Split-Plot Designs and Response Surface Designs

Total Page:16

File Type:pdf, Size:1020Kb

Split-Plot Designs and Response Surface Designs Chapter 8: Split-Plot Designs Split-plot designs were originally developed by Fisher (1925,Statistical Methods for Research Workers) for use in agricultural experiments where factors are differentiated with respect to the ease with which they can be changed from experimental run to experimental run. This may be due to the fact that a particular treatment is expensive or time-consuming to change or it may be due to the fact that the experiment is to be run in a large batches and batches can be subdivided later for additional treatments. A split-plot design may be considered as a special case of the two-factor randomized design where one wants to obtain more precise information about one factor and also about the interaction between the two factors. The second factor being of secondary importance to the experimenter. Example 1 Consider a study of the effects of four irrigation methods (A1, A2, A3, and A4) and three fertilizers (B1, B2, and B3). Two replicates are planned to carried out for each of the 12 combinations. A completely randomized design will randomly assign a combination of irrigation and fertilization methods to one of the 24 fields. This may be easy for fertilization but it is hard for irrigation where sprinklers are arranged in a certain way. What is usually done is to use just eight plots of land, chosen so that each can be irrigated at a set level without affecting the irrigation of the others. These large plots are called the whole plots. Randomly assign irrigation levels to each whole plot so that each irrigation level is assigned to exactly two plots. Irrigation is called the whole-plot factor. Divide each whole plot into three plots. Each plot is called a subplot. Within each whole plot, randomly assign the three fertilizers to the three subplots. Fertilizer is called the subplot factor. Note that, in this split-plot design, the first randomization assigns the four irrigation types to eight fields (whole plots), then, the second randomization is conducted to assign the three fertilizers to three subplots within each field. 1 whole plot factor A3 A1 A2 A4 A2 A1 A4 A3 subplot factor subplot B3 B2 B1 B1 B2 B3 B1 B2 B2 B1 B2 B3 B1 B2 B3 B3 B1 B3 B3 B2 B3 B1 B2 B1 Example 2 Box et al. (2005, Statistics for experimenters: Design, inovation, and Discovery, 2nd) described a prototypical split-plot experiment with one easy- to-change factor and one hard-to-change factor. The experiment was designed to study the corrosion resistance of steel bars treated with four coatings, C1, o o o C2, C3, and C4, at three furnance temperatures, 360 C, 370 C, and 380 C. Furnance temperature is the hard-to-change factor because of the time it takes to reset the furnance and reach new equilibrium temperature. Once the equilibrium temperature is reached, four steel bars with randomly assigned coatings C1, C2, C3, and C4 are randomly positioned in the furnance and heated. Here are the data. 2 Temperature Coatings Whole-plot (oC) (randomized order) 1 360 C2 C3 C1 C4 73 83 67 89 2 370 C1 C3 C4 C2 65 87 86 91 3 380 C3 C1 C2 C4 147 155 127 212 4 380 C4 C3 C2 C1 153 90 100 108 5 370 C4 C1 C3 C2 150 140 121 142 6 360 C1 C4 C2 C3 33 54 8 46 Split-Plot Experiments with CRD in Whole Plots (CRSP) Split-plot designs have three main characteristics: 1. The levels of all the factors are not randomly determined and reset for each experimental run. Usually the experimenter holds a factor at a particular setting and then run all the combinations of the other factors. 2. The size of the experimental unit is not the same for all experimental factors. Usually the experimenter applies one factor to a larger unit or group of units involving combinations of the other factors. 3. There is a restriction on the random assignment of the treatment combina- tions to the experimental units. Usually, there is something that prevents assigning the treatments to the units completely randomly. Example 3 Recipes for chocolate and orange cookies include exactly the same ingredients up to the point where the syrup was added to the batch. However, after the cookies were baked, the chocolate cookies had an appealing round and plump appearance, while the orange cookies spread during the baking process and became thin, flat, and unappealing. A factorial experiment was devised to determine if there was a way to change the process of making the orange cookies that would reduce the spreading during baking. The factors that were chosen to be varied were A: the amount of shortening in the dough batch (80% of what the recipe called for or 100%), B: the baking temperature (below, at or above the temperature called for by the