Engineering topics
Publications and source records attributed to Holms, A. G..
Backward elimination procedure minimizes prediction errors of equations fitted to factorial experiments. Problem of minimizing prediction error solved for two-level full- or fractional-factorial experiment with 16 factorial points and 0 to 6 center points. Such experiment provides 16 orthogonal coefficient estimates for empirical polynomial model. Each coefficient estimated with low variance error estimated from 16 observations.
A previous report described a backward deletion procedure of model selection that was optimized for minimum prediction error and which used a multiparameter combination of the F - distribution and an order statistics distribution of Cochran's. A computer program is described that applies the previously optimized procedure to real data. The use of the program is illustrated by examples.
Population model coefficients were chosen to simulate a saturated 2 to the fourth power fixed effects experiment having an unfavorable distribution of relative values. Using random number studies, deletion strategies were compared that were based on the F distribution, on an order statistics distribution of Cochran's, and on a combination of the two. Results of the comparisons and a recommended strategy are given.
Population model coefficients were chosen to simulate a saturated 2 to the 4th fixed-effects experiment having an unfavorable distribution of relative values. Using random number studies, deletion strategies were compared that were based on the F-distribution, on an order statistics distribution of Cochran's, and on a combination of the two. The strategies were compared under the criterion of minimizing the maximum prediction error, wherever it occurred, among the two-level factorial points. The strategies were evaluated for each of the conditions of 0, 1, 2, 3, 4, 5, or 6 center points. Three classes of strategies were identified as being appropriate, depending on the extent of the experimenter's prior knowledge. In almost every case the best strategy was found to be unique according to the number of center points. Among the three classes of strategies, a security regret class of strategy was demonstrated as being widely useful in that over a range of coefficients of variation from 4 to 65%, the maximum predictive error was never increased by more than 12% over what it would have been if the best strategy had been used for the particular coefficient of variation. The relative efficiency of the experiment, when using the security regret strategy, was examined as a function of the number of center points, and was found to be best when the design used one center point.
As many as three iterated statistical model deletion procedures are considered for an experiment. Population model coefficients were chosen to simulate a saturated factorial experiment having an unfavorable distribution of parameter values. Using random number studies, three model selection strategies were developed, namely, (1) a strategy to be used in anticipation of large coefficients of variation (neighborhood of 65 percent), (2) strategy to be used in anticipation of small coefficients of variation (4 percent or less), and (3) a security regret strategy to be used in the absence of such prior knowledge.
Tables are given for the numbers of center points to be used with blocked sequential designs of composite response surface experiments as used in empirical optimum seeking. The tables also give the star point radii for exact orthogonal blocking. The center point options vary from a lower limit of one to an upper limit equal to the numbers proposed by Box and Hunter for approximate rotatability and uniform variance, and exact orthogonal blocking. Some operating characteristics of the proposed options are described.
Tables are given for the numbers of center points to be used with blocked sequential designs of composite response surface experiments as used in empirical optimum seeking. The star point radii for exact orthogonal blocking is presented. The center point options varied from a lower limit of one to an upper limit equal to the numbers proposed by Box and Hunter for approximate rotatability and uniform variance, and exact orthogonal blocking. Some operating characteristics of the proposed options are described.
As many as three iterated statistical model deletion procedures were considered for an experiment. Population model coefficients were chosen to simulate a saturated 2 to the 4th power experiment having an unfavorable distribution of parameter values. Using random number studies, three model selection strategies were developed, namely, (1) a strategy to be used in anticipation of large coefficients of variation, approximately 65 percent, (2) a strategy to be sued in anticipation of small coefficients of variation, 4 percent or less, and (3) a security regret strategy to be used in the absence of such prior knowledge.
A statistical decision procedure called chain pooling had been developed for model selection in fitting the results of a two-level fixed-effects full or fractional factorial experiment not having replication. The basic strategy included the use of one nominal level of significance for a preliminary test and a second nominal level of significance for the final test. The subject has been reexamined from the point of view of using as many as three successive statistical model deletion procedures in fitting the results of a single experiment. The investigation consisted of random number studies intended to simulate the results of a proposed aircraft turbine-engine rotor-burst-protection experiment. As a conservative approach, population model coefficients were chosen to represent a saturated 2 to the 4th power experiment with a distribution of parameter values unfavorable to the decision procedures. Three model selection strategies were developed.
Based on published results on rotor burst containment with single materials, and on body armor using composite materials, a set of hypotheses is established as to what variables might control the design of a weight-efficient protective device. Based on modern concepts for the design and analysis of small optimum seeking experiments, a particular experiment for evaluating the hypotheses and materials was designed. The design and methods for the analysis of results are described.
Published results on rotor burst containment with single materials, and on body armor using composite materials were used to establish a set of hypotheses about what variables might control the design of a weight-efficient protective device. Based on modern concepts for the design and analysis of small optimum seeking experiments, a particular experiment for evaluating the hypotheses and materials was designed. The design and methods for the analysis of results are described. The consequence of such hypotheses is that the device should consist of as many as four concentric rings, each to consist of a material uniquely chosen for its position in the penetration sequence.
The existing theory of subset regression is examined, taking into account optimality criteria, small experiments, nonlinear models, colinearities, and special techniques. Approaches based on chain pooling coupled with principal components regression are discussed, giving attention to a comparison of half-normal plotting with chain pooling, a procedure based on prior ordering, deletion under the F-test, the largest of a set of chi-square variates, and principal components regression and model deletion. The choice of a true (population) model for simulations is considered along with the evaluation of the decision procedure and suitable computer programs.
Monte Carlo studies using population models intended to represent response surface applications are reported. Simulated experiments were generated by adding pseudo random normally distributed errors to population values to generate observations. Model equations were fitted to the observations and the decision procedure was used to delete terms. Comparison of values predicted by the reduced models with the true population values enabled the identification of deletion strategies that are approximately optimal for minimizing prediction errors.
The maximum fuel pin temperature for nominal operation in an advanced power reactor is 1370 K. Because of possible nitrogen embrittlement of the clad, the fuel temperature was limited to 1622 K. Assuming simultaneous occurrence of the most adverse conditions a deterministic analysis gave a maximum fuel temperature of 1610 K. A statistical analysis, using a synthesized estimate of the standard deviation for the highest fuel pin temperature, showed probabilities of 0.015 of that pin exceeding the temperature limit by the distribution free Chebyshev inequality and virtually nil assuming a normal distribution. The latter assumption gives a 1463 K maximum temperature at 3 standard deviations, the usually assumed cutoff. Further, the distribution and standard deviation of the fuel-clad gap are the most significant contributions to the uncertainty in the fuel temperature.
A computer program is described that performs a statistical multiple-decision procedure called chain pooling. It uses a number of mean squares assigned to error variance that is conditioned on the relative magnitudes of the mean squares. The model selection is done according to user-specified levels of type 1 or type 2 error probabilities.
Development of optimal design procedures by posing experimental design problem as finite decision problem
Water radiolysis measurement in nuclear reactor tests, discussing experiment design as doubly telescoping sequences of blocks
Optimal design procedure as experimental design finite decision problem, using Bayes and minimax techniques