Optimal Designs for Quadratic Regression on One Variable and Blocks of Size Two

2002 ◽  
pp. 139-154
Author(s):  
Peter Goos
Keyword(s):  
Optimal Designs ◽  
Quadratic Regression

D-Optimal Designs for Quadratic Regression Models

2000 ◽  
pp. 189-195 ◽  
Author(s):  
E. E. M. van Berkum ◽  
B. Pauwels ◽  
P. M. Upperman
Keyword(s):  
Regression Models ◽  
Optimal Designs ◽  
Quadratic Regression

Optimal designs for quadratic regression

1987 ◽  
Vol 16 ◽  
pp. 213-218
Author(s):  
Franz Preitschopf ◽  
Friedrich Pukelsheim
Keyword(s):  
Optimal Designs ◽  
Quadratic Regression

On optimization methods in the problems of experiment design

2019 ◽  
Vol 85 (1(I)) ◽  
pp. 72-77 ◽  
Author(s):  
S. M. Ermakov ◽  
D. N. Semenchikov
Keyword(s):  
Random Search ◽  
Optimization Methods ◽  
Optimal Designs ◽  
Global Maximum ◽  
Quadratic Regression ◽  
Global Extremum ◽  
Computational Costs ◽  
Wide Range ◽  
Function Domain ◽  
Proposed Modification

A new known modification for simulation of annealing to search the global extremum of the functions of many variables uses the fact that the function whenn→ ¥ converges to the δ-function concentrated at the point of global maximum off(x). The case when the function has many equal extrema is discussed in detail. Problems of this type are often present, particularly in the design of regression experiments. Here we introduce the reader to an extremum search method that is effective in solving a wide range of applied problems, and also illustrate the use of the method in some of the simplest problems of designing the regression experiments. The proposed modification of simulated annealing uses quasi-random search at the intermediate stages. This is not the most rapid, but very reliable method which provide a complete exploring of the function domain. When solving numerical examples, the so-called exactD-optimal designs are constructed, which are very difficult to be obtained by other methods. Although with the increase in the number of variables, the complexity of the method (as well as the complexity of other well-known methods) increases dramatically due to an increase in the order of the determinant, the proposed algorithm is simple, reliable, and easily parallelized. It is known that the gain from using optimal designs in some cases can justify any computational costs of developing those designs. Using the described technique, the reader will be able to construct (even using the laptop capacity) the optimal designs in different areas at moderate values of the parameters (for example, for quadratic regression for s variables in variables fors≤ 10).

Q-optimal experimental designs and close to them experimental designs for polynomial regression on the interval

2020 ◽  
Vol 86 (5) ◽  
pp. 65-72
Author(s):  
Yu. D. Grigoriev
Keyword(s):  
Optimal Design ◽  
Polynomial Regression ◽  
Direct Consequence ◽  
Experimental Designs ◽  
Optimal Designs ◽  
Software Systems ◽  
General Character ◽  
Support Points ◽  
Optimal Weights ◽  
Universal Expression

The problem of constructing Q-optimal experimental designs for polynomial regression on the interval [–1, 1] is considered. It is shown that well-known Malyutov – Fedorov designs using D-optimal designs (so-called Legendre spectrum) are other than Q-optimal designs. This statement is a direct consequence of Shabados remark which disproved the Erdős hypothesis that the spectrum (support points) of saturated D-optimal designs for polynomial regression on a segment appeared to be support points of saturated Q-optimal designs. We present a saturated exact Q-optimal design for polynomial regression with s = 3 which proves the Shabados notion and then extend this statement to approximate designs. It is shown that when s = 3, 4 the Malyutov – Fedorov theorem on approximate Q-optimal design is also incorrect, though it still stands for s = 1, 2. The Malyutov – Fedorov designs with Legendre spectrum are considered from the standpoint of their proximity to Q-optimal designs. Case studies revealed that they are close enough for small degrees s of polynomial regression. A universal expression for Q-optimal distribution of the weights pi for support points xi for an arbitrary spectrum is derived. The expression is used to tabulate the distribution of weights for Malyutov – Fedorov designs at s = 3, ..., 6. The general character of the obtained expression is noted for Q-optimal weights with A-optimal weight distribution (Pukelsheim distribution) for the same problem statement. In conclusion a brief recommendation on the numerical construction of Q-optimal designs is given. It is noted that in this case in addition to conventional numerical methods some software systems of symbolic computations using methods of resultants and elimination theory can be successfully applied. The examples of Q-optimal designs considered in the paper are constructed using precisely these methods.

Constrained optimal designs

1984 ◽  
Author(s):  
Moun-Shen Carl Lee
Keyword(s):  
Optimal Designs
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