral array. The first equation that interpolates the three, four-point diamonds that generate an octahedron is based on Eq. (5) in Ref. [6]. The cross-product coefficients in the first equation for the octahedron do not utilize an estimated or measured datum D at the center point of the design. These three coefficients terms are listed as Eqs. (1)–(3) below (see Fig. 1). xyc ¼ ðC EÞðB F ÞðF þ B E CÞ=½ðB þ C F EÞðB C F þ EÞ;
ð1Þ
xzc ¼ ðC EÞðA GÞðG þ A E CÞ=½ðA þ C G EÞðA C G þ EÞ; yzc ¼ ðB F ÞðA GÞðG þ A F BÞ=½ðA þ B G F ÞðA B G þ F Þ.
ð2Þ ð3Þ
An expression for the center point of a four-point diamond array appears as Eq. (2) in Ref. [6]. In the case of three mutually perpendicular, four-point diamond arrays, the center point is response estimated by Eq. (4) below: 2
2
D ¼ ½ðF BÞ ðE þ CÞ ðE CÞ ðF þ BÞ=½6ðB C þ E F ÞðB þ C E F Þ 2
2
2
2
þ ½ðG AÞ ðE þ CÞ ðE CÞ ðG þ AÞ=½6ðA C þ E GÞðA þ C E GÞ þ ½ðG AÞ ðF þ BÞ ðF BÞ ðG þ AÞ=½6ðA B þ F GÞðA þ B F GÞ.
ð4Þ
The first interpolating equation for the octahedron illustrated in Fig. 1 is based on Eqs. (1)–(4). It appears as Eq. (5). The letter R represents the interpolated value for any combination of the x-, y-, and z-coordinates in the 1 .. 1 coordinate system. The subscript 1, as in R1, means that Eq. (5) is the first operational equation for the six-point octahedral design. D is estimated by means of Eq. (4).
Fig. 1. Skeleton of the octahedral design. Only the three mutually-perpendicular axes are shown.
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R1 ¼ D þ ðE CÞx=2 þ ðF BÞy=2 þ ðG AÞz=2 þ ðxycÞxy þ ðxzcÞxz þ ðyzcÞyz þ ðC 2D þ EÞx2 =2 þ ðB 2D þ F Þy 2 =2 þ ðA 2D þ GÞz2 =2.
ð5Þ
The cross-product coefficients in Eq. (5) can also be estimated by means of Eq. (4a) in Ref. [7]. When they are estimated by the cited Eq. (4a), they take the forms of Eqs. (6)–(8) below. These coefficients can also be used in Eq. (5) above. xyc ¼ ½ðF BÞðE CÞðF þ B þ C þ E 4DÞ=½ðF BÞ2 þ ðE CÞ2 ;
ð6Þ
2
2
ð7Þ
2
2
ð8Þ
xzc ¼ ½ðG AÞðE CÞðG þ A þ C þ E 4DÞ=½ðG AÞ þ ðE CÞ ; yzc ¼ ½ðG AÞðF BÞðG þ A þ B þ F 4DÞ=½ðG AÞ þ ðF BÞ .
The second equation that interpolates the octahedron in Fig. 1 uses Eqs. (6)–(8). The new equation is denoted R2. Eq. (R2) takes the same form as Eq. (5). The cross-product coefficients can also be based on Eq. (4b) in Ref. [7]. They are listed as Eqs. (9)–(11) below. 2
2
2
2
ð9Þ
2
2
2
2
ð10Þ
2
2
2
2
ð11Þ
xyc ¼ ½ðE þ CÞðF BÞ þ ðF þ BÞðE CÞ 2DððF BÞ þ ðE CÞ Þ=½2ðF BÞðE CÞ; xzc ¼ ½ðE þ CÞðG AÞ þ ðG þ AÞðE CÞ 2DððG AÞ þ ðE CÞ Þ=½2ðG AÞðE CÞ; yzc ¼ ½ðF þ BÞðG AÞ þ ðG þ AÞðF BÞ 2DððG AÞ þ ðF BÞ Þ=½2ðG AÞðF BÞ.
The third equation that interpolates the octahedron in Fig. 1 uses Eqs. (9)–(11). It is denoted R3. Eq. (R3) also takes the same form as Eq. (5). Eqs. (R1), (R2), and (R3) are exact on trilinear data and on the squares of trilinear data. 3. The octahedron and the cube The trilinear equation represents the traditional method for interpolating the prismatic array. It requires eight data to estimate eight coefficients: the constant term, three linear-term coefficients, three two-member cross-product coefficients, and one three-member cross-product coefficient. Each equation for the octahedron estimates ten coefficients using only six data. The three equations do not estimate the coefficient of the threeway interaction but each operational equation estimates three quadratic-term coefficients. Operational equations for the six-point octahedron estimate two more coefficients than the trilinear equation for the eight-point cube. If three curvature coefficients are more significant than one three-way cross-product coefficient, the octahedral design may be a better choice than the trilinear equation. Eight-point equations for the cube use data obtained where the x-, y-, and z-coordinates are combinations of (±1, ±1, ±1), respectively. The six-point equations for the octahedron use data from the same kinds of experiments but they are evaluated at different combinations of the x-, y-, and z-coordinates: (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1), respectively. It is interesting to compare the trilinear equation for the cubic design to the three operational equations for the octahedral design. In the case of the cube, trial data are obtained by applying the functions listed in the first column of Table 1 to (5 + x/2 + y + 5z/2). In the 1 .. 1 coordinate system, this means that the cited functions are applied to the numbers (1, 2, 3, 4, 6, 7, 8, 9) at A .. I in Fig. 2, respectively. For example, the function M2 means that the trial data are derived from (5 + x/2 + y + 5z/2)2 so they are (1, 4, 9, 16, 36, 49, 64, 81) at A .. I, respectively. In this case, the true surface is R = (5 + x/2 + y + 5z/2)2. The first interpolating surface for the cited data is obtained from the trilinear equation. The sum of the squares of the deviations of the interpolating surface from the true surface is found by triple integration. These sums appear as the numerical entries in the second column of Table 1. The third column in Table 1 lists the sums of squares of deviations as obtained by Eq. (10) in Ref. [1]. That equation applies to the eight-point cube but it differs from the trilinear equation in that it also estimates quadratic-term coefficients. The six measurements needed by the three, six-point operational equations for the octahedron are M(5/2), M(4), M(9/2), M(11/2), M(6) and M(15/2) at vertices A, B, C, E, F, and G in Fig. 1, respectively. The sums of the squares of the deviations of the new interpolating surfaces are obtained from the three operational equations for the octahedron, R1, R2, and R3. Those sums appear in the fourth, fifth, and sixth columns of Table 1,
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Table 1 Sums-of-squares-of-deviations between functions representing trial surfaces and five surfaces that model the trial surfaces Functiona
Trilinear equation
Eq. (10) [1]
Eq. (R1)
Eq. (R2)
Eq. (R3)
M4/10 M3 M2 M(3/2) (1000)sin(10M°) (1000)cos(10M°) sinh(M/2) cosh (M/2) (M 1/M)3 ln(M!)
66,421 52,621 229 7.06 29,320 20,907 265 270 52,846 2.32
5092 1201 0 0.0651 408 551 36.7 36.0 1187 0.0696
1739 387 0 0.0130 115 157 9.28 9.15 387 0.0131
1672 374 0 0.0129 114 156 9.09 8.95 373 0.0130
1674 377 0 0.0130 114 156 9.02 8.88 376 0.0131
The equations R1, R2, and R3 are equations that model the six-point octahedron as described in the text. Center point response D is an estimation. a M = (5 + x/2 + y + 5z/2).
Fig. 2. The nine-point cube.
respectively. The revealing property of Table 1 is the observation that the smallest numerical entries appear in the fourth, fifth, and sixth columns of the table. Table 2 is similar to Table 1. This table also lists sums of squares of deviations. It is generated by adding a measurement at the center of the cube to the six previously available data. There is no popular nine-point analog of the trilinear equation, so the second column in the table represents Eqs. (1) and (9)–(13) in Ref. [3]. The cited equation applies to the nine-point cube and it estimates quadratic-term coefficients. The third, fourth, and fifth columns of Table 2 list the sums of squares of deviations of the interpolating surfaces as obtained from the three operational equations R1, R2, and R3 in which D represents a measurement instead of an esti-
Table 2 Sums-of-squares-of-deviations between functions representing trial surfaces and five surfaces that model the trial surfaces Functiona 4
M /10 M3 M2 M(3/2) (1000)sin(10M°) (1000)cos(10M°) sinh(M/2) cosh(M/2) (M 1/M)3 ln(M!)
Eqs. (1), (9)–(13) [3]
Eq. (R1)
Eq. (R2)
Eq. (R3)
4957 1221 0 0.0553 373 527 33.0 32.2 1208 0.0586
1721 385 0 0.0130 114 157 9.20 9.06 385 0.0132
1555 357 0 0.0130 111 155 8.63 8.48 355 0.0132
1372 340 0 0.0132 108 153 7.75 7.57 336 0.0135
The equations R1, R2, and R3 are equations that model the seven-point octahedron as described in the text. Center point response D is a measurement. a M = (5 + x/2 + y + 5z/2).
