|Asymptotically optimal design
Hong Kong Polytechnic University -- Dissertations
|iii, 146 p. : ill. ; 31 cm
|In many industrial and agricultural applications, people often face practical problems that have the following mathematical relationship y = X帣 + 庰 where y are some output variables, dependent on some input variables X and 帣 are some parameters of the relationship, 庰 are some random errors of the function. In order to get the optimal y values, the optimal X values have to be chosen, this is the optimal design problem and it is equivalent to choosing the optimal X values so as to make the estimates of the parameters 帣 as accurate as possible. The objective of this dissertation is to study the optimality of regression experimental design. Iterative procedures are devised to achieve such optimality. Then these procedures are implemented using computer programs and the results of running these programs on some example models are shown and graphed. This is to show that these iterative procedures can find the optimal design asymptotically. There are different regression models such as linear models, mixture models, Fourier models, Bayesian models and other non-linear models etc, but we will focus on 2 of them, namely, linear models and Fourier models. Also, there are different criteria in choosing an optimal design, such as D-optimality (maximizing the determinant of the information matrix), E-optimality (minimizing the largest eigenvalue of the dispersion matrix (the inverse of the information matrix) ) and A-optimality (minimizing the trace of the dispersion matrix), here we will concentrate on D-optimality because D-optimality is the most important one. This dissertation includes some new methods, more specifically, a new optimization method for single independent variable problems which performs better than some existing methods and a new modified steepest ascent method for finding the optimal design which provides a higher convergence rate than the plain steepest ascent method as proposed by Bates(1983).
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