The four-point diamond configuration

Applied Mathematical Modelling 25 (2001) 629±634

www.elsevier.nl/locate/apm

The four-point diamond con®guration G.L. Silver

*

Los Alamos National Laboratory 1, P.O. Box 1663, MS E502, Los Alamos, NM 87545, USA Received 3 November 1999; received in revised form 17 November 2000; accepted 4 December 2000

Abstract It is widely regarded as impossible to estimate interaction and quadratic coecients from four data taken at the end points of a cross. This manuscript illustrates four methods for approximating the desired coecients, and compares them to the coecients obtained from Taylor expansions of the generating functions. Ó 2001 Elsevier Science Inc. All rights reserved.

1. Introduction It is a common misconception that interaction and quadratic terms cannot be estimated on four-points taken at the end points of a cross. Let the equally spaced data be denoted by the letters B; D; F ; H , as indicated below. If straight lines bound the ®gure, a square standing on point B is formed. D

H B

F

This article illustrates how the xy-, x2 -, and y 2 -coecient terms can be estimated by operational equations [1]. It is convenient to let the data form a bilinear set: B ˆ 2; D ˆ 4; F ˆ 6; H ˆ 8, so that predicted interaction and curvature coecients can be compared to the like coecients obtained from Taylor expansions of the generating functions. The bilinear set is described by the function z ˆ 5 ‡ x ‡ 3y. 2. Operational equations An operational, polynomial equation for the four data is Eq. (1). The coecients are de®ned by Eqs. (2)±(7). z ˆ CN ‡ …XC†x ‡ …YC†y ‡ …XYC†xy ‡ …X 2†x2 ‡ …Y 2†y 2 ;

*

…1†

Tel.: +1-505-667-5656; fax: +1-505-665-4775. E-mail address: [email protected] (G.L. Silver). 1 Los Alamos National Laboratory is operated by the University of California for the U.S. Department of Energy under contract No. W-7405-ENG-36. 0307-904X/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 1 ) 0 0 0 1 1 - 7

630

G.L. Silver / Appl. Math. Modelling 25 (2001) 629±634

CN ˆ ‰…H

B†2 …F ‡ D†

D†2 …H ‡ B†Š=‰2…B

…F

D‡F

H†…B ‡ D

F

H †Š;

…2†

XC ˆ …F

D†=2;

…3†

YC ˆ …H

B†=2;

…4†

XYC ˆ …D

F †…B

H†…H ‡ B

2

F

D†=‰…B ‡ D

H

F †…B

D

H ‡ F †Š;

…5†

X 2 ˆ …D

F † …H ‡ B

F

D†=‰2…B ‡ D

H

F †…B

D

H ‡ F †Š;

…6†

Y 2 ˆ …B

H †2 …H ‡ B

F

D†=‰2…B ‡ D

H

F †…B

D

H ‡ F †Š:

…7†

Eq. (1) can be obtained from [1, Eq. (7)]. This equation applies to the four-point rectangle with vertices denoted A; C; G; I as shown in the diagram directly above Eq. (6) therein, or in the diagram shown [5, Fig. 1]. To obtain Eq. (1) above, change A to B, C to F , I to H , G to D, x to …y ‡ x†, and y to …y x† in the cited Eq. (7). After the substitutions have been made, expand the substituted equation, and then collect the linear, cross-product, and quadratic terms to obtain Eq. (1) above. The cited data, substituted into Eq. (1), yield z ˆ …5 ‡ x ‡ 3y†. If the data are squared …B ˆ 4; D ˆ 16; F ˆ 36; H ˆ 64† Eq. (1) yields z ˆ 25 ‡ 10x ‡ 30y ‡ 6xy ‡ x2 ‡ 9y 2 . This ex2 pression can be rewritten as z ˆ …5 ‡ x ‡ 3y† . Eq. (1) is exact on all bilinear data and on the squares of all bilinear data. The interaction and quadratic coecients are accurately predicted for these special cases. Eq. (1) can be used to approximate the xy-, x2 -, and y 2 -coecients for other functions operating on the original data, including unknown functions that arise in the collection of experimental measurements. Four typical test functions are …5 ‡ x ‡ 3y†2 , …5 ‡ x ‡ 3y†3 , ln…6 ‡ x ‡ 3y†, and 1=…5 ‡ x ‡ 3y†. Assuming these functions, the data become M 2 , M 3 , ln…1 ‡ M†, and 1=M, respectively, where M represents any of the original bilinear numbers. Taylor expansion of each test function yields the reference values of the interaction and quadratic coecients. Operational equations yield estimates of the same coecients. Comparing them to the reference coecients illustrates the advantage of operational equations. Table 1 illustrates that the interaction and curvature terms obtained from Eq. (1) are approximations to the similar terms obtained by Taylor expansions of the generating functions. Eq. (1) needs only four measurements, yet it approximates the three higher terms on monotonic data described by polynomials of modest order. Sometimes measurements follow exponential laws. The second operational approach is to apply a four-point, exponential equation to the data, Eq. (8). z ˆ J x K y …E ‡ T † E ˆ ‰JK…J ‡ K

T;

…8†

2†…B ‡ D† ‡ …J …2K

1†

K†…F ‡ H†Š=‰2…J 2 K ‡ J …K 2

1†

K†Š;

…9†

Table 1

Interaction plus quadratic terms of test functions and Eq. (1) Function 2

…5 ‡ x ‡ 3y† …5 ‡ x ‡ 3y†3 ln…6 ‡ x ‡ 3y† 1=…5 ‡ x ‡ 3y†

xy ‡ x2 ‡ y 2 (Taylor-series)

xy ‡ x2 ‡ y 2 (Eq. (1))

6xy ‡ x2 ‡ 9y 2 90xy ‡ 15x2 ‡ 135y 2 0:083xy 0:014x2 0:13y 2 0:048xy ‡ 0:008x2 ‡ 0:072y 2

6xy ‡ x2 ‡ 9y 2 80xy ‡ 12x2 ‡ 132y 2 0:088xy 0:013x2 0:14y 2 0:049xy ‡ 0:005x2 ‡ 0:11y 2

G.L. Silver / Appl. Math. Modelling 25 (2001) 629±634

T ˆ ‰JK…B ‡ D ‡ F ‡ H †

E…J 2 K ‡ J …K 2 ‡ 1† ‡ K†Š=‰J 2 K ‡ J …K 2

H †Š…1=2† =‰B…D ‡ F †

K ˆ ‰…H

F †…D

J ˆ …B

F †K=…D

DF

B2 Š…1=2† ;

H†:

631

4K ‡ 1† ‡ KŠ;

…10† …11† …12†

Table 2 lists the same test functions in the left-hand column. In the middle column of the table are the equations that result from substituting the trial data into Eq. (8). The right-hand column lists the coecients of the interaction and curvature terms as obtained from a Taylor expansion of the equations that appear in the middle of the table. When the entries in the third column of Table 2 are compared to the Taylor-series values obtained from the generating functions (the middle column of Table 1), we ®nd that Eq. (8) also provides estimates of the interaction and quadratic coecients. Care should be exercised when using operational equations. It should be veri®ed that the equations reproduce the original data. The four rotations of the data around their center point should always yield the same response at the center point. If the equations generate complex numbers, interpolate with their real parts (see Appendix A and [7]). It is not customary to represent the data as powers of linear expressions, but there seems to be no law of nature denying the potential advantage of this representation. The third operational method is to solve Eq. (13) for the exponent n. If Eq. (13) has a solution, take the nth roots of the original data. The data so reduced form a bilinear set
n F: …13† B…1=n† ‡ H …1=n† D…1=n† Substitute the reduced data into Eq. (1). The interaction and quadratic terms disappear, or represent round-o€ errors that can be discarded. The interpolating equation is the remaining bilinear expression raised to the nth power, where a; b; k are constants, and the interpolating equation takes the form of Eq. (14) n

z ˆ …k ‡ ax ‡ by† :

…14†

The equations that result from the trial data and Eq. (14) are listed in the middle column of Table 3. The xy-, x2 -, and y 2 -terms that result from the Taylor expansion of these equations are listed in the third column of the table. Again, it is seen that the coecients in the third column of Table 3 compare favorably to the like coecients of the generating functions in the middle column of Table 1. If Eq. (14) is rewritten with a translating term, it usually permits a selection of exponents [2]. Experimental data can be represented by functions that depend on the circular or hyperbolic sine and cosine as shown in Eq. (15) z ˆ M sinh…Px ‡ Qy† ‡ N cosh…Px ‡ Qy†:

