- Department of Mathematical Sciences
Montana State University
Bozeman, MT 59717 - +01 406 994 4606
John Borkowski
Montana State University - Bozeman, Mathematical Sciences, Department Member
- Thammasat University, Mathematics and Statistics, Department Memberadd
- I am a Professor of Statistics with instructional emphases in experimental design, response surface methodology, stat... moreI am a Professor of Statistics with instructional emphases in experimental design, response surface methodology, statistical quality control, and sampling. My recent research foci have been in the areas of optimal design, space-filling designs, mixture experiments, and spatial sampling plans. I also enjoy multidisciplinary collaborations and consulting.edit
- James Lucasedit
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The purpose of this study was to examine Principal Component Sizing System (PCSS) methodology as an alternative approach to advancing the mathematical efficiency and effectiveness of apparel sizing for women 55 and older (Salusso-Deonier,... more
The purpose of this study was to examine Principal Component Sizing System (PCSS) methodology as an alternative approach to advancing the mathematical efficiency and effectiveness of apparel sizing for women 55 and older (Salusso-Deonier, 1982). The 1994 American Societyfor Testing and Materials (ASTM) national body measurement databasefor Women 55 and Older was classified using the PCSS method (ASTM, 2001). PCSS-55+ has a thickness-by-length structure similar to the current domestic sizing system. Only 25 sizes were needed to encompass the same range as compared to the 55 sizes within the current sizing standard. The PCSS method correctly classified 95% of subjects within 25 size categories and demonstrates potential as an alternative methodfor creating a simplified and marketable apparel sizing systems. With appropriate methods and databases, revision of US. apparel sizingfor women of all ages can provide long awaited valid and reliable sizing.
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Research Interests: Geography, Population Dynamics, Environmental Monitoring, Bovine Brucellosis Epidemiology, Medicine, and 15 moreMultidisciplinary, Bison, Population, Animals, Animal migration, Research article, PLoS one, Cattle, National Park, HERD, Bayesian model, Ecosystem, Bayes Theorem, Environment, and Epidemiological monitoring
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G-optimal designs are those which minimize the worst-case prediction variance. Thus, such designs are of interest if prediction is a primary component of the post-experiment analysis and decision making. G-optimal designs have not... more
G-optimal designs are those which minimize the worst-case prediction variance. Thus, such designs are of interest if prediction is a primary component of the post-experiment analysis and decision making. G-optimal designs have not attained widespread use in practical applications, in part, because they are difficult to compute. In this paper, we review the last two decades of algorithm development for generating exact G-optimal designs. To date, Particle Swarm Optimization (PSO) has not been applied to construct exact G-optimal designs for small response surface scenarios commonly encountered in industrial settings. We were able to produce improved G-optimal designs for the second-order model and several sample sizes under experiments with K=1,2,3,4, and 5 design factors using an adaptation of PSO. Thereby, we publish updated knowledge on the best-known exact G-optimal designs. We compare computing cost/time and algorithm efficacy to all previous published results including those ge...
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The objective of the research is to study and compare response surface designs: Central composite designs (CCD), Box- Behnken designs (BBD), Small composite designs (SCD), Hybrid designs, and Uniform shell designs (USD) over sets of... more
The objective of the research is to study and compare response surface designs: Central composite designs (CCD), Box- Behnken designs (BBD), Small composite designs (SCD), Hybrid designs, and Uniform shell designs (USD) over sets of reduced models when the design is in a spherical region for 3 and 4 design variables. The two optimality criteria ( D and G ) are considered which larger values imply a better design. The comparison of design optimality criteria of the response surface designs across the full second order model and sets of reduced models for 3 and 4 factors based on the two criteria are presented.
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In this study we presented a new class of probability sampling designs, simple latin cubic sampling +1 sampling designs that were developed from simple latin square sampling designs by focus on three-dimensional, with the specific goals... more
In this study we presented a new class of probability sampling designs, simple latin cubic sampling +1 sampling designs that were developed from simple latin square sampling designs by focus on three-dimensional, with the specific goals of deriving an estimator of the population total, true variances of these estimators, and estimators of these variances. And the Horvitz-Thompson estimation method will be the primary method used to generate these estimator. These designs when compared with simple random, stratified, and systematic sampling will provide estimator with smaller variance for simulation population with spatial correlation and assume that the survey region can be partitioned into three-dimensional grid of d 3 equalized three-dimensional.
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Mixture experiments with the presence of process variables are commonly encountered in the manufacturing industry. The experimenter who plans to conduct mixture experiments in which a process involves the combination of machines, methods,... more
Mixture experiments with the presence of process variables are commonly encountered in the manufacturing industry. The experimenter who plans to conduct mixture experiments in which a process involves the combination of machines, methods, and other resources will try to find condition of design factors which make the product/process insensitive or robust to the variability transmitted into the response variable. We propose the genetic algorithm (GA) for generating robust mixture‐process experimental designs involving control and noise variables. When the noise variables, which are extremely difficult to control or not routinely controlled during the manufacturing process and may change without warning, are considered in a mixture experiment, we propose the robust design setting. When considering a robust design, the design that has a lower and flatter faction of design space curves for all levels of the controllable process variables at varying noise interaction is preferable. We ev...
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In manufacturing processes, a vector of multiple responses is often monitored to assess if the process is in-control. A multivariate Shewhart control chart is one method of monitoring the mean vector. If the chi-square statistic exceeds... more
In manufacturing processes, a vector of multiple responses is often monitored to assess if the process is in-control. A multivariate Shewhart control chart is one method of monitoring the mean vector. If the chi-square statistic exceeds an upper control limit (UCL), then an out-of-control signal occurs. The average run length (ARL) is used to determine the UCL value. ARLs have been estimated under an assumption of a multivariate normal (MVN) assumption. This research explores the sensitivity of ARLs when the MVN assumption is incorrect. ARLs for data from Multivariate t, lognormal, uniform, and beta distributions are estimated and compared to ARLs under the MVN assumption.
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Abstract: We monitored the behavioral responses of bison (Bison bison), elk (Cervus elaphus), and trumpeter swans (Olor buccinator) to motorized winter recreation by repeatedly surveying seven groomed or plowed road ... INTRODUCTION... more
Abstract: We monitored the behavioral responses of bison (Bison bison), elk (Cervus elaphus), and trumpeter swans (Olor buccinator) to motorized winter recreation by repeatedly surveying seven groomed or plowed road ... INTRODUCTION National parks protect ...
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Missing observation is a common problem in scientific and industrial experiments, particularly in a small-scale experiment. They often present significant challenges when experiment repetition is infeasible. In this research, we propose a... more
Missing observation is a common problem in scientific and industrial experiments, particularly in a small-scale experiment. They often present significant challenges when experiment repetition is infeasible. In this research, we propose a multi-objective genetic algorithm as a practical alternative for generating optimal mixture designs that remain robust in the face of missing observation. Our algorithm prioritizes designs that exhibit superior D-efficiency while maintaining a high minimum D-efficiency due to missing observation. The focus on D-efficiency stems from its ability to minimize the impact of missing observations on parameter estimates, ensure reliability across the experimental space, and maximize the utility of available data. We study problems with three mixture components where the experimental region is an irregularly shaped polyhedral within the simplex. Our designs have proven to be D-optimal designs, demonstrating exceptional performance in terms of D-efficiency ...
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Research Interests: Biomass, Biology, Invasive Species, Grasses, Range Management, and 15 moreMontana, Regression Analysis, Mathematical Models, Species Diversity, Species Richness, Endemic species, Introduced species, Forbs, Plant Community, Plant Communities, Equations, Integrated Weed Management, Picloram, Perennial Plant, and Plant density
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Results of the sequential approach for evaluating various forms of the quantitative covariates.
In this paper, new plots, called factorwise variance dispersion graphs (FVDGs) with accompanying coordinate trace plots (CTPs), are introduced. FVDGs display prediction variances throughout the des...
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Nonlinear models pervade the statistical literature on drug development, and specifically in pharmacokinetics (PK), pharmacodynamics (PD), and the biological and physical sciences in general. Obtaining efficient experimental designs for... more
Nonlinear models pervade the statistical literature on drug development, and specifically in pharmacokinetics (PK), pharmacodynamics (PD), and the biological and physical sciences in general. Obtaining efficient experimental designs for such models is non-trivial due to the well-documented parametersensitivity problem. Bayesian methods, which integrate prior information about the model parameters into the design process, have been proposed as a solution to the problem. In implementing such methods, the assumption is made that a single prior distribution exists for the parameters which may not be the case. In this research, we discuss situations in which there may be multiple (or competing) prior distributions and propose a robust design criterion for obtaining efficient designs in such cases.
