G.E.P. Box, and K.B. Wilson, "On the experimental attainment of optimum conditions," Journal of the Royal Statistics Society, B13, 1-38, 1951 171  Home/Search   Document Not in Database   Summary   Related Articles   Check

....immediately generalize to bound constrained problems: in x5.2 we will present a counterexample that defeats G.E.P. Box s method of evolutionary operation using two level factorial designs [1] 3, 12] and show how the convergence theory guides us to a remedy that uses composite designs [2], instead of the simpler factorial or fractional factorial designs. The multidirectional search algorithm of Dennis and Torczon [7, 13] also requires us to augment the pattern used for the algorithm; again we find a straightforward extension, but one that reveals much about the interesting ....

....nonsingular basis matrix) in addition, we must augment the columns of the generating matrix to ensure a sufficient set of search directions. In the case of evolutionary operation, we also must augment the columns of the generating matrix, which we do using a classical variant of factorial designs [2]. The difference between pattern search methods for unconstrained problems and bound constrained problems lies in the two additional conditions discussed in x2.5. First, pattern search methods for bound constrained problems must start with a feasible iterate and choose feasible trial steps. ....

[Article contains additional citation context not shown here]

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), pp. 1--45.

....distinguish direct search methods within the larger class of derivative free methods. We have already remarked that response surface methodology constructs local approximations of f by regression. Response surface methodology was proposed in 1951, in a seminal paper by G.E.P. Box and K.B. Wilson [4], as a variant of steepest descent (actually steepest ascent, since the authors were maximizing) In 1957, concerned with the problem of improving industrial processes and the shortage of technical personnel, Box [3] outlined a less sophisticated procedure called evolutionary operation.Response ....

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), pp. 1--45.

....until the steps are deemed sufficiently small. If we consider the pattern of points from which the function can be sam pled (shown in Figure 1 for problems with only two variables) we see something that looks much like a two factor composite design as described by G. E. P. Box and K. B. Wilson [4] on the use of experimental designs to attain optimum conditions. FIGURE 1: Pattern for coordinate search in = In fact, the patterns associated with pattern search methods share many features with some of the early orthogonal designs suggested for experimental design. Our own work for the ....

....be required for a full two level factorial design. This loose definition of pattern search methods captures a variety of other long standing direct search methods. These include optimization methods based on the response surface methodology work first intro duced by G. E. P. Box and K. B. Wilson [4] and subsequently developed by Box and other researchers [2, 3, 5] the original pattern search algorithm of Robert Hooke and T. A. Jeeves [14] and more recently the multidirectional search algorithm and its variants de veloped by the authors [8, 19, 20] 2.2 A Formal Abstraction The analysis ....

G. E. P. Box and K. B. Wilson. On the ex- perimental attainment of optimum conditions. Journal of the Royal Statistical Society, Series B, XIII(1):1-45, 1951.

.... that modify the search directions at the end of each iteration, as exemplified by the methods of Box [2] Nelder and Mead [15] Powell [17] Rosenbrock [18] and Zangwill [31] and those that use a fixed set of search directions at all iterations, as exemplified by the methods of Box and Wilson [1], Hooke and Jeeves [10] Spendley, Hext, and Himsworth [23] and Dennis and Torczon [6, 25] Further studies of these and related methods are presented in [3, 4, 7, 8, 24, 26, 29, 30] 1.1 Motivation for the new method Three previously proposed direct search methods are based on the intriguing ....

....n. This m is denoted by m k . If we subsequently return to Step 1 from Step 5, then l(F (S) F (S k ) could possibly be less than n, and hence m cannot be chosen as freely. If f is convex or more generally, quasiconvex (see [13] in the sense that f(x #(y x) # max f(x) f(y) ## # [0, 1], #x, y # # n , 2.14) then we have l(F (S) F (S k ) # n every time we return to Step 1, so that m can always be chosen freely. This is because when we shrink the simplex S in Step 5 towards the best vertex 7 s best to obtain S # = s best # s (S s best ) we have f(s best ) # ....

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), 1--45.

....of the approach, we show that it is equivalent to using the delta method proposed by Carter, Chinchilli and Campbell (1990) KEY WORDS: Response Surface Methodology, Canonical Analysis, Confidence Intervals, Delta Method. 2 1. INTRODUCTION Response surface methodology, first introduced by Box and Wilson (1951), is frequently used to find conditions that either maximize or minimize a given quality characteristic or more broadly for the exploration of near stationary conditions including ridges. When near stationary conditions are reached, second order polynomial models are typically fit to the data ....

....identification and exploration of ridge systems so important. In two or three dimensions, a stationary ridge can be easily identified and exploited using a contour plot like those discussed above. When more than three factors are involved, plots are difficult to make and other methods are needed. Box and Wilson (1951) pointed out that the shape of the response surface locally can be interpreted by performing a canonical analysis where the signs and magnitudes of the eigenvalues of B , the matrix of second order parameters in (1) are considered. In particular, when all of the eigenvalues are positive the ....

[Article contains additional citation context not shown here]

Box, G. E. P., and Wilson, K. B. (1951), "On the Experimental Attainment of Optimum Conditions," Journal of the Royal Statistical Society, Ser. B, 13, 1-45.

....and uses the delta method and nonlinear least squares methodology. KEY WORDS: canonical analysis, eigenvalues, joint confidence intervals, extra sum of squares, parameterization, orthogonal matrix, nonlinear least squares, delta method. 1. INTRODUCTION Traditional response surface methodology (Box and Wilson, 1951) advocates the fitting of a quadratic response surface in the latter stages of response surface exploration. Much literature has been devoted to the identification of ridge behavior in fitted quadratic surfaces since this behavior can often be exploited to simultaneously optimize the primary ....

Box, G. E. P., and Wilson, K. B. (1951), "On the Experimental Attainment of Optimum Conditions," Journal of the Royal Statistical Society, Ser. B, 13, 1-45.

....distinguish direct search methods within the larger class of derivative free methods. We have already remarked that response surface methodology constructs local approximations of f by regression. Response surface methodology was proposed in 1951, in a seminal paper by G.E.P. Box and K.B. Wilson [4], as a variant of steepest descent (actually steepest ascent, since the authors were maximizing) In 1957, concerned with the problem of improving industrial processes and the shortage of technical personnel, Box [3] outlined a less sophisticated procedure called evolutionary operation. Response ....

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), pp. 1--45.

....literature on the design of experiments for stochastic optimization, a branch of statistics called response surface methodology. It is interesting to note that many popular direct search methods can be traced back to this literature. See [18] for details. In a seminal paper, Box and Wilson [3] proposed that steepest descent (actually ascent, as the authors were maximizing) in the presence of random noise might be based on estimating linear models of the objective function by performing regression experiments. Where the curvature of f is substantial, one might likewise construct a ....

G. E. P. Box and K. B. Wilson. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, Series B, 13:1--45, 1951. Includes discussion.

....asymmetric composite designs is also considered. The methodology is illustrated via a paper helicopter experiment. KEY WORDS: Asymmetric composite design; Central composite design; Fractional factorial; Sequential experimentation; Simplex design; Steepest ascent 16 June 2000 2 1. INTRODUCTION Box and Wilson (1951) is a seminal paper containing most essential elements of current response surface methodology (RSM) practice: sequential experimentation, steepest ascent, central composite designs, canonical analyses of quadratic models, etc. Where sequential experimentation is possible, the success of RSM is ....

....block around one of the factorial points from the first block, with at least one center point run per block. Section 4. 1 describes another family of second order designs that can be constructed by augmentation of two level factorial designs, the asymmetric composite designs first suggested by Box and Wilson (1951) and Box (1954) and later studied by Lucas (1974) Section 4.2 provides an intuitive understanding of asymmetric, noncentral and central composite design alternatives, while Section 4.3 briefly compares the efficiency of central and noncentral composite designs of the same size. Section 4.4 makes ....

[Article contains additional citation context not shown here]

Box, G.E.P. and Wilson, K.B. (1951). "On the Experimental Attainment of Optimum Conditions," Journal of the Royal Statistical Society, Ser. B 13, pp. 1-38.

....designs are simple to construct. It is also noted that the k treatment combinations in each simplex design . form a subset of the saturated resolution V designs proposed by Rechtschaffner (1967) coincide with the axial points in the second order, asymmetric composite designs proposed by Box and Wilson (1951) and later discussed by Lucas (1974) Including the base point permits inclusion of a block effect in the fitted second order model. Completed October 29, 1999 Dr. Mee is Professor in the Department of Statistics. He has been department head since 1997. He is a member of ASQ. 2 Introduction ....

....to complete the resolution V design. For k = 5, 6, and 7, n = 11, 16, and 22 more runs, respectively, are required. Except when k = 5, this is not a regular 2 k p fraction. However, for k = 6 and 7, the designs offer reasonable efficiency. Simplex to Complete an Asymmetric Composite Design Box and Wilson (1951) proposed two types of second order composite designs consisting of a two level factorial or fractional factorial design augmented with axial points. One of these alternatives, the popular central composite design (CCD) augments with 2k axial points arranged symmetrically about the center of the ....

[Article contains additional citation context not shown here]

BOX, G.E.P. and WILSON, K.B. (1951). "On the Experimental Attainment of Optimum Conditions," Journal of the Royal Statistical Society, Ser. B 13, pp. 1-38.

....use of plots may still facilitate the detection of active factors if correctly interpreted. This is due to the fact that bias caused by interaction terms may have very systematic influence on the estimated contrasts. The general way of obtaining the alias pattern for any design was given by Box and Wilson (1951). Suppose we try to fit the regression model E(y) Xfi (2.1) by least squares, but that the true expectation is given by E(y) Xfi X 1 fi 1 : 2.2) Then the estimator of the regression coefficient vector is given by fi = X 0 X) Gamma1 X 0 y, and it follows that E( fi) fi Afi ....

Box, G. E. P. and Wilson, K. B. (1951). On the Experimental Attainment of Optimum Conditions, Journal of the Royal Statistical Society, Series B 13(1), 1--38.

....immediately generalize to bound constrained problems: in x5.2 we will present a counterexample that defeats G.E.P. Box s method of evolutionary operation using two level factorial designs [1] 3, 12] and show how the convergence theory guides us to a remedy that uses composite designs [2], instead of the simpler factorial or fractional factorial designs. The multidirectional search algorithm of Dennis and Torczon [7, 13] also requires us to augment the pattern used for the algorithm; again we find a straightforward extension, but one that reveals much about the interesting ....

....nonsingular basis matrix) in addition, we must augment the columns of the generating matrix to ensure a sufficient set of search directions. In the case of evolutionary operation, we also must augment the columns of the generating matrix, which we do using a classical variant of factorial designs [2]. The difference between pattern search methods for unconstrained problems and bound constrained problems lies in the two additional conditions discussed in x2.5. First, pattern search methods for bound constrained problems must start with a feasible iterate and choose feasible trial steps. ....

[Article contains additional citation context not shown here]

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), pp. 1--45.

....minimization immediately generalize to bound constrained problems: in 2 we present a counterexample that defeats G.E.P. Box s method of evolutionary operation using two level factorial designs [1] 3, 14] and show how the convergence theory guides us to a remedy that uses composite designs [2], instead of the simpler factorial or fractional factorial designs. The multidirectional search algorithm of Dennis and Torczon [7, 15] also requires us to augment the pattern used for the algorithm; again we find a straightforward extension, but one that reveals much about the interesting ....

....evolutionary operation is not compatible with the geometry of the feasible region. A moment s reflection reveals that the problem is that the pattern does not allow us to move parallel to the bounds. This problem goes away if, for instance, we augment the pattern using the idea of composite design [2] (as opposed to factorial design) An example of such a design is shown in Fig. 2.3. We now have a feasible step along the active constraint x 2 # 0 that will produce descent. This simple example captures the essential idea for the generalization of pattern search methods to bound constrained ....

[Article contains additional citation context not shown here]

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), pp. 1--45.

....increasing number of processors solving for the predictor and corrector steps becomes more and more significant computationally. To wrap up, Table 1. 2 shows an overview of a few problems that enable us to get an impression of the differences between WGPP and MMD (for more computational results see [1]) We see that MMD and WGPP are very comparable in the first 2 problems and that WGPP performs significantly better on the PDS problems. In fact, PDS 20 could not be solved on less than 8 processors using the minimum degree ordering due to memory limitations. The speedup of the factorization is ....

....Investigate algorithmic variants that allow concurrent solution of triangular systems with multiple right hand sides. This report is based on a talk given by the third author at the Eighth SIAM conference on Parallel Processing for Scientific Computing in Minneapolis, MN, March 1997. REFERENCES [1] T. F. Coleman, J. Czyzyk, C. Sun, M. Wagner, and S. J. Wright, pPCx: Parallel Software for Linear Programming, Proc. of the Eighth SIAM Conference on Parallel Processing in Scientific Computing, Minneapolis, MN, March 1997 [2] J. Czyzyk, S. Mehrotra, and S. J. Wright, PCx User Guide, Technical ....

[Article contains additional citation context not shown here]

G. E. P. Box and K. B. Wilson. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, Series B, 13:1--45, 1951. Includes discussion.

....until the steps are deemed sufficiently small. If we consider the pattern of points from which the function can be sampled (shown in Figure 1 for problems with only two variables) we see something that looks much like a two factor composite design as described by G. E. P. Box and K. B. Wilson [4] on the use of experimental designs to attain optimum conditions. r r r r r r r r r oe 6 Figure 1: Pattern for coordinate search in IR 2 . In fact, the patterns associated with pattern search methods share many features with some of the early orthogonal designs suggested for experimental ....

....be required for a full two level factorial design. This loose definition of pattern search methods captures a variety of other long standing direct search methods. These include optimization methods based on the response surface methodology work first introduced by G. E. P. Box and K. B. Wilson [4] and subsequently developed by Box and other researchers [2, 3, 5] the original pattern search algorithm of Robert Hooke and T. A. Jeeves [14] and more recently the multidirectional search algorithm and its variants developed by the authors [8, 19, 20] 2.2 A Formal Abstraction The analysis of ....

G. E. P. Box and K. B. Wilson. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, Series B, XIII(1):1--45, 1951.

....methods, i.e. methods that model the objective function without recourse to derivative information, is an interesting and important class of algorithms. For optimization in the presence of noise, methods that model the objective function by quadratic regression include Box and Wilson [1] and their legacy of response surface methodology, Glad and Goldstein [9] and Elster and Neumaier [7] For numerical optimization, similar methods that model the objective function without recourse to derivative information include Buckley and Ma [2] Powell [12, 13, 14] and Conn and Toint [5] ....

G. E. P. Box and K. B. Wilson. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, Series B, 13:1--45, 1951. Includes discussion.

....4 ) for those subsequent iterations where S k changes little and no rank deficiencies occur. For n 12, computing time for the linear algebra remains in the seconds on a SUN SPARC station, and does not matter for the applications to expensive function minimization. The methods of Box Wilson [1], Dixon [3] and Glad Goldstein [7] avoid expensive linear algebra by evaluating the function on well chosen experimental designs for which the least squares problem can be solved more cheaply explicitly; but this requires an excessive number O(n 2 ) of additional function evaluations) For ....

Box, G.E.P. and Wilson, K.B., On the experimental attainment of optimum conditions, J. Royal Stat. Soc., Ser. B, 13 (1951), 1-45.

.... aware that the same style of argument used to prove global convergence for the multidirectional search algorithm could be applied, individually, to such classical algorithms as coordinate search, with fixed step sizes, variants of response surface methodology, first developed by Box and Wilson [4] and later popularized by Box [2, 3] and the original pattern search algorithm of Hooke and Jeeves [10] The challenge was to develop an abstraction that both allowed for a general convergence theory and explained why such algorithms, often viewed as disparate direct search methods, could be ....

....stationary point convergence, but to do so requires placing stronger conditions on the specifications for generalized pattern search methods. These stronger conditions are immediately satisfied by only one of the pattern search methods we will present (that due to G. E. P. Box and Wilson [4]) We could certainly impose these stronger conditions on the remaining pattern search methods presented in x5 none of them are unreasonable to suggest or to enforce but we would do so at the expense of attractive algorithmic features found in the original methods. 3.1. The Algebraic ....

[Article contains additional citation context not shown here]

G. E. P. Box and K. B. Wilson, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society, Series B, XIII (1951), pp. 1--45.

....conformation had an energy less than 0.0. 2.2 Direct Search Methods Direct search methods belong to a class of optimization methods that do not compute derivatives. Examples of direct search methods are the Nelder Mead Simplex method [13] Hooke and Jeeves pattern search [14] the box method [15], and Dennis and Torczon s parallel direct search algorithm (PDS) The PDS algorithm can be viewed as an intelligent adaptive grid search algorithm employing a multi sided simplex. Starting from an initial simplex S o , the function value at each of the vertices in S o is computed and the vertex ....

G.E.P. Box and K.B. Wilson. On the experimental attainment of optimum conditions. J. Royal Statistical Society, Series B, 13:1--45, 1951.

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G.E.P. Box, and K.B. Wilson, "On the experimental attainment of optimum conditions," Journal of the Royal Statistics Society, B13, 1-38, 1951 171

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G.E.P. Box and K.B. Wilson. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, 13:1--45, 1951.

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) Box, G. E. P., and K. B. Wilson (1951). On the experimental attainment of optimum conditions, J. Roy. Stat. Soc., Ser. B, 13,1.

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Box, G.E.P. and Wilson, K.B. "On the Experimental Attainment of Optimum Conditions." Journal of Royal Statistical Society, B13, 1-38, Discussion 38-45. 1951.

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Box, G.E.P. and K.B. Wilson, Experimental attainment of optimum conditions, Journal of the Royal Statistical Society, 13, pp.1-45, 1951

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Box, G. E. P. and K. B. Wilson, Experimental attainment of optimum conditions, Journal of the Royal Statistical Society, 13, pp.1-45, 1951

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