recipe), and C: The temperature of the cookies sheet upon which the cookies were placed to be baked (hot out of the oven or cooled to room temperature). A response that could quantify the objective of the experiment was the diameter of the baked cookies. A completely randomized design for this 3 22 factorial experiment with- out replicates needs 12 experimental units. The× experimental units is a batch of cookie doughs. However, the cooking-making process consists of two steps: 3 mixing cookie dough batch and baking cookies. The amount of shortening was a hard-to-change factor because each time it was changed it required making a new batch of cookie dough, while the baking temperature and tray temperature were relative easier to vary. Therefore, an alternative way of designing the experiments would be to follow a two-step plan. First, plan to make four batches of cookie dough and randomly assign two batches to use 80% of the recipe recommended amount of shorten and two batches to recieve the full amount of shortening recommended by the recipe. This represents a completely randomized design in one factor and the experimental unit is a batch of cookie dough. This first step is called the whole plot design and the whole-plot factor is A: the amount of shortening. Next, bake six trays of cookies from each batch of dough and completely ran- domized the six combinations of bake temperature and tray temperature to the six trays of cookies within each batch. This is a randomized block 3 2 facto- rial within each batch of cookie dough and is called the sub-plot design× . The sub-plot block is the batch of cookies, the sub-plot experimental unit is a tray of cookies and the sub-plot factors are B: the bake temperature and C: the tray temperature. The combination of the whole-plot and sub-plot design is called a split-plot design with CRD in the whole plots or CRSP. By designing the experiment in this way only four batches of cookies are required in tead of 12 and there are replicate whole-plot and sub-plot experimental units from which variances of two experimental errors can be estimated. It will be showns that there is less power for testing the shortening effect than there would be with the 12 batch completely randomized designed, but there is actually more power for detecting the baking temperature, tray temperature, and interaction due to the blocking by batch. Here are more examples from Kowalski, S.M. and Potcner, K.J. (2003, How to recognize a split-plot experiment, Quality Progress, 36(11), 60-66). In the printing press process, blanket type is the hard-to-change factor. 4 The Printing Press In the experiment of water resistance property of wood, pretreatment type is the hard-to-change factor. 5 Water Resistance Property of Wood In the baking process of plastic strength, oven temperature is the hard-to-change factor. 6 Strength of Plastic in Baking Process Chapter 10. Response Surface Methodology Response surface methodology (RSM) is a collection of statistical and mathe- matical techniques useful for developing, improving, and optimizing processes. It is also important applications in the design, development, and formulation of new products, as well as in the improvement of existing product designs. Response surface methodology was developed by Box and Wilson (1951, On the experimental attainment of optimum conditions, JRSS-B, 13, 1-45) to aid the improvement of manufacturing processes in the chemical industry. The purpose was to optimize chemical reactions to obtain, for example, high yield and purity at low cost. This was accomplished through the use of sequential experimentation involving factors such as temperature, pressure, duration of reaction, and proportion of reactants. The same methodology can be used to model or optimize any response that is affected by the levels of one or more quantitative factors. Response surface methods generally refer to a complete package of statistical design and analysis tools that are used for the following steps. 1. Design and collection of data to fit an equation to approximate the rela- tionship between factors and responses 7 2. Regression analysis to fit a model to describe the data 3. Examination of the fitted relationship through graphical and numerical techniques. The most extensive applications of RSM are in the industrial world, par- ticularly in situations where several input variables potentially influence some performance measure or quality characteristic of the product or process. The general scenario is as follows. The response is a quantitative continuous vari- able (e.g., yield, purity, cost), and the mean response is a smooth but unknown function of the levels of p factors (e.g., temperature, pressure), and the levels are real-valued and accurately controllable.
Recommended publications
  • Recommendations for Design Parameters for Central Composite Designs with Restricted Randomization
    Recommendations for Design Parameters for Central Composite Designs with Restricted Randomization Li Wang Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics Geoff Vining, Chair Scott Kowalski, Co-Chair John P. Morgan Dan Spitzner Keying Ye August 15, 2006 Blacksburg, Virginia Keywords: Response Surface Designs, Central Composite Design, Split-plot Designs, Rotatability, Orthogonal Blocking. Copyright 2006, Li Wang Recommendations for Design Parameters for Central Composite Designs with Restricted Randomization Li Wang ABSTRACT In response surface methodology, the central composite design is the most popular choice for fitting a second order model. The choice of the distance for the axial runs, α, in a central composite design is very crucial to the performance of the design. In the literature, there are plenty of discussions and recommendations for the choice of α, among which a rotatable α and an orthogonal blocking α receive the greatest attention. Box and Hunter (1957) discuss and calculate the values for α that achieve rotatability, which is a way to stabilize prediction variance of the design. They also give the values for α that make the design orthogonally blocked, where the estimates of the model coefficients remain the same even when the block effects are added to the model. In the last ten years, people have begun to realize the importance of a split-plot structure in industrial experiments. Constructing response surface designs with a split-plot structure is a hot research area now. In this dissertation, Box and Hunters’ choice of α for rotatablity and orthogonal blocking is extended to central composite designs with a split-plot structure.
    [Show full text]
  • Interventions for Controlling Myopia Onset and Progression Report
    Special Issue IMI – Interventions for Controlling Myopia Onset and Progression Report Christine F. Wildsoet,1 Audrey Chia,2 Pauline Cho,3 Jeremy A. Guggenheim,4 Jan Roelof Polling,5,6 Scott Read,7 Padmaja Sankaridurg,8 Seang-Mei Saw,9 Klaus Trier,10 Jeffrey J. Walline,11 Pei-Chang Wu,12 and James S. Wolffsohn13 1Berkeley Myopia Research Group, School of Optometry and Vision Science Program, University of California Berkeley, Berkeley, California, United States 2Singapore Eye Research Institute and Singapore National Eye Center, Singapore 3School of Optometry, The Hong Kong Polytechnic University, Hong Kong 4School of Optometry and Vision Sciences, Cardiff University, Cardiff, United Kingdom 5Erasmus MC Department of Ophthalmology, Rotterdam, The Netherlands 6HU University of Applied Sciences, Optometry and Orthoptics, Utrecht, The Netherlands 7School of Optometry and Vision Science and Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, Australia 8Brien Holden Vision Institute and School of Optometry and Vision Science, University of New South Wales, Sydney, Australia 9Saw Swee Hock School of Public Health, National University of Singapore, Singapore 10Trier Research Laboratories, Hellerup, Denmark 11The Ohio State University College of Optometry, Columbus, Ohio, United States 12Department of Ophthalmology, Kaohsiung Chang Gung Memorial Hospital and Chang Gung University College of Medicine, Kaohsiung, Taiwan 13Ophthalmic Research Group, Aston University, Birmingham, United Kingdom Correspondence: Christine F. Wild- Myopia has been predicted to affect approximately 50% of the world’s population based soet, School of Optometry, Universi- on trending myopia prevalence figures. Critical to minimizing the associated adverse ty of California Berkeley, 588 Minor visual consequences of complicating ocular pathologies are interventions to prevent or Hall, Berkeley, CA 94720-2020, USA; delay the onset of myopia, slow its progression, and to address the problem of mechanical [email protected].
    [Show full text]
  • The Usefulness of Optimal Design for Generating Blocked and Split-Plot Response Surface Experiments Peter Goos
    The Usefulness of Optimal Design for Generating Blocked and Split-Plot Response Surface Experiments Peter Goos Universiteit Antwerpen Abstract This article provides an overview of the recent literature on the design of blocked and split-plot experiments with quantitative experimental variables. A detailed literature study introduces the ongoing debate between an optimal design approach to constructing blocked and split-plot designs and approaches where the equivalence of ordinary least squares and generalized least squares estimates are envisaged. Examples where the competing approaches lead to totally different designs are given, as well as examples in which the proposed designs satisfy both camps. Keywords: equivalence of OLS and GLS, ordinary least squares, generalized least squares, orthogonal blocking, D-optimal design 1 Introduction Blocked experiments are typically used when not all the runs of an experiment can be conducted under homogeneous circumstances. In such experiments, the runs are par- titioned in blocks so that the within-block variability is considerably smaller than the between-block variability. Split-plot experiments are used for a different reason, that is because the levels of some of the experimental variables are not independently reset for every experimental run as this is often impractical, costly or time-consuming. As a con- sequence, the runs of a split-plot experiment are also partitioned in groups, and within each group the levels of the so-called hard-to-change variables are constant. Split-plot experiments can thus be interpreted as special cases of blocked experiments. Despite the similarity between the two types of experiments, the terminology utilized in the context of blocked and split-plot experiments is different: the groups of observations in a blocked experiment are called blocks, whereas those in split-plot experiments are named whole plots.
