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Browsing by Author "Robinson, Timothy J."

Now showing 1 - 7 of 7
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    Cost Penalized Estimation and Prediction Evaluation for Split-Plot Designs
    Liang, Li; Anderson-Cook, Christine M.; Robinson, Timothy J. (Virginia Tech, 2005-02-02)
    The use of response surface methods generally begins with a process or system involving a response y that depends on a set of k controllable input variables (factors) x₁, x₂,…,xk. To assess the effects of these factors on the response, an experiment is conducted in which the levels of the factors are varied and changes in the response are noted. The size of the experimental design (number of distinct level combinations of the factors as well as number of runs) depends on the complexity of the model the user wishes to fit. Limited resources due to time and/or cost constraints are inherent to most experiments, and hence, the user typically approaches experimentation with a desire to minimize the number of experimental trials while still being able to adequately estimate the underlying model.
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    Dual Model Robust Regression
    Robinson, Timothy J. (Virginia Tech, 2004-07-30)
    In typical normal theory regression, the assumption of homogeneity of variances is often not appropriate. Instead of treating the variances as a nuisance and transforming away the heterogeneity, the structure of the variances may be of interest and it is desirable to model the variances. Aitkin (1987) proposes a parametric dual model in which a log linear dependence of the variances on a set of explanatory variables is assumed. Aitkin's parametric approach is an iterative one providing estimates for the parameters in the mean and variance models through joint maximum likelihood. Estimation of the mean and variance parameters are interrelatedas the responses in the variance model are the squared residuals from the fit to the means model. When one or both of the models (the mean or variance model) are misspecified, parametric dual modeling can lead to faulty inferences. An alternative to parametric dual modeling is to let the data completely determine the form of the true underlying mean and variance functions (nonparametric dual modeling). However, nonparametric techniques often result in estimates which are characterized by high variability and they ignore important knowledge that the user may have regarding the process. Mays and Birch (1996) have demonstrated an effective semiparametric method in the one regressor, single-model regression setting which is a "hybrid" of parametric and nonparametric fits. Using their techniques, we develop a dual modeling approach which is robust to misspecification in either or both of the two models. Examples will be presented to illustrate the new technique, termed here as Dual Model Robust Regression.
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    Graphical Tools, Incorporating Cost and Optimizing Central Composite Designs for Split-Plot Response Surface Methodology Experiments
    Liang, Li (Virginia Tech, 2005-03-28)
    In many industrial experiments, completely randomized designs (CRDs) are impractical due to restrictions on randomization, or the existence of one or more hard-to-change factors. Under these situations, split-plot experiments are more realistic. The two separate randomizations in split-plot experiments lead to different error structure from in CRDs, and hence this affects not only response modeling but also the choice of design. In this dissertation, two graphical tools, three-dimensional variance dispersion graphs (3-D VDGs) and fractions of design space (FDS) plots are adapted for split-plot designs (SPDs). They are used for examining and comparing different variations of central composite designs (CCDs) with standard, V- and G-optimal factorial levels. The graphical tools are shown to be informative for evaluating and developing strategies for improving the prediction performance of SPDs. The overall cost of a SPD involves two types of experiment units, and often each individual whole plot is more expensive than individual subplot and measurement. Therefore, considering only the total number of observations is likely not the best way to reflect the cost of split-plot experiments. In this dissertation, cost formulation involving the weighted sum of the number of whole plots and the total number of observations is discussed and the three cost adjusted optimality criteria are proposed. The effects of considering different cost scenarios on the choice of design are shown in two examples. Often in practice it is difficult for the experimenter to select only one aspect to find the optimal design. A realistic strategy is to select a design with good balance for multiple estimation and prediction criteria. Variations of the CCDs with the best cost-adjusted performance for estimation and prediction are studied for the combination of D-, G- and V-optimality criteria and each individual criterion.
