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Computing D-Optimum Weighing Designs where Statistics, Combinatorics, and Computation Meet

IP.com Disclosure Number: IPCOM000127976D
Original Publication Date: 1983-Dec-31
Included in the Prior Art Database: 2005-Sep-14
Document File: 13 page(s) / 41K

Publishing Venue

Software Patent Institute

Related People

Zvi Galil: AUTHOR [+3]

Abstract

This paper surveys results and techniques for computing D-optimum weighing designs. 1. Introduction. In this paper we survey results and techniques for computing D-optimum weighing designs. The paper summarizes the results of my work with Jack Kiefer ([1]-[6]), and des-cribes some related results. This paper is written with two hopes in mind. We hope that the work on D-optimum design will be continued. Although we know now much more than before, the picture is far from complete. We also hope that some of the techniques we developed will be found useful elsewhere: for finding optimum designs using other optimality criteria or for solving other optimization problems. Let k and n be positive integers with k S n, and let )C =_ X(k,n) denote the set of all n x k matrices [Equation ommitted] consisting entirely of entries [Equation ommitted] If X maximizes [Equation ommitted] is said to be D-optimum. The problem of characterizing such X arises in two statistical settings, both with uncorrelated homoscedastic observations. In both cases [Equation ommitted] is proportional to the generalized variance of the least squares estimators of the parameters 81~ 92"'" ek of interest. Firstly, there is the setting of finding the weights [Equation ommitted] of k objects with n wieghings. In one model, in which a chemical balance is used with each object present on each weighing, we let [Equation ommitted] depending on whether the j-th object is on-the left or right pan in the ith weighing. That weighing model may be altered to allow the [Equation ommitted] i. e. all k objects need not be present in each weighing. It can easily be shown that every X optimum for the previous model is optimum for this one. Also, when [Equation ommitted] the optimality results for [Equation ommitted] are well-known to correspond to optimality results for [Equation ommitted] or 1, the "spring-balance" model; see Mood (1946). The equivalences of the various D-optimality problems for the two settings is also treated by Hedayat and Wallis (1978), when k = n. Secondly, there is the setting of estimating the para-meters of the first order regression model on the p-dimensional cube [-1,1]p with [Equation ommitted] , the ith observation being at [Equation ommitted] with expectation [Equation ommitted] which we can write [Equation ommitted] by defining [Equation ommitted] It can easily be shown that there is a D-optimum X in x.. Conversely, each X in N can be transformed into an element of X, with the same determinant and all [Equation ommitted] if [-1, 1] p is replaced by (-1, 1)p in the above, we obtain. the even simpler correspondence of the weighing problem to the first order (resolution III) fractional 2p-factorial problem.

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Page 1 of 13

THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Computing D-Optimum Weighing Designs where Statistics, Combinatorics, and Computation Meet

Zvi Galil* Tel-Aviv University and Columbia University

August 1983

*Research supported by NSF Grant MCS-83-03139. This paper was also delivered at a Statistical Research Conference dedicated to the memory of Jack Kiefer and Jacob Wolfowitz. The converence was hald at Cornell University, July 6-9, 1983. It was a special meeting of the Institute of Mathe-matical Statistics cosponsored by the American Statistical Association. Financial support was provided by the National Science Foundation, Office of Naval Research the Army Research Office--Durham and units of Cornell University.

Abstract:

This paper surveys results and techniques for computing D-optimum weighing designs.

1. Introduction. In this paper we survey results and techniques for computing D-optimum weighing designs. The paper summarizes the results of my work with Jack Kiefer ([1]-[6]), and des-cribes some related results. This paper is written with two hopes in mind. We hope that the work on D-optimum design will be continued. Although we know now much more than before, the picture is far from complete. We also hope that some of the techniques we developed will be found useful elsewhere: for finding optimum designs using other optimality criteria or for solving other optimization problems. Let k and n be positive integers with k S n, and let )C =_ X(k,n) denote the set of all n x k matrices

(Equation Omitted)

consisting entirely of entries

(Equation Omitted)

If X maximizes

(Equation Omitted)

is said to be D-optimum. The problem of characterizing such X arises in two statistical settings, both with uncorrelated homoscedastic observations. In both cases

(Equation Omitted)

is proportional to the generalized variance of the least squares estimators of the parameters 81~ 92"'" ek of interest. Firstly, there is the setting of finding the weights

(Equation Omitted)

Columbia University Page 1 Dec 31, 1983

Page 2 of 13

Computing D-Optimum Weighing Designs where Statistics, Combinatorics, and Computation Meet

of k objects with n wieghings. In one model, in which a chemical balance is used with each object present on each weighing, we let

(Equation Omitted)

depending on whether the j-th object is on-the left or right pan in the ith weighing. That weighing model may be altered to allow the

(Equation Omitted)

i. e. all k objects need not be present in each weighing. It can easily be shown that every X optimum for the previous model is optimum for this one. Also, when

(Equation Omitted)

the optimality results for

(Equation Omitted)

are well-known to correspond to optimality results for

(Equation Omitted)

or 1, the "spring-balance" model; see Mood (1946). The equivalences of the various D- optimality problems for the two settings is also treated by Hedayat and Wallis (1978), when k =
n. Secondly, there is the setting of estimating th...