Restricted Randomized Two-level Fractional Factorial Designs using Gray Code
|
|
- Mitchell Newton
- 5 years ago
- Views:
Transcription
1 Restricted Randomized Two-level Fractional Factorial Designs using Gray Code Dr. Veena Budhraja Department of Statistics, Sri Venkateswara College, University of Delhi, Delhi Mrs. Puja Thapliyal Assistant Professor, Department of Statistics., University of Delhi ABSTRACT Traditionally, randomization is used to generate sequence of run orders. But in situations like where the factors of interest are correlated with trend or where it is expensive to frequently change the levels of factors in the design, randomization may not give the desired results. Hence one is interested in obtaining the run order in which the factor effects are trend free and the design has minimum level changes. In other words a systematic run order with trend free effects and with minimum number of level changes is preferred. This paper provides a different approach to construct n and n-p fractional factorial designs satisfying both design criteria of trend freeness and minimum level change using GRAY CODE. The designs are further compared with designs given by Coster and Cheng(1988) using the uniformity measures. Keywords: Systematic designs, uniformity, fractional factorial designs. 1. ITRODUCTIO The choice of a good design in order to run an experiment is an important issue in experimental design. There are not only the statistical considerations ( the proposed model, the experimenter s objective etc.) but also practical considerations ( financial limitations, time considerations etc. ) that can influence their choice. Traditionally, the usual advice given to the experimenter is to randomize the run order of the experiments. But sometimes randomizing the run order does not achieve the desired effect of neutralizing the influence of unknown factors or it may lead to designs that may have factor effects aliased with the time trend or the run order may lead to designs that have excessive changes in the factor levels. Therefore it is essential to sequence the runs such that a) Factor effects are orthogonal to the time trend. b) Factors with expensive or difficult to vary levels are minimally varied during an experimentation, The problem of obtaining run order satisfying the properties a) and b) was studied by Cox(1951), Hill(1960). Wilcoxon (1966) dealt with problem (a) whereas Dic kinson (1974) considered (b) for 4 and 5 factorial design. Joiner and Campbell (1976) gave example to emphasize the importance of a) and b). Cheng (1985) and Coster and Cheng (1988) described Generalized Foldover Scheme (GFS) to select run order to achi eve requirements a) and b). Sometimes in practical situations, it is easier for experimenter to choose the factors and interactions of interest than to decide on the defining contrast at the design stage. However, most of the methods (Cheng (1985) and Coster and Cheng (1988)) involve finding defining contrasts under the constraints first and then selecting an appropriate run order. For this reason the methods are not easy to use for experimenter whowant to arrange both experiments and appropriate run orders. The paper presents a method to construct run orders of n and n-p fractional factorial designs with factor effects orthogonal to time trend and minimal level change during an experimentation, using Gray codes. 1 Dr. Veena Budhraja, Mrs. Puja Thapliyal
2 The method of constructionis different from the one given by Coster and Cheng (1988) in the sense that the former first give the defining relationship to construct the run order whereas in the method the run orders are generated first and the defining relationship are determined from the run orders. We further compared the constructed designs with the designs given by Coster and Cheng (1988) using the different uniformity measures e.g. Wrap around Discrepancy, Central discrepancy.. PRELIMIARIES Definition 1: A binary Gray code of length n is a sequence s 0, s 1, s,, s n -1 of the n distinct n-bit strings (or words) of 0 s and 1 s, with the property that the Hamming distance between successive code words is one (where Hamming distance between s i and s i+1 is the number of positions where they differ). Definition : Time Count of a specific effect is calculated as the inner product of the column that corresponds to this effect and the row number. The effects with zero time count are said to have linear trend free (LTF) property. 3. MIIMUM LEVEL CHAGE n, n, FACTORIAL DESIGS One of the most important criteria in an experimental process is the number of level changes for factors if cost/time element is considered in the experiment. The minimum number of factor level changes is obtained when the level of only one factor is changed between two consecutive runs. Thus, in a n design, a minimum number of level changes equals n -1. Such type of designs can easily be obtained using Gray Codes. ( Gilbert(1958) ). METHOD OF COSTRUCTIO Step 1 Let (1) = {0,1} and (1) ={1,0} be the mirror image of (1). Step For, (i) = where (i) denotes the mirror image of (i) and and are null vector and unit vector of order (i-1) 1. Step 3 Assign the factors. to the n columns in (n). Step 4 A n factorial experiment with n-1 + n = n -1 number of level changes which is minimum possible. Let us call these designs as Gray designs. Remark 1.The method given above can be extended easily to obtain q n ; (q>)minimum level change Gray designs.. This technique can be generalised to construct n factorial experiment with n-1 + n- + n = n -1 number of level changes, which is the minimum possible. ILLUSTRATIO 1 In order to construct 3 factorial designs we begin with S (1) = {0,1} then its mirror image (1) ={1,0}. Appending a column of -1 = zeroes and ones to ( (1) (1)) we get () = The mirror image of () is Dr. Veena Budhraja, Mrs. Puja Thapliyal
3 () = Appending a column of = 4 zeroes and 4 ones we get (3), a 3 factorial design given in Table 1. Table 1: 3 factorial design A B C C LC TC The run order obtained in table 1 is 1, a, ab, b, bc, abc, ac, c. We observe that the number of level change be made linear trend free (LTF) using following mapping. Let denotes the mapped factor. Step 3 For the first half of, Replace first zeroes by 0, next zeroes by 1 and next ges for the factors A, B and C are in the order, 1, 0 respectively. The total number of level changes is 7 which is the minimum possible. 4. TRED FREE n FULL FACTORIAL DESIG The method described above can also be used to construct linear trend free full factorial design. METHOD OF COSTRUCTIO Step1 Construct n factorial design with minimum number of level changes using the above method Step The time count (TC) for the n th factor A n is non-zero which has one level change. This factor can zeroes by 0. For the second half of the A n containing n-1 ones, replace first ones by1, next ones by 0 and next ones by 1. We may note that the number of level changes for the new mapped factor will always be 5. Thus, using the method described above, a n factorial design is obtained with all main effects linear trend free and number of level changes is n +3. ILLUSTRATIO Consider the design in Illustration 1. In this run order, the main effect of all factors except 3 rd factor C are linear trend free as the time counts for the factors A and B are zero. To make the factor C linear trend free, we use the 3 Dr. Veena Budhraja, Mrs. Puja Thapliyal
4 mapping as described in Step 3 of the above method. The first half of C contains zeroes and second half contains ones. Thus, the first half of it using Step 3, is replaced by and the second half is replaced by Let C denotes the mapped factor, Then, the new run order is 1, ac, abc, b, bc, ab, a, c. The number of level changes for C is 5 which is also linear trend free and is displayed in last column of Table 1.Thus, a 3 factorial design in which all the main effects are linear trend free and the number of level change is 11, is constructed. 5. TRED FREE n-p FRACTIOAL FACTORIAL DESIG The above procedure yields n factorial design with minimum number of level changes in which (n-1) main effects are linear trend free. We observe here that the interactions involving these (n-1) factors are also linear trend free. ow we construct n-p fractional factorial design with all main effects linear trend free. The procedure to construct such designs is given below. METHOD OF COSTRUCTIO Step 1 Consider a n-p, full factorial design constructed using the above method. The design has (n-p-1) factors that are with LTF property and the (n-p) th factor, a mapped factor which is made LTF through mapping. StepAssign the p factors to the interaction of the first (n-p-1) factors that are having LTF property i.e. a new factor is generated by the interaction involving only unmapped factors, then LC in the new factor will be equal to the sum of the LC of the factors in the interaction i.e. for = ; i,j = 1,...(n-p-1), i j and k = n-p+1,...,n. LC(A k)= LