A. C. Atkinson and A. N. Donev. Optimum experimental designs. Oxford Science Publishers, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....binary repetition code (Chapter 4) In this case, the dual code is the even weight code. Definition 17 The even weight code is the linear [n, n 1, 2] code consisting of all elements of T = 0, 1 n with even weight (n # 2) The solution set of the equation Hx = 0 for the 1 n matrix H = [1 1 1] is the dual of the binary repetition code 0 n , 1 n and is easily seen to consist of all the codewords of even weight. This observation leads to the following result. CHAPTER 6. THE BINARY SETTING 52 010 110 011 111 000 100 001 101 Figure 6.1: Three Dimensional Simplex Code Corollary ....

....(T, d) may be of interest to experimenters with specific problems in mind. Also, evaluating these designs in more traditional ways (rather than on their space filling properties alone) such as resolution, aberration, and the standard numerical optimality criteria (A , D , E , and G optimality [1]) may increase their appeal to those unfamiliar with the concepts of space filling designs. Less is know about codes for which q 2; however, a thorough investigation of the q 2 results may prove useful in determining additional space filling designs for the case of factors with more than ....

[Article contains additional citation context not shown here]

A.C. Atkinson and A.N. Donev. Optimum Experimental Designs. Oxford University Press, 1992.

....) Factor Experiment 1 2 3 4 1 1 1 1 1 2 1 2 2 3 3 1 3 3 2 4 2 1 2 2 5 2 2 3 1 6 2 3 1 3 7 3 1 3 3 8 3 2 1 2 9 3 3 2 1 ble design matrices, it is necessary to define a suitable measure of quality and then find good designs. There is an extensive literature on this subject e.g. 3] 4] [5], 6] 7] Here two types of designs are considered: orthogonal designs and uniform designs. An orthogonal design has the property that each possible pair of levels for any two factors appears exactly the same number of times. An example is the design in Table I. If any two columns of the design ....

A. C. Atkinson and A. N. Donev, Optimum Experimental Designs, Clarendon Press, Oxford, 1992.

....with respect to the improvement of the student s performance. facilitate training by removing redundancy from the training set. 3. 1 Variations on Active Learning The basic idea of active learning may be traced back to the statistical theory of optimum experimental design (OED) cf. e.g. [5, 6, 7]. Limited experimental resources and increasing costs in experimental investigations led to the need of extracting an increased quantity of data from processes under study while only nite resources were placed at disposal. This motivated the development of statistical tools for the design of ....

A. C. Atkinson and A. N. Donev. Optimum Experimental Designs. Clarendon Press, Oxford, 1992.

....p j r i j i i l t t G , 1 , 1 0 ) 1 = J = J J J h J h = diag 1 r w w W = This is the information matrix for a transformed variable l y u h = for which 2 ) var( s = u and, technically, the problem simplifies to the constant variance case. See Atkinson and Donev (1992) for wide discussion of optimum design or He, Studden and Sun (1996) for detailed consideration of optimum design in rational models. Furthermore, let us note that if the number of support points of a design is equal to the number of parameters, i.e. r = p, then the weights i w are all equal and ....

.... designs, D x , for constant error variance (l = 0) compared to the optimum designs, l D h x , for non constant error variance (l 0) As an efficiency measure we use the square root of the ratio of the values of the D optimality criterion for non constant variance at the compared designs, cf. Atkinson and Donev (1992): 2 1 ) 0 0 J x F J x F = x x h h l l Eff l D D D D . The numerator and denominator involve the same function (and the same parameter values 0 J , l) evaluated at different design times: the denominator is the optimal value of the criterion function for the ....

Atkinson, A. C., Donev, A. N. (1992), Optimum Experimental Designs. Oxford University Press.

....of the least squares estimator b . Let (dx) be a probability distribution on X describing the frequency of taking x as an observation point. Then the covariance matrix k cov( b i ; b j )k equals 2 M( 1 , where M( Z f(x) f(x) dx) is the information matrix, see, e.g. (Atkinson and Donev, 1992) for details. The measure that maximises det M( or, equivalently, minimises f( log det M( algo sub.tex; 1 11 2000; 15:52; p.5 6 Molchanov and Zuyev is called D optimal design measure (taking logarithm makes it a convex minimisation problem) The gradient d f (x; f(x)M 1 ( f ....

....f( log det M( algo sub.tex; 1 11 2000; 15:52; p. 5 6 Molchanov and Zuyev is called D optimal design measure (taking logarithm makes it a convex minimisation problem) The gradient d f (x; f(x)M 1 ( f (x) then becomes the standardised variance of the predicted response at point x (Atkinson and Donev, 1992), also called the sensitivity function (Fedorov and Hackl, 1997) Typically, the only constraint is that is a probability measure, while further constraints can be naturally incorporated if such need arises. For example, let X = R d and e i ; i = 1; d be the unit coordinate vectors. ....

Atkinson, A. C. and A. N. Donev: 1992, Optimum Experimental Designs. Oxford: Clarendon Press.

.... applied directly to the present situation of increasing interest in additional parameters there is a bulk of criteria which can be regarded as a generalization of the common D criterion of minimizing the generalized variance (for the related generalization problem in a Bayesian approach see e.g. Atkinson and Donev (1992), p. 214) 2 Let I 1 (ffi 1 ) be the information matrix of ffi 1 associated with the parameters fi 1 of interest in the starting experiment and let I 2 (ffi 2 ) be the information matrix of ffi 2 associated with the (larger) set of parameters fi 2 which are to be investigated in the ....

Atkinson, A. C. and A. N. Donev (1992): Optimum Experimental Designs. Clarendon Press, Oxford.

....: f k (x) and unknown coefficients = 1 ; k ) The optimal design is described as a probability measure on X that determines frequencies 1 2 making an observation at particular points. The information matrix is given by M( Z f(x) f(x) dx) 1. 1) see, e.g. (Atkinson and Donev, 1992) for details. The measure that minimises det M 1 ( is called D optimal design measure. To create a convex optimisation criterion it is convenient to minimise (M) log(det M) which is a convex functional on the space of matrices. In general, a measure that minimises a given ....

....have a direct interpretation within the design framework for an arbitrary . To circumvent this difficulty, it is quite typical in the optimal design literature to replace (1.3) by the following definition of the directional derivative: D ( lim t#0 t 1 ( 1 t) t ) 1. 4) (Atkinson and Donev, 1992; Wynn, 1972) Now (1 t) t is a probability measure for t 2 [0; 1] if both and are probability measures. This definition of the directional derivative is used to construct the steepest (with respect to differential operator D) descent algorithms for finding optimal designs when it is ....

[Article contains additional citation context not shown here]

Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs. Clarendon Press, Oxford.

....60G55, 62K05 1 Introduction Functionals that depend on measures naturally appear in many areas of science. In this paper we give a few examples mainly from statistics and probability. In particular, the corresponding optimisation problems constitute the well developed optimal design theory, see [1] and references therein. In probability theory such problems appear, for instance, when optimising functionals of Poisson point processes [12] whose distributions are determined by the corresponding intensity measure. Such point processes naturally appear in approximation problems for sets and ....

....of a probability measure . Let (dx) be a discrete probability distribution on X describing the frequency of taking x as an observation point. Then the above covariance matrix equals 2 M( 1 , where M( Z f(x) f(x) dx) 6. 2) is the so called information matrix (see, e.g. [1] for details) The measure that maximises det M( or, equivalently, minimises f( log det M( 6.3) is called D optimal design measure (taking logarithm makes it a convex minimisation problem) In this case d f (x; f(x)M 1 ( f (x) Typically, the only constraint is that is a ....

A. C. Atkinson and A. N. Donev. Optimum Experimental Designs. Clarendon Press, Oxford, 1992.

....by r n . In particular, r n with r = 0 is called an all bias design that minimizes J b ( n ) alone, and r n with r = 1 is called an all variance design that minimizes J v ( n ) alone. It is known that the all variance design is L optimal if the assumed linear model is exactly correct (Atkinson and Donev (1992)) The meaning of the all bias design has analogy with the uniform design introduced by Fang and Wang (1994) Further, in order to compare the behaviour of different designs, such as all variance and all bias designs described above, we define the efficiency of a design, n , as follows: Eff( ....

Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs. Oxford Science Publications, Oxford.

....b fi i ; b fi j )k. Speci cally, let (dx) be a probability distribution on X describing the frequency of taking x as an observation point. Then it can be shown that the above covariance matrix equals oe 2 M( Gamma1 , where M( Z f(x) f(x) dx) is the information matrix (cf. [2] for details) The measure that maximises det M( is called D optimal design measure. Note that the objective functional is a concave functional RR n# 32 I. Molchanov and S. Zuyev on the space of measures, so that any maximum found will certainly be global. The general equivalence theorem by ....

Atkinson, A. and Donev, A. (1992) Optimum Experimental Designs. Clarendon Press, Oxford.

....and general analysis of query by committee, and show that such an exponential decrease is guaranteed for a general class of learning problems. The problem of selecting the optimal examples for learning is closely related to the problem of experimental design in statistics (see e.g. Fedorov, 1972, Atkinson and Donev, 1992] Experimental design is the analysis of methods for selecting sets of experiments, which correspond to membership queries in the context of learning theory. The goal of a good design is to select experiments in a way that their outcomes, which correspond to labels, give sufficient information ....

A. C. Atkinson and A. N. Donev. Optimum Experimental Designs. Oxford science publications, 1992.

....x2X d 0 ( c; x) r 0 on the precision which guarantees a prespecified performance in case the constrained model is true. To be more specific, let us recall some general results in the theory of op 8 Norbert Benda, Rainer Schwabe timal designs (for further readings on this topic we refer to Atkinson and Donev, 1992, and Pukelsheim, 1993) If and 0 are G optimal in the full model or in the constrained model, respectively, and if the associated least squares estimators b fi resp. b fi 1 are applied, then the standardized maximal mean squared errors equal the corresponding numbers of parameters ....

Atkinson, A.C. and Donev, A.N. (1992). Optimum Experimental Designs.

....a more complete and general analysis of query by committee, and show that such an exponential decrease is guaranteed for a general class of learning problems. The problem of selecting the optimal examples for learning is closely related to the problem of experimental design in statistics (see e.g. [Fed72, AD92]) Experimental design is the analysis of methods for selecting sets of experiments, which correspond to membership queries in the context of learning theory. The goal of a good design is to select experiments in a way that their outcomes, which correspond to labels, give sufficient information ....

A. C. Atkinson and A. N. Donev. Optimum Experimental Designs. Oxford science publications, 1992.

No context found.

A. C. Atkinson and A. N. Donev. Optimum experimental designs. Oxford Science Publishers, 1992.

No context found.

A.C. Atkinson and A.N. Donev. Optimum Experimental Design. Clarendon Press Oxford, 1992.

No context found.

A.C. Atkinson and A.N. Donev. Optimum Experimental Designs. Oxford University Press, 1992.

No context found.

A.C. Atkinson and A.N. Donev, Optimum Experimental Designs. Clarendon Press, Oxford,1992.

No context found.

Atkinson, A. and Donev A., 1992, Optimum Experimental Designs, Oxford Science Publications.

No context found.

A. C. Atkinson and A. N. Donev, Optimum experimental designs, Clarendon Press, 1992.

No context found.

Atkinson, A.C. and A. N. Donev (1992). Optimum Experimental Design. Clarendon Press, Oxford.

No context found.

Atkinson,A.C.,andA.N.Donev,Optimum Experimental Designs, Oxford Science Publications 1992.

No context found.

A. C. Atkinson and A. N. Donev. Optimum experimental designs. Oxford Science Publishers, 1992.

No context found.

Atkinson, A. C. and Donev, A. N. (1992), Optimum Experimental Designs, Oxford Science Publications, Oxford.

No context found.

Atkinson A. C. and Donev A. N. Optimum Experimental Design. Clarendon Press, Oxford, 1992.

No context found.

A. C. Atkinson and A. N. Donev. Optimum Experimental Designs. Oxford science publications, 1992.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC