Optimum designs for optimum mixtures
TL;DR: In this paper, the authors try to find optimum designs for the estimation of optimum mixture combination on the assumption that the response function is quadratic concave over the simplex region.
About: This article is published in Statistics & Probability Letters.The article was published on 2006-07-15. It has received 28 citations till now.
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TL;DR: In this article, the minimax criterion has been employed to find a solution to the problem of estimating the optimum proportion of mixture components, assuming prior knowledge about the optimum mixing proportions.
18 citations
TL;DR: In this paper, the deficiency criterion due to Chatterjee and Mandal (1981) has been used as a measure for comparing the performance of competing mixture models, and the problem of estimating the optimum proportion of mixture components is of great practical importance.
Abstract: In a mixture experiment the measured response is assumed to depend only on the relative proportion of ingredients or components present in the mixture. Scheffe (1958, 1963) first systematically considered this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation of parameters of different mixture models are available in the literature. The problem of estimating the optimum proportion of mixture components is of great practical importance. Pal and Mandal (2006, 2007) attempted to find a solution to this problem by adopting a pseudo-Bayesian approach and using the trace criterion. Subsequently, Pal and Mandal (2008) solved the problem using minimax criterion. In this article, the deficiency criterion due to Chatterjee and Mandal (1981) has been used as a measure for comparing the performance of competing designs.
16 citations
Cites background or methods from "Optimum designs for optimum mixture..."
...The Problem and the Perspectives As in Pal and Mandal (2006), we assume the response function to be quadratic concave in the components x1 x2 xq in the factor space = x1 x2 xq xi ≥ 0 i = 1 1 q ∑ xi = 1 and to have the form E Y x = x = ∑ i iix 2 i + ∑ i<j ijxixj = f ′ x (2....
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...1) with inverse M−1 It is shown in Pal and Mandal (2006) that A can be expressed as...
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...In order to find the optimal values of the masses, we use an alternative representation of the model considered by Pal and Mandal (2007), which simplifies algebraic computation substantially....
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...Pal and Mandal (2006) showed that for the optimum WCD, 3 = 0, i....
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TL;DR: In this paper, the authors consider the estimation of the optimum factor combination in a response surface model and find the optimum design using the trace optimality criterion using a pseudo-Bayesian approach.
Abstract: In this paper, we consider the estimation of the optimum factor combination in a response surface model. Assuming that the response function is quadratic concave and there is a linear cost constraint on the factor combination, we attempt to find the optimum design using the trace optimality criterion. As the criterion function involves the unknown parameters, we adopt a pseudo-Bayesian approach to resolve the problem.
9 citations
Cites background or methods from "Optimum designs for optimum mixture..."
...Pal and Mandal [14] first attempted to find optimum designs for the estimation of optimum mixture combination....
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...Within the class of WCDs, Pal and Mandal [14], obtained the optimum mixture designs in the two- and three-component cases....
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...It was shown by Pal and Mandal [14] that A(γ ) can be expressed as...
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...Pal and Mandal [14] studied this problem for the two- and three-component cases and obtained the optimum mixture design by a pseudo-Bayesian approach, using the trace criterion....
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...Hence, to resolve this problem, we adopt a pseudo-Bayesian approach, as in Pal and Mandal [14], and assume that γ0, and hence γ , is random with...
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TL;DR: In this article, a mixture-amount model was proposed, which is quadratic both in the proportions of mixing components and the amount of mixture, and the optimality of the derived designs in the entire class of competing designs has been investigated through equivalence theorem.
Abstract: The authors propose a mixture-amount model, which is quadratic both in the proportions of mixing components and the amount of mixture. They attempt to find the A- and D-optimal designs for the estimation of the model parameters within a subclass of designs. The optimality of the derived designs in the entire class of competing designs has been investigated through equivalence theorem.