    [Show full text]
  • Experiment Design and Analysis Reference
    Experiment Design & Analysis Reference ReliaSoft Corporation Worldwide Headquarters 1450 South Eastside Loop Tucson, Arizona 85710-6703, USA http://www.ReliaSoft.com Notice of Rights: The content is the Property and Copyright of ReliaSoft Corporation, Tucson, Arizona, USA. It is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. See the next pages for a complete legal description of the license or go to http://creativecommons.org/licenses/by-nc-sa/4.0/legalcode. Quick License Summary Overview You are Free to: Share: Copy and redistribute the material in any medium or format Adapt: Remix, transform, and build upon the material Under the following terms: Attribution: You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. See example at http://www.reliawiki.org/index.php/Attribution_Example NonCommercial: You may not use this material for commercial purposes (sell or distribute for profit). Commercial use is the distribution of the material for profit (selling a book based on the material) or adaptations of this material. ShareAlike: If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. Generation Date: This document was generated on April 29, 2015 based on the current state of the online reference book posted on ReliaWiki.org. Information in this document is subject to change without notice and does not represent a commitment on the part of ReliaSoft Corporation.
    [Show full text]
  • Graphical Tools, Incorporating Cost and Optimizing Central Composite Designs for Split- Plot Response Surface Methodology Experiments
    GRAPHICAL TOOLS, INCORPORATING COST AND OPTIMIZING CENTRAL COMPOSITE DESIGNS FOR SPLIT- PLOT RESPONSE SURFACE METHODOLOGY EXPERIMENTS by Li Liang Dissertation submitted to the faculty of the Virginia Polytechnic Institute and State University in fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics C.M. Anderson-Cook, Co-Chair T.J. Robinson, Co-Chair E.P. Smith G.G. Vining K. Ye March 28, 2005 Blacksburg, Virginia Keywords: 3-dimensional variance dispersion graph, fraction of design space, cost penalized evaluation, multiple criteria, optimal factorial levels, central composite structure GRAPHICAL TOOLS, INCORPORATING COST AND OPTIMIZING CENTRAL COMPOSITE DESIGNS FOR SPLIT- PLOT RESPONSE SURFACE METHODOLOGY EXPERIMENTS Li Liang (Abstract) 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.
    [Show full text]
  • Design and Analysis of Experiments Course Notes for STAT 568
    Design and Analysis of Experiments Course notes for STAT 568 Adam B Kashlak Mathematical & Statistical Sciences University of Alberta Edmonton, Canada, T6G 2G1 March 27, 2019 cbna This work is licensed under the Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-nc-sa/4.0/. Contents Preface1 1 One-Way ANOVA2 1.0.1 Terminology...........................2 1.1 Analysis of Variance...........................3 1.1.1 Sample size computation....................6 1.1.2 Contrasts.............................7 1.2 Multiple Comparisons..........................8 1.3 Random Effects..............................9 1.3.1 Derivation of the F statistic................... 11 1.4 Cochran's Theorem............................ 12 2 Multiple Factors 15 2.1 Randomized Block Design........................ 15 2.1.1 Paired vs Unpaired Data.................... 17 2.1.2 Tukey's One DoF Test...................... 18 2.2 Two-Way Layout............................. 20 2.2.1 Fixed Effects........................... 20 2.3 Latin Squares............................... 21 2.3.1 Graeco-Latin Squares...................... 23 2.4 Balanced Incomplete Block Designs................... 24 2.5 Split-Plot Designs............................ 26 2.6 Analysis of Covariance.......................... 30 3 Multiple Testing 33 3.1 Family-wise Error Rate......................... 34 3.1.1 Bonferroni's Method....................... 34 3.1.2 Sidak's Method.......................... 34 3.1.3 Holms' Method.......................... 35 3.1.4 Stepwise Methods........................ 35 3.2 False Discovery Rate........................... 36 3.2.1 Benjamini-Hochberg Method.................. 37 4 Factorial Design 39 4.1 Full Factorial Design........................... 40 4.1.1 Estimating effects with regression................ 41 4.1.2 Lenth's Method.......................... 43 4.1.3 Key Concepts..........................
