James R. Simpson

A Genetic Algorithm Approach to Multiple-Response Optimization

Many designed experiments require the simultaneous optimization of multiple responses. A common approach is to use a desirability function combined with an optimization algorithm to find the most desirable settings of the controllable factors. However, as the problem grows even moderately in either the number of factors or the number of responses, conventional optimization algorithms can fail to find the global optimum. An alternative approach is to use a heuristic search procedure such as a genetic algorithm (GA). This paper proposes and develops a multiple-response solution technique using a GA in conjunction with an unconstrained desirability function. The GA requires that several parameters be determined in order for the algorithm to operate effectively. We perform a robust designed experiment in order to tune the genetic algorithm to perform well regardless of the complexity of the multiple-response optimization problem. The performance of the proposed GA method is evaluated and compared with the performance of the method that combines the desirability with the generalized reduced gradient (GRG) optimization. The evaluation shows that only the proposed GA approach consistently and effectively solves multiple-response problems of varying complexity.

Statistical Process Monitoring of Nonlinear Profiles Using Wavelets

Many modern industrial processes are capable of generating rich and complex data records that do not readily permit the use of traditional statistical process-control techniques. For example, a “single observation” from a process might consist of n pairs of (x, y) data that can be described as y = f (x) when the process is in control. Such data structures or relationships between y and x are called profiles. Examples of profiles include calibration curves in chemical processing, oxide thickness across wafer surfaces in semiconductor manufacturing, and radar signals of military targets. In this paper, a semiparametric wavelet method is proposed for monitoring for changes in sequences of nonlinear profiles. Based on a likelihood ratio test involving a changepoint model, the method uses the spatial-adaptivity properties of wavelets to accurately detect profile changes taking nearly limitless functional forms. The method is used to differentiate between different radar profiles and its performance is assessed with Monte Carlo simulation. The results presented indicate the method can quickly detect a wide variety of changes from a given, in-control profile.

A comparative analysis of multiple outlier detection procedures in the linear regression model

Abstract We evaluate several published techniques to detect multiple outliers in linear regression using an extensive Monte Carlo simulation. These procedures include both direct methods from algorithms and indirect methods from robust regression estimators. We evaluate the impact of outlier density and geometry, regressor variable dimension, and outlying distance in both leverage and residual on detection capability and false alarm (swamping) probability. The simulation scenarios focus on outlier configurations likely to be encountered in practice and use a designed experiment approach. The results for each scenario provide insight and limitations to performance for each technique. Finally, we summarize each procedures performance and make recommendations.

ESTIMATING THE CHANGE POINT OF A POISSON RATE PARAMETER WITH A LINEAR TREND DISTURBANCE

Knowing when a process changed would simplify the search and identification of the special cause. In this paper, we compare the maximum likelihood estimator (MLE) of the process change point designed for linear trends to the MLE of the process change point designed for step changes when a linear trend disturbance is present. We conclude that the MLE of the process change point designed for linear trends outperforms the MLE designed for step changes when a linear trend disturbance is present. We also present an approach based on the likelihood function for estimating a confidence set for the process change point. We study the performance of this estimator when it is used with a cumulative sum (CUSUM) control chart and make direct performance comparisons with the estimated confidence sets obtained from the MLE for step changes. The results show that better confidence can be obtained using the MLE for linear trends when a linear trend disturbance is present. Copyright

Estimating the Change Point of the Process Fraction Non-conforming with a Monotonic Change Disturbance in SPC

Knowing when a process has changed would simplify the search for and identification of the special cause. In this paper, we propose a maximum-likelihood estimator for the change point of the process fraction non-conforming without requiring knowledge of the exact change type a priori. Instead, we assume the type of change present belongs to a family of monotonic changes. We compare the proposed change-point estimator to the maximum-likelihood estimator for the process change point derived under a simple step change assumption. We do this for a number of monotonic change types and following a signal from a binomial cumulative sum (CUSUM) control chart. We conclude that it is better to use the proposed change point estimator when the type of change present is only known to be monotonic. The results show that the proposed estimator provides process engineers with an accurate and useful estimate of the time of the process change regardless of the type of monotonic change that may be present. Copyright

Response Surface Methods for Efficient Complex Aircraft Configuration Aerodynamic Characterization

A response surface methodology approach to wind-tunnel testing of aircraft with complex configurations is being investigated at the Langley full-scale tunnel as part of a series of tests using design of experiments. An exploratory study was conducted using response surface methodology and a 5% scale blended-wing-body model in an effort to efficiently characterize aerodynamic behavior as a function of attitude and multiple control surface inputs. This paper provides a direct comparison of the design of experiments/response surface methodology and one factor at a time methods for a low-speed wind-tunnel test of a blended-wing-body aircraft configuration with 11 actuated control surfaces. A modified fractional factorial design, augmented with center points and axial points, produced regression models for the characteristic aerodynamic forces and moments over a representative design space as a function of model attitude and control surface inputs. Model adequacy and uncertainty levels were described using robust statistical methods inherent to the response surface methodology practice. Experimental goals included the capture of fundamental stability and control data for simulation models and comparisons to baseline data from recent one factor at a time tests. Optimization is demonstrated for control surface allocation for a desired response. A discussion of highlights and problems associated with the test is included.

CHANGE POINT ESTIMATION FOR MONOTONICALLY CHANGING POISSON RATES IN SPC

Knowing when a process has changed would simplify the search for and identification of the special cause. Although several change point methods have been suggested, many of them rely on the assumption that the effect present in the process output follows some known form (e.g. sudden shift or linear trend). Since processes are often influenced by several input factors, sudden shifts and linear trends do not always adequately describe the true nature of the process behavior. In this paper, we propose a maximum likelihood estimator for the change point of a Poisson rate parameter without requiring exact a priori knowledge regarding the form of the effect present. Instead, we assume the form of effect present can be characterized as belonging to the set of monotonic effects. We compare the proposed change point estimator to the commonly used maximum likelihood estimator for the process change point derived under a sudden and persistent shift assumption. We do this for a number of monotonic effects and following a signal from a Poisson CUSUM control chart. We conclude that it is better to use the proposed change point estimator when the form of the effect present is only known to be monotonic. The results show that the proposed estimator provides process engineers with an accurate and useful estimate of the last observation obtained from the unchanged process regardless of the form of monotonic effect that may be present.

Resampling methods for variable selection in robust regression

With the inundation of large data sets requiring analysis and empirical model building, outliers have become commonplace. Fortunately, several standard statistical software packages have allowed practitioners to use robust regression estimators to easily fit data sets that are contaminated with outliers. However, little guidance is available for selecting the best subset of the predictor variables when using these robust estimators. We initially consider cross-validation and bootstrap resampling methods that have performed well for least-squares variable selection. It turns out that these variable selection methods cannot be directly applied to contaminated data sets using a robust estimation scheme. The prediction errors, inflated by the outliers, are not reliable measures of how well the robust model fits the data.As a result, new resampling variable selection methods are proposed by introducing alternative estimates of prediction error in the contaminated model. We demonstrate that, although robust estimation and resampling variable selection are computationally complex procedures, we can combine both techniques for superior results using modest computational resources. Monte Carlo simulation is used to evaluate the proposed variable selection procedures against alternatives through a designed experiment approach. The experiment factors include percentage of outliers, outlier geometry, bootstrap sample size, number of bootstrap samples, and cross-validation assessment size. The results are summarized and recommendations for use are provided.

A High Performance Aircraft Wind Tunnel Test using Response Surface Methodologies

A Response Surface Methodology (RSM) approach to wind tunnel testing of high performance aircraft is being investigated at the Langley Full-Scale Tunnel (LFST). In an effort to better characterize an aircrafts aerodynamic behavior as a function of attitude and control inputs, and also decrease testing time required, an exploratory study was completed using RSM on a 19 percent scale modified X-31 model. The X-31 model was chosen based on its non-linear aerodynamic behavior at high angle of attack that is representative of modern fighter design and a substantial pre-esisting data base. A five-level nested fractional factorial design, augmented with center points and axial points, produced regression models including pure cubic terms for the characteristic aerodynamic forces and moments over a cuboidal design space as a function of model attitude and control surface inputs. Model adequacy and uncertainty levels were described using robust statistical methods inherent to RSM practice. Comparisons to baseline data and sample lateral-directional and longitudinal aerodynamic characteristic are given.

The development and evaluation of alternative generalized m-estimation techniques

Least squares estimation is a widely used regression technique. The presence of outliers in data can have an adverse effect on least squares estimates, resulting in a model that does not adequately fit the bulk of the data. Robust regression techniques have been proposed as an alternative to least squares when outliers are present. We develop and evaluate new robust regression procedures and compare their performance to the best alternatives currently available in terms of efficiency, breakdown, and bounded influence. We offer the better performing alternatives as possible methods for use in a robust regression scenario.