…15†

Table 2

Substituted Eq. (8) and interaction and quadratic terms from its Taylor-series Function

xy ‡ x2 ‡ y 2 (Taylor-series)

Eq. (8) 2

…5 ‡ x ‡ 3y† …5 ‡ x ‡ 3y†3 ln…6 ‡ x ‡ 3y† 1=…5 ‡ x ‡ 3y†

x

y

44:9…1:247† …1:871† 20 163:8…1:567†x …3:374†y 40:5 0:97…0:842†x …0:584†y ‡ 2:76 0:1…0:667†x …0:25†y ‡ 0:1

6:2xy ‡ 1:1x2 ‡ 8:8y 2 89xy ‡ 16x2 ‡ 121y 2 0:090xy 0:014x2 0:14y 2 0:056xy ‡ 0:0082x2 ‡ 0:096y 2

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G.L. Silver / Appl. Math. Modelling 25 (2001) 629±634

Table 3

Substituted Eq. (14) and interaction and quadratic terms from its Taylor-series Function

xy ‡ x2 ‡ y 2 (Taylor-series)

Eq. (14) 2

2

…5 ‡ x ‡ 3y† …5 ‡ x ‡ 3y†3 ln…6 ‡ x ‡ 3y† 1=…5 ‡ x ‡ 3y†

6xy ‡ x2 ‡ 9y 2 90xy ‡ 15x2 ‡ 135y 2 0:076xy 0:013x2 0:11y 2 0:048xy ‡ 0:008x2 ‡ 0:072y 2

…5 ‡ x ‡ 3y† …5 ‡ x ‡ 3y†3 …4:77 ‡ 1:19x ‡ 3:49y†0:373 1=…5 ‡ x ‡ 3y†

Table 4

Substituted Eq. (15) and interaction and quadratic terms from its Taylor-series Function 2

…5 ‡ x ‡ 3y† …5 ‡ x ‡ 3y†3 ln…6 ‡ x ‡ 3y† 1=…5 ‡ x ‡ 3y†

Eq. (15)

xy ‡ x2 ‡ y 2 (Taylor-series)

24:8 cosh…0:307x ‡ 0:836y† ‡ 32:0 sinh…0:307x ‡ 0:836y† 122 cosh…0:543x ‡ 1:39y† ‡ 133 sinh…0:543x ‡ 1:39y† 1:79 cos…0:120x ‡ 0:401y† ‡ 1:40 sin…0:120x ‡ 0:401y† 0:201 cosh…0:261x ‡ 1:01y† 0:158 sinh…0:261x ‡ 1:01y†

6:4xy ‡ 1:2x2 ‡ 8:7y 2 92xy ‡ 18x2 ‡ 118y 2 0:086xy 0:013x2 0:14y 2 0:053xy ‡ 0:007x2 ‡ 0:10y 2

This method uses Eq. (15) as applied to the rectangle with corner points A; C; G; I. Point A is at the lower left-hand corner of the ®gure, equidistant from D and B, whereas I is at the diagonally opposite corner [2]. From the data at B; D; F ; H , we ®nd the squares of the responses at A; C; G; I. The equation for the square of the response at corner point A is A2 ˆ …D2