    [Show full text]
  • Visual Performance of Center-Distance Multifocal Contact Lenses Fit Using a Myopia Control Paradigm
    VISUAL PERFORMANCE OF CENTER-DISTANCE MULTIFOCAL CONTACT LENSES FIT USING A MYOPIA CONTROL PARADIGM By Hannah R. Gregory, BS A thesis submitted to the Graduate Program, University of Houston College of Optometry In partial fulfillment of the requirements for the degree of MASTER OF SCIENCES In PHYSIOLOGICAL OPTICS & VISION SCIENCE Chair of Committee: Eric R. Ritchey, OD, PhD Co-chair of Committee: David A. Berntsen, OD, PhD Committee Member: Han Cheng, PhD, OD University of Houston May 2021 Acknowledgments I would like to thank my advisor, Dr. Eric Ritchey, for his many hours of teaching and directing me throughout this entire process. His passion for teaching and research is evident in all that he does. He welcomed me into his new lab when it first began and made sure I understood what it really meant to be a researcher, explaining how it is relevant to the optometric world, especially in the field of myopia progression. My research project would not be where it is without all of his help. He always made himself available when I needed help, especially when aiming for deadlines throughout this process. His guidance truly pushed me to become the professional I am today. I am also so thankful for the support of his wife, Dr. Moriah Chandler, and his children. I would also like to thank my co-advisor, Dr. David Berntsen, for his support and help throughout my schooling. He also spent countless hours, aiding in analyzing data and explaining it in a way that I could understand to utilize in my project. His dedication to the field of myopia management is inspiring and has made me want to pursue this throughout my career.
    [Show full text]
  • Fitting Success for Three Multifocal Designs Multicentre Randomised Trial
    Contact Lens and Anterior Eye xxx (xxxx) xxx–xxx Contents lists available at ScienceDirect Contact Lens and Anterior Eye journal homepage: www.elsevier.com/locate/clae Fitting success for three multifocal designs: Multicentre randomised trial ⁎ Emilia Novillo-Díaza, , César Villa-Collarb, Mónica Narváez-Peñac, José Luis R. Martínc a María Cristina Optometry Center, Tomelloso, Ciudad Real, Spain b Universidad Europea de Madrid, Madrid, Spain c Simplifying Research Institute, Layos, Toledo, Spain ARTICLE INFO ABSTRACT Keywords: Purpose: To determine if the discontinuation of commercially-available simultaneous vision Multifocal Soft Multifocal design Contact Lenses is independent from the multifocal design. To determine causes for discontinuation and psy- Simultaneous vision chosocial factors involved. Discontinuation Methods: Multicentre single-blinded randomised controlled trial with external blinded evaluation for a three Psychosocial factors months follow-up period for three intervention groups. 150 single-vision soft wearers were randomly assigned a Quality of life spherical near centred lens (S-CN), distance centred lens (CD) or aspherical near centred lens (A-CN). Cases of discontinuation, anxiety and quality of life were measured at one week and one month. Results: 120 females and 30 males were included with an age range of 40–62 (48.79 ± 5.23). At one month, the S-CN design had a statistically significant higher risk of discontinuation than the other two OR: 6.12 (95%CI 2.5–14.9). Twenty-eight subjects discontinued wearing S-CN at first week (56%), while discontinuation of CD and A-CD were 15 (30%) and 11 (22%), with a statistically significant difference between S-CN design and the other two (p = 0.001).
    [Show full text]
  • Spherical Designs and Non-Convex Minimization for Recovery of Sparse Signals on the Sphere∗
    SPHERICAL DESIGNS AND NON-CONVEX MINIMIZATION FOR RECOVERY OF SPARSE SIGNALS ON THE SPHERE∗ XIAOJUN CHENy AND ROBERT S. WOMERSLEYzz Abstract. This paper considers the use of spherical designs and non-convex minimization for 2 recovery of sparse signals on the unit sphere S . The available information consists of low order, 2 potentially noisy, Fourier coefficients for S . As Fourier coefficients are integrals of the product of a function and spherical harmonics, a good cubature rule is essential for the recovery. A spherical 2 t-design is a set of points on S , which are nodes of an equal weight cubature rule integrating exactly all spherical polynomials of degree ≤ t. We will show that a spherical t-design provides a sharp error bound for the approximation signals. Moreover, the resulting coefficient matrix has orthonormal 2 rows. In general the `1 minimization model for recovery of sparse signals on S using spherical harmonics has infinitely many minimizers, which means that most existing sufficient conditions for sparse recovery do not hold. To induce the sparsity, we replace the `1-norm by the `q-norm (0 < q < 1) in the basis pursuit denoise model. Recovery properties and optimality conditions are discussed. Moreover, we show that the penalty method with a starting point obtained from the re- weighted `1 method is promising to solve the `q basis pursuit denoise model. Numerical performance on nodes using spherical t-designs and t-designs (extremal fundamental systems) are compared with tensor product nodes. We also compare the basis pursuit denoise problem with q = 1 and 0 < q < 1.
    [Show full text]
  • Design and Analysis of Response Surface Designs with Restricted Randomization Wayne R
    Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2006 Design and Analysis of Response Surface Designs with Restricted Randomization Wayne R. Wesley Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING DESIGN AND ANALYSIS OF RESPONSE SURFACE DESIGNS WITH RESTRICTED RANDOMIZATION By Wayne R. Wesley A Dissertation submitted to the Department of Industrial Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2006 Copyright © 2006 Wayne R. Wesley All Rights Reserved The members of the Committee approve the dissertation of Wayne R. Wesley defended on July 6, 2006. ______________________________ James R. Simpson Professor Directing Dissertation ______________________________ Anuj Srivastava Outside Committee Member ______________________________ Peter A. Parker Outside Committee Member ______________________________ Joseph J. Pignatiello, Jr. Committee Member Approved: _________________________________ Chuck Zhang, Chair, Department of Industrial and Manufacturing Engineering _________________________________ Ching-Jen Chen, Dean, College of Engineering The Office of Graduate Studies has verified and approved the above named committee members. ii I dedicate this dissertation to my son and daughter Jowayne and Jowaynah Wesley. iii ACKNOWLEDGEMENTS I must first give thanks to the almighty God for good health, strength and wisdom to successfully complete my dissertation. It is with a deep sense of gratitude that I express my sincere appreciation to Dr. James Simpson and Dr. Peter Parker for their invaluable advice and contributions in directing the dissertation. Special thanks to Dr. Joseph Pignatiello and Dr. Anuj Srivastava for their meaningful comments and suggestions that helped to improve the quality of the dissertation.
    [Show full text]
  • 2004/12/31-Sandia Report No. SAND2004-6258, "Guidance On
    SANDIA REPORT SAND2004-6258 Unlimited Release Printed December 2004 Guidance on Risk Analysis and Safety Implications of a Large Liquefied Natural Gas (LNG) Spill Over Water Mike Hightower, Louis Gritzo, Anay Luketa-Hanlin, John Covan, Sheldon Tieszen, Gerry Wellman, Mike Irwin, Mike Kaneshige, Brian Melof, Charles Morrow, Don Ragland Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. Approved for public release; further dissemination unlimited. Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government, nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represent that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof, or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof, or any of their contractors.
    [Show full text]
  • Nonparametric Regression for Spherical Data
    Nonparametric regression for spherical data Marco Di Marzio Agnese Panzera DMQTE, DiSIA, Universit`adi Chieti-Pescara, Universit`adi Firenze, Viale Pindaro 42, Viale Morgagni 59, 65127 Pescara, 50134 Firenze, Italy. Italy. email: [email protected] email: [email protected] Charles C. Taylor Department of Statistics, University of Leeds, Leeds LS2 9JT, UK email: [email protected] October 21, 2013 1 Authors’ Footnote: Marco Di Marzio is Associate Professor of Statistics, DMQTE, University of Chieti-Pescara, Italy. Agnese Panzera is researcher in Statistics at DISIA, University of Florence, Italy. Charles Taylor is Professor of Statistics, School of Mathematics, University of Leeds, Leeds LS2 9JT (E-mail: [email protected]). We are grateful to an Associate Editor and two anonymous Referees for a careful reading of the manuscript and helpful comments, which led to an improved version of this article. We also thank Isabella Fabbri for having read and discussed some parts of the manuscript. 2 Abstract We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression.
    [Show full text]