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mate. The seven measurements used by the three, seven-point operational equations for the octahedron are M(5/2), M(4), M(9/2), M(5), M(11/2), M(6) and M(15/2) at vertices A, B, C, D, E, F, and G in Fig. 1, respectively. 4. Discussion The trilinear equation interpolates the cubical space bounded by eight data collected at the vertices of the cube. The octahedron uses six data to interpolate the space bounded by the centers of the faces of the cube. The numerical entries in Tables 1 and 2 are obtained by triple integration over the cubical space spanned by the eight-point equations. For the eight-point equations, the experimental space represents measured and interpolated responses. When Eqs. (R1), (R2), and (R3) are applied to the same functions, the experimental space represents measured, interpolated, and extrapolated responses. If the region near the center point of the interpolated space is the primary interest, the octahedral design may be preferred. This remark is based on the observation that the center point datum can be more accurately estimated by the octahedral design than by the cubic design. For example, let the trial data be generated by the function R = (5 + x/2 + y + 5x/2)4. The trilinear equation estimates the center point response to be 1839, the eight-point operational equation estimates it to be 681, while Eqs. (R1), (R2), and (R3) all estimate it to be 646. The true response at the center point of both designs is 625. Table 1 is constructed using commonplace, monotonic-increasing expressions to generate trial data. Laboratory data are not always of this kind but the table illustrates that the six-point octahedron is capable of rendering interpolating surfaces that are more accurate than the like surfaces generated by the trilinear equation. A lower sum-of-squares-of-deviations within the cubical space is the criterion by which superiority is judged. The numerical entries in columns 4–6 are smaller than the entries in column 2 of Table 1. Table 2 is similar to Table 1 but it applies to seven data in octahedral array and nine data in cubical array. In each case, the extra datum is a measurement at the center point of both arrays. The entries in Table 2 also suggest that the octahedral array can be a better choice than the cubic array even when the experimental spaces are the same. The numerical entries in Table 2 are commonly smaller than the like entries in Table 1. In other words, a measurement at the center point of the octahedral array is preferable to an estimate of what that measurement might be. The sums-of-squares-of-deviations in Tables 1 and 2 can be compared to the like entries in Table 4 of Ref. [2] and Table 1 of Ref. [4]. The comparison suggests that eight- and nine-point equations containing thirdorder coefficients are likely to be superior to Eqs. (R1), (R2), and (R3). Seen in this light, the principal advantage of the octahedral design is lower costs because of the need for only six measurements. The expenses associated with the eight-point cubical design can be reduced 25% by choosing the six-point octahedral design. The later design implies potential loss of accuracy if all points within the eight-point space are equally important or if regions near the vertices of the cube are the primary interest. The loss of accuracy may be an acceptable exchange if laboratory expenses are high and if the region near the center point of the design is the principal interest. The three operational equations for the octahedron do not permit estimations of the three-way interaction coefficient. This loss is compensated by the estimation of three quadratic-term coefficients. One interpretation of the entries in Tables 1 and 2 is that the three-for-one exchange is potentially beneficial. Eqs. (R1), (R2), and (R3) are based on polynomial representations of four- and five-point diamonds. Diamond arrays can be represented by equations based on exponential and trigonometric forms [6,7]. Those representations are potentially useful alternatives to the methods illustrated herein. Linear-term coefficients for a polynomial-type interpolating equation for the eight-point design are listed as Eqs. (6)–(8) in Ref. [2]. Their forms are complicated but they can be simplified by rewriting them as Eqs. (12)– (14) below. See also Eqs. (2)–(4) in Ref. [4]. nxc ¼ xc y2x z2x x3;
ð12Þ
nyc ¼ yc x2y z2y y3;
ð13Þ
nzc ¼ zc x2z y2z z3.
ð14Þ
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G.L. Silver / Applied Mathematics and Computation 179 (2006) 553–558
References [1] [2] [3] [4]
G.L. Silver, Analysis of three-dimensional grids: the eight-point cube, Appl. Math. Comput. 153 (2004) 467–473. G.L. Silver, Analysis of three-dimensional grids: cubes and cubic coefficients, Appl. Math. Comput. 166 (2005) 196–203. G.L. Silver, Analysis of three-dimensional grids: five- and nine-point cubes, Appl. Math. Comput. 160 (2005) 133–140. G.L. Silver, Analysis of three-dimensional grids: cubic equations for nine-point prismatic arrays, Appl. Math. Comput. 170 (2005) 752– 760. [5] G.L. Silver, Analysis of three-dimensional grids: interpolation of data in prismatic array and the estimation of a missing datum, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.05.047. [6] G.L. Silver, Analysis of four-point grids: the diamond configuration, Appl. Math. Comput. 131 (2002) 215–221. [7] G.L. Silver, Analysis of five-point grids: the diamond configuration, Appl. Math. Comput. 144 (2003) 389–395.