FD

B2 ‡ HB†2 …BD

FH †=‰…B

H ‡D

F †…D

F ‡H

B†…DH

BF †Š: …16†

Rotation of the data about the center point of the diamond design yields analogous equations for C 2 , G2 , and I 2 . Each equation produces a positive and a negative number for the response at a vertex. From these 16 values, the combination that reproduces the original data is selected [3]. For most positive data, all corner point responses are also positive, or three are positive and the numerically smallest is negative. Table 4 reiterates the test functions, the equations obtained by substituting Eq. (15) with the test data, and the interaction and curvature coecients obtained from expanding these equations into a Taylor-series. It is observed that the entries in the third column of the table are approximations to the entries in the middle column of Table 1. 3. Discussion In the 16th century, Viete developed many of the identities found in textbooks of trigonometry [4]. Application of the shifting operator ‰exp…x†F …x† ˆ F …x ‡ h†Š to the Euler forms of trigonometric identities generates ®nite-di€erence equations [5]. These equations underlie the four classes of operational equations whose properties are summarized in Tables 1±4. As a method for developing interpolating equations for geometric ®gures, this approach apparently escaped notice for about 400 years. Four-point operational equations are useful for many data sets that derive from monotonic, di€erentiable surfaces. Such surfaces are commonly found in tabular data, but they also arise in experimental design methodology. When experiments are costly, it is advantageous to be able to model response surfaces at minimum cost. This can be done with four-points in rectangular array, in the diamond array, and in other arrays, by means of four-point operational equations. The operational equations usually give better approximations to known surfaces than the

G.L. Silver / Appl. Math. Modelling 25 (2001) 629±634

633

standard bilinear equation that also requires four data. The judgment ``better approximations'' can be assessed by comparing the sums of the squares of the deviations of the bilinear and operational surfaces from known surfaces [6]. The operational equations usually have lower sum of squares. If ridges and troughs are absent from the true surface, and if they are also absent from an operational approximation to the true surface, then the operational surface is usually a better choice than the bilinear surface. The bilinear model is the commonly encountered model for surfaces sampled at only four-points, but operational equations provide four selections: a polynomial model including quadratic terms, an exponential model, a model that uses circular or hyperbolic sines and cosines, and a model that uses a power of the bilinear expression. In the last option, the exponent in the modeling equation can usually be adjusted within modest limits, as has been illustrated by an example in [2]. The operational equations have the advantage that they provide curvature estimates, something that is not available with the bilinear equation. A nine-point Latin square is suggested if an extremum, a ridge, or a trough is suspected to be present in the true surface or if the opposite sides of the four-point design have slopes of contrary sign [2]. Operational predictions for the response at the center of the data are usually closer to the true value than the arithmetic average, the common measure of central tendency for the four data. Appendix A There is another method for using Eq. (8). If one or both of J and K are imaginary, change them to real numbers by removing the operator I, where I ˆ sqrt… 1†. If one or both of J or K are negative, change their signs so that after the changes only real, positive numbers are found in the ®rst three terms of Eq. (8). These terms are multiplied by a factor that is chosen from Table 5. For example, if …B; D; F ; H † are …4; 16; 256; 64†, then Eq. (8) changes from Eq. (A.1) to Eq. (A.2). z ˆ …25:88I†… 9:17I†x … 1:75I†y ‡ 18:82;

…A:1†

z ˆ …x ‡ y†…25:88†…9:17†x …1:75†y ‡ 18:82:

…A:2†

The advantage of the new equation is that it often provides a simpler surface through the data, one that minimizes ridges and troughs that arise by the exponentiation of negative or imaginary numbers. Table 5

Coecients for the Real Form of Eq. (8)a

a

Leading term

J

K

Multiplier

)Im )Im )Im )Im +Im +Im +Im +Im )Re )Re +Re +Re

)Im )Im +Im +Im +Im +Im )Im )Im )Re +Re )Re +Re

)Im +Im )Im +Im +Im )Im +Im )Im +Re )Re +Re )Re

… x y† …y x† …x y† …x ‡ y† … x y† …y x† …x y† …x ‡ y† …x2 y 2 † …y 2 x2 † …y 2 x2 † …x2 y 2 †

Im represents an imaginary number. Re represents a real number.

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G.L. Silver / Appl. Math. Modelling 25 (2001) 629±634

References [1] [2] [3] [4]

G.L. Silver, Comput. Stat. Data Anal. 28 (1998) 211±215. G.L. Silver, Quality Eng. 10 (1998) 625±631. G.L. Silver, Quality Eng. 9 (1996±1997) 7±10. A. Tuller, Trigonometry (Section 5, History), Encyclopedia Americana, vol. 27, Grolier Inc., Danbury, Ct., 1998, pp. 103±110. [5] G.L. Silver, J. Comput. Chem. 6 (1985) 229±236. [6] G.L. Silver, Quality Eng. 6 (1993±1994) 307±310. [7] G.L. Silver, Los Alamos National Laboratory publication LA-UR-00-5063, October 2000.