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    Long-term population dynamics of dreissenid mussels (Dreissena polymorpha and D. rostriformis): a cross-system analysis
    Strayer, David L.; Adamovich, Boris, V.; Adrian, Rita; Aldridge, David C.; Balogh, Csilla; Burlakova, Lyubov E.; FriedPetersen, Hannah B.; G-Toth, Laszlo; Hetherington, Amy L.; Jones, Thomas S.; Karatayev, Alexander Y.; Madill, Jacqueline B.; Makarevich, Oleg A.; Marsden, J. Ellen; Martel, Andre L.; Minchin, Dan; Nalepa, Thomas F.; Noordhuis, Ruurd; Robinson, Timothy J.; Rudstam, Lars G.; Schwalb, Astrid N.; Smith, David R.; Steinman, Alan D.; Jeschke, Jonathan M. (Ecological Society of America, 2019-04)
    Dreissenid mussels (including the zebra mussel Dreissena polymorpha and the quagga mussel D. rostriformis) are among the world's most notorious invasive species, with large and widespread ecological and economic effects. However, their long-term population dynamics are poorly known, even though these dynamics are critical to determining impacts and effective management. We gathered and analyzed 67 long-term (>10 yr) data sets on dreissenid populations from lakes and rivers across Europe and North America. We addressed five questions: (1) How do Dreissena populations change through time? (2) Specifi- cally, do Dreissena populations decline substantially after an initial outbreak phase? (3) Do different measures of population performance (biomass or density of settled animals, veliger density, recruitment of young) follow the same patterns through time? (4) How do the numbers or biomass of zebra mussels or of both species combined change after the quagga mussel arrives? (5) How does body size change over time? We also considered whether current data on long-term dynamics of Dreissena populations are adequate for science and management. Individual Dreissena populations showed a wide range of temporal dynamics, but we could detect only two general patterns that applied across many populations: (1) Populations of both species increased rapidly in the first 1-2 yr after appearance, and (2) quagga mussels appeared later than zebra mussels and usually quickly caused large dedines in zebra mussel populations. We found little evidence that combined Dreissena populations declined over the long term. Different measures of population performance were not congruent; the temporal dynamics of one life stage or population attribute cannot generally be accurately inferred from the dynamics of another. We found no consistent patterns in the long-term dynamics of body size. The long-term dynamics of Dreissena populations probably are driven by the ecological characteristics (e.g., predation, nutrient inputs, water temperature) and their temporal changes at individual sites rather than following a generalized time course that applies across many sites. Existing long-term data sets on dreissenid populations, although dearly valuable, are inadequate to meet research and management needs. Data sets could be improved by standardizing sampling designs and methods, routinely collecting more variables, and increasing support.
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    Robust Parameter Design: A Semi-Parametric Approach
    Pickle, Stephanie M.; Robinson, Timothy J.; Birch, Jeffrey B.; Anderson-Cook, Christine M. (Virginia Tech, 2005)
    Parameter design or robust parameter design (RPD) is an engineering methodology intended as a cost-effective approach for improving the quality of products and processes. The goal of parameter design is to choose the levels of the control variables that optimize a defined quality characteristic. An essential component of robust parameter design involves the assumption of well estimated models for the process mean and variance. Traditionally, the modeling of the mean and variance has been done parametrically. It is often the case, particularly when modeling the variance, that nonparametric techniques are more appropriate due to the nature of the curvature in the underlying function. Most response surface experiments involve sparse data. In sparse data situations with unusual curvature in the underlying function, nonparametric techniques often result in estimates with problematic variation whereas their parametric counterparts may result in estimates with problematic bias. We propose the use of semi-parametric modeling within the robust design setting, combining parametric and nonparametric functions to improve the quality of both mean and variance model estimation. The proposed method will be illustrated with an example and simulations.
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    A Semiparametric Approach to Dual Modeling
    Robinson, Timothy J.; Birch, Jeffrey B.; Starnes, B. Alden (Virginia Tech, 2006)
    In typical normal theory regression, the assumption of homogeneity of variances is often not appropriate. When heteroscedasticity exists, instead of treating the variances as a nuisance and transforming away the heterogeneity, the structure of the variances may be of interest and it is desirable to model the variances. Modeling both the mean and variance is commonly referred to as dual modeling. In parametric dual modeling, estimation of the mean and variance parameters are interrelated. When one or both of the models (the mean or variance model) are misspecified, parametric dual modeling can lead to faulty inferences. An alternative to parametric dual modeling is nonparametric dual modeling. However, nonparametric techniques often result in estimates that are characterized by high variability and ignore important knowledge that the user may have regarding the process. We develop a dual modeling approach [Dual Model Robust Regression (DMRR)], which is robust to user misspecification of the mean and/or variance models. Numerical and asymptotic results illustrate the advantages of DMRR over several other dual model procedures.
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    Semiparametric Techniques for Response Surface Methodology
    Pickle, Stephanie M. (Virginia Tech, 2006-06-28)
    Many industrial statisticians employ the techniques of Response Surface Methodology (RSM) to study and optimize products and processes. A second-order Taylor series approximation is commonly utilized to model the data; however, parametric models are not always adequate. In these situations, any degree of model misspecification may result in serious bias of the estimated response. Nonparametric methods have been suggested as an alternative as they can capture structure in the data that a misspecified parametric model cannot. Yet nonparametric fits may be highly variable especially in small sample settings which are common in RSM. Therefore, semiparametric regression techniques are proposed for use in the RSM setting. These methods will be applied to an elementary RSM problem as well as the robust parameter design problem.