9 citations
01 Jan 2008
TL;DR: In this article, Mandal et al. derived a pseudo-Bayesian approach with invariance property of the second order moments of the optimum mixing proportions for estimating the optimum proportion of mixture components when the factor space is constrained.
Abstract: Scheffe (1958, 1963) first introduced models and designs suitable for a mixture experiment where the mean response is assumed to depend only on the relative proportions of the ingredients or components. Extensive literature on optimum designs for the estimation of parameters of different mixture models is available. The specific problem of characterization of optimal designs for estimating the op- timum proportion of mixture components has been recently considered by Pal and Mandal (2006, 2008) as also by Mandal and Pal (2008) using different optimality criteria. Generalizing the work of Pal and Mandal (2006), who had dealt with the trace criterion and adopted a pseudo-Bayesian approach with invariance property of the second order moments of the optimum mixing proportions, Mandal et al. (2008) relaxed the invariance property of the second order moments of the op- timum mixing proportions. In this paper, optimum designs are derived for the problem of estimating the optimum proportion of mixture components when the factor space is constrained.
9 citations
Cites background or methods from "Optimum designs for optimum mixture..."
...It has been shown in Pal and Mandal (2006) that A(γ) can be expressed as A(γ) = d 2 6 6 6 6 6 6 6 6 6 4 −2(q − 1)γ1 2γ2 . . . 2γq γ1 − (q − 1)γ2 . . . γq−1 + γq 2γ1 −2((q − 1)γ2 . . . 2γq γ2 − (q − 1)γ1 . . . γq−1 + γq 2γ1 2γ2 . . . 2γq γ1 + γ2 . . . γq−1 + γq . . . . . . . . . . . . . . . . . . .…...
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...Here we restrict our study to the class of non-singular information matrices, as in Pal and Mandal (2006). It has been shown in Pal and Mandal (2006) that A(γ) can be expressed as...
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...Generalizing the work of Pal and Mandal (2006), who had dealt with the trace criterion and adopted a pseudo-Bayesian approach with invariance property of the second order moments of the optimum mixing proportions, Mandal et al. (2008) relaxed the invariance property of the second order moments of…...
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...2 Problem and the perspectives As in Pal and Mandal (2006), we assume the response function to be quadratic concave in the components x1, x2, ....
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...Pal and Mandal (2006), assumed a prior distribution of γ with E(γ2i ) = v, i = 1, 2, . . . , q and E(γiγj) = w, i, j = 1, 2, . . . , 1; i < j and minimized E[φ(γ, M(ξ))], expectation being taken with respect to the prior distribution of γ....
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References
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TL;DR: The work described in this article is the result of a study extending over the past few years by a chemist and a statistician, which has come about mainly in answer to problems of determining optimum conditions in chemical investigations, but they believe that the methods will be of value in other fields where experimentation is sequential and the error fairly small.
Abstract: The work described is the result of a study extending over the past few years by a chemist and a statistician. Development has come about mainly in answer to problems of determining optimum conditions in chemical investigations, but we believe that the methods will be of value in other fields where experimentation is sequential and the error fairly small.
4,359 citations
TL;DR: In this paper, the concept of the variance function for an experimental design is introduced, and the problem of selecting practically useful designs is discussed, and in this connection, the notion of variance function is introduced.
Abstract: Suppose that a relationship $\eta = \varphi(\xi_1, \xi_2, \cdots, \xi_k)$ exists between a response $\eta$ and the levels $\xi_1, \xi_2, \cdots, \xi_k$ of $k$ quantitative variables or factors, and that nothing is assumed about the function $\varphi$ except that, within a limited region of immediate interest in the space of the variables, it can be adequately represented by a polynomial of degree $d$. A $k$-dimensional experimental design of order $d$ is a set of $N$ points in the $k$-dimensional space of the variables so chosen that, using the data generated by making one observation at each of the points, all the coefficients in the $d$th degree polynomial can be estimated. The problem of selecting practically useful designs is discussed, and in this connection the concept of the variance function for an experimental design is introduced. Reasons are advanced for preferring designs having a "spherical" or nearly "spherical" variance function. Such designs insure that the estimated response has a constant variance at all points which are the same distance from the center of the design. Designs having this property are called rotatable designs. When such arrangements are submitted to rotation about the fixed center, the variances and covariances of the estimated coefficients in the fitted series remain constant. Rotatable designs having satisfactory variance functions are given for $d = 1, 2$; and $k = 2, 3, \cdots, \infty$. Blocking arrangements are derived. The simplification in the form of the confidence region for a stationary point resulting from the use of a second order rotatable design is discussed.
1,332 citations
847 citations
TL;DR: In this paper, the authors consider the problem of finding an optimum design of experiments in regression problems, where the desired inference concerns one of the regression coefficients, and illustrative examples will be given in Section 3.
Abstract: Although regression problems have been considered by workers in all sciences for many years, until recently relatively little attention has been paid to the optimum design of experiments in such problems. At what values of the independent variable should one take observations, and in what proportions? The purpose of this paper is to develop useful computational procedures for finding optimum designs in regression problems of estimation, testing hypotheses, etc. In Section 2 we shall develop the theory for the case where the desired inference concerns just one of the regression coefficients, and illustrative examples will be given in Section 3. In Section 4 the theory for the case of inference on several coefficients is developed; here there is a choice of several possible optimality criteria, as discussed in [1]. In Section 5 we treat the problem of global estimation of the regression function, rather than of the individual coefficients. We shall now indicate briefly some of the computational aspects of the search for optimum designs by considering the problem of Section 2 wherein the inference concerns one of $k$ regression coefficients. For the sake of concreteness, we shall occasionally refer here to the example of polynomial regression on the real interval $\lbrack -1, 1\rbrack$, where all observations are independent and have the same variance. The quadratic case is rather trivial to treat by our methods, so we shall sometimes refer here to the case of cubic regression. In the latter case we suppose all four regression coefficients to be unknown, and we want to estimate or test a hypothesis about the coefficient $a_3$ of $x^3$. If a fixed number $N$ of observations is to be taken, we can think of representing the proportion of observations taken at any point $x$ by $\xi(x)$, where $\xi$ is a probability measure on $\lbrack -1, 1\rbrack$. To a first approximation (which is discussed in Section 2), we can ignore the fact that in what follows $N\xi$ can take only integer values. We consider three methods of attacking the problem of finding an optimum $\xi$: A. The direct approach is to compute the variance of the best linear estimator of $a_3$ as a function of the values of the independent variable at which observations are taken or, equivalently, as a function of the moments of $\xi$. Denoting by $\mu_i$ the $i$th moment of $\xi$, and assuming $\xi$ to be concentrated entirely on more than three points (so that $a_3$ is estimable), we find easily that the reciprocal of this variance is proportional to $$\frac{\mu^2_5(\mu^2_1 - \mu_2) + 2\mu_5(\mu^2_2 \mu_3 + \mu_3 \mu_4 - \mu_1 \mu^2_3 - \mu_1 \mu_2 \mu_4)\\- \mu^3_4 + \mu^2_4(\mu^2_2 + 2\mu_1 \mu_3) - 3\mu_4 \mu_2 \mu^2_3 + \mu^4_3}{\mu_4(\mu_2 - \mu^2_1) - \mu^2_3 - \mu^3_2 + 2\mu_1 \mu_2 \mu_3} + \mu_6$$ in the case of cubic regression. The problem is to find a $\xi$ on $\lbrack -1, 1\rbrack$ which maximizes this expression. Thus, this direct approach leads to a calculation which appears quite formidable. This is true even if one uses the remark on symmetry of the next paragraph and restricts attention to symmetrical $\xi$, so that $\mu_i = 0$ for $i$ odd. For polynomials of higher degree or for regression functions which are not polynomials, the difficulties are greater. B. The results of Section 2 yield the following approach to the problem: Let $c_0 + c_1x + c_2x^2$ be a best Chebyshev approximation to $x^3$ on $\lbrack -1, 1\rbrack$, i.e., such that the maximum over $\lbrack -1, 1\rbrack$ of $|x^3 - (c_0 + c_1x + c_2x^2)|$ is a minimum over all choices of the $c_i$, and suppose $B$ is the subset of $\lbrack -1, 1\rbrack$ where the maximum of this absolute value is taken on. Then $\xi$ must give measure one to $B$, and the weights assigned by $\xi$ to the various points of $B$ (there are four in this case) can be found either by solving the linear equations (2.10) or by computing these weights so as to make $\xi$ a maximum strategy for the game discussed in Section 2. Two points should be mentioned: (1) In the general polynomial case, where there are $k$ parameters ($k = 4$ here), the results described in [10], p. 42, or in Section 2 below imply that there is an optimum $\xi$ concentrated on at most $k$ points. Thus, even if we use this result with the approach of the previous paragraph, we obtain the following comparison in a $k$-parameter problem in Section 2: Method A: minimize a nonlinear function of $2k - 1$ real variables. Method B: solve the Chebyshev problem and then solve $k - 1$ simultaneous linear equations. The fact that the solution of the Chebyshev problem can often be found in the literature (e.g., [2]) makes the comparison of the second method with the first all the more favorable. (2) Although the computational difficulty cannot in general be reduced further, in the case of polynomial regression on $\lbrack -1, 1\rbrack$ there is present a kind of symmetry (discussed in Section 2) which implies that there is an optimum $\xi$ which is symmetrical about 0 and which is concentrated on four points; thus, in the case of cubic regression, this fact reduces the computation under Method A to a minimization in 3 variables, but Method B involves only the solution of a single linear equation. C. A third method, which rests on the game-theoretic results of Section 2, and which is especially useful when one has a reasonable guess of what an optimum $\xi$ is, involves the following steps: first guess a $\xi$, say $\xi^{\ast}$, and compute the minimum on the left side of (2.8); second, if this minimum is achieved for $c = c^{\ast}$, compute the square of the maximum on the right side of (2.9); then, if these two computations yield the same number, $\xi^{\ast}$ is optimum. If one has a guess of a class of $\xi$'s depending on one or several parameters, among which it is thought that there is an optimum $\xi$, then one can maximize over that class at the end of the first step and, the maximum being at $\xi^{\ast}$, go through the same analysis as above. This method is illustrated in Example 3.5 and Example 4. Of course, the remarks (1) and (2) of the previous paragraph can be used in applying Method C, as in these examples. In the example of cubic regression just cited, the optimum procedure turns out to be $\xi(-1) = \xi(1) = \frac{1}{6}, \xi(\frac{1}{2}) = \xi(-\frac{1}{2}) = \frac{1}{3}$. It is striking that any of the commonly used procedures which take equal numbers of observations at equally spaced points on $\lbrack -1, 1\rbrack$ requires over 38% more observations than this optimum procedure in order to yield the same variance for the best linear estimator of $a_3$ (see Example 3.1); the comparison is even more striking for higher degree regression. The unique optimum procedure in the case of degree $h$ is given by (3.3). The comparison of a direct computational attack, analogous to that of A above, with the methods developed in Sections 4 and 5 for the problems considered there, indicates even more the inferiority of the direct attack. In particular cases, e.g., Example 5.1, special methods may prove useful. Among recent work in the design of experiments we may mention the papers of Elfving [3], [4], Chernoff [5], Williams [11], Ehrenfeld [12], Guest [13], and Hoel [15]. Only Guest and Hoel explicitly consider computational problems of the kind discussed below. Our methods of employing Chebyshev and game theoretic results seem to be completely new. The results obtained in the examples below are also new, except for some slight overlap with results of [13] and [15], which is explicitly described below. We shall consider elsewhere some further problems of the type considered in this paper.
803 citations