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The goal of a space-filling design is to uniformly scatter the design points in the experimental region of interest. For mixture experiment designs of reasonable size, as the dimensionality of the experimental region increases,... more
The goal of a space-filling design is to uniformly scatter the design points in the experimental region of interest. For mixture experiment designs of reasonable size, as the dimensionality of the experimental region increases, space-filling design criteria (such as maximin and minimax) place most, if not all, design points at or near the boundary of the constrained region. This article introduces two number-theoretic methods for generating space-filling (specifically uniform) designs for constrained mixture experiments defined by single- and multiple-component constraints. The two methods are illustrated for a simple 3-component mixture problem and a more complicated 16-component waste-glass mixture problem. The uniform scatter of the points in the resulting designs is evaluated using three distance-based criteria.
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Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to... more
Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to determine which units to sample within the study region. Adaptive cluster sampling designs Thompson (1990) are sampling designs that are
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Page 1. Designs of Mixed Resolution for Process Robustness Studies John J. BORKOWSKI Department of Mathematical Sciences Montana State University Bozeman, MT 59717 James M. LUCAS JM Lucas and Associates Wilmington, DE 19808 ...
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For quadratic regression on the hypercube, a single-number criterion, such as a G efficiency that is based on the prediction variance, is often included as one of the criteria when selecting a response surface design, As an alternative to... more
For quadratic regression on the hypercube, a single-number criterion, such as a G efficiency that is based on the prediction variance, is often included as one of the criteria when selecting a response surface design, As an alternative to the single-number-criterion approach, the variance dispersion graph, presented by Giovannitti-Jensen and Myers, is a graphical technique for evaluating prediction-variance properties throughout the experimental region. Three properties of interest are the maximum, minimum, and average spherical prediction variances, given the spherical radius. As an alternative to the computer-based approach requiring an optimization algorithm to evaluate these properties, the maximum, minimum, and spherical prediction variances for central composite and Box–Behnken designs can be determined analytically and are functions only of the radius and the design parameters. These functions yield the exact values of the spherical prediction-variance properties of central composite and Box–Behnke...
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Living archosaurs (crocodilians and birds) share several reproductive features, including hard-shelled eggs 1, parental care 2, 3, assembly-line oviducts 4 and luteal morphology 5. Nevertheless, crocodilians produce many small eggs that... more
Living archosaurs (crocodilians and birds) share several reproductive features, including hard-shelled eggs 1, parental care 2, 3, assembly-line oviducts 4 and luteal morphology 5. Nevertheless, crocodilians produce many small eggs that they ovulate, shell and deposit ...
Research Interests: Reproduction, Biology, Multidisciplinary, Nature, Phylogeny, and 3 moreCretaceous, Mesozoic, and Phanerozoic
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If spatial correlation is present, sampling designs which ensure the sample is well-distributed over the study region will, in general, improve estimation of population abundance relative to designs that do not. Simple latin square... more
If spatial correlation is present, sampling designs which ensure the sample is well-distributed over the study region will, in general, improve estimation of population abundance relative to designs that do not. Simple latin square sampling ±k designs, a new class of probability sampling designs that were developed for this situation, will be presented. When the goal is estimation of the
Research Interests: Statistics, Experimental Design, Stratification, Multivariate Analysis, Sampling, and 11 moreVariance, Sampling design, Random sampling, Spatial Correlation, Covariance analysis, Simple Random Sampling, Variance Analysis, Statistical Regression, Systematic sampling, Stratified sampling, and Probability Distribution
Single value design optimality criteria are often considered when selecting a response surface design. An alternative to a single value criterion is to evaluate prediction variance properties throughout the experimental region and to... more
Single value design optimality criteria are often considered when selecting a response surface design. An alternative to a single value criterion is to evaluate prediction variance properties throughout the experimental region and to graphically display the results in a variance dispersion graph (VDG) (Giovannitti-Jensen and Myers (1989)). Three properties of interest are the spherical average, maximum, and minimum prediction variances. Currently, a computer-intensive optimization algorithm is utilized to evaluate these prediction variance properties. It will be shown that the average, maximum, and minimum spherical prediction variances for central composite designs and Box-Behnken designs can be derived analytically. These three prediction variances can be expressed as functions of the radius and the design parameters. These functions provide exact spherical prediction variance values eliminating the implementation of extensive computing involving algorithms which do not guarantee convergence. This resea...
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Good sample coverage is often an essential component to observational studies of biological species or communities. Whether the goals of the survey center on pattern recognition and prediction or parameter estimation, designs which ensure... more
Good sample coverage is often an essential component to observational studies of biological species or communities. Whether the goals of the survey center on pattern recognition and prediction or parameter estimation, designs which ensure the sample is spread over the region of interest may provide more information than schemes that do not. This additional information could be qualitative in nature; for example, the investigator gleans a deeper understanding of the biological system. Alternatively, good coverage could translate to better precision in estimating parameters associated with the system. We present a new class of spatial sampling designs, simple latin square sampling + 1. Our approach is quadrat-based in that the study region is partitioned into nonoverlapping quadrats or sampling units from which a sample or subset of units is selected. The design falls into the classical sampling framework in that the sample selection probabilities are independent of the underlying variable(s) of interest. We illustrate that estimators generated by the design are generally more efficient than those from simple random samples and certain systematic designs when spatial autocorrelation among units is suspected or known to exist. Further, our design can provide better sample coverage than either of the latter designs.
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The DeGroot model for opinion diffusion over social networks dates back to the 1970s and models the mechanism by which information or disinformation spreads through a network, changing the opinions of the agents. Extensive research exists... more
The DeGroot model for opinion diffusion over social networks dates back to the 1970s and models the mechanism by which information or disinformation spreads through a network, changing the opinions of the agents. Extensive research exists about the behavior of the DeGroot model and its variations over theoretical social networks; however, research on how to estimate parameters of this model using data collected from an observed network diffusion process is much more limited. Existing algorithms require large data sets that are often infeasible to obtain in public health or social science applications. In order to expand the use of opinion diffusion models to these and other applications, we developed a novel genetic algorithm capable of recovering the parameters of a DeGroot opinion diffusion process using small data sets, including those with missing data and more model parameters than observed time steps. We demonstrate the efficacy of the algorithm on simulated data and data from...
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Output from exploratory analyses of the elk model best supported by the data, including odds ratios and their reciprocals.
The DeGroot model for opinion diffusion over social networks dates back to the 1970s and models the mechanism by which information or disinformation spreads through a network, changing the opinions of the agents. Extensive research exists... more
The DeGroot model for opinion diffusion over social networks dates back to the 1970s and models the mechanism by which information or disinformation spreads through a network, changing the opinions of the agents. Extensive research exists about the behavior of the DeGroot model and its variations over theoretical social networks; however, research on how to estimate parameters of this model using data collected from an observed network diffusion process is much more limited. Existing algorithms require large data sets that are often infeasible to obtain in public health or social science applications. In order to expand the use of opinion diffusion models to these and other applications, we developed a novel genetic algorithm capable of recovering the parameters of a DeGroot opinion diffusion process using small data sets, including those with missing data and more model parameters than observed time steps. We demonstrate the efficacy of the algorithm on simulated data and data from...
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Experimentation for achieving a robust process often involves signal variables which are controllable and internal to the process and noise variables which are generally external and routinely uncontrollable. To achieve a robust process,... more
Experimentation for achieving a robust process often involves signal variables which are controllable and internal to the process and noise variables which are generally external and routinely uncontrollable. To achieve a robust process, designs based on a combined array have been suggested by many authors. Many of these designs allow parameter estimation for the linear-quadratic (LQ) response surface model when the experimental design region is the hypercube. The LQ model contains the full quadratic model terms in the Q signal variables, the linear model terms in the L noise variables, and the signal by noise variable interaction terms. Because the quadratic regression model is just a special case of the LQ model when there are L = 0 noise variables, this article extends the optimal design theory regarding regression on the hypercube. An approach similar to that of Farrell, Kiefer, and Walbran [13] will be taken in this article. A support of D- and G-optimal designs for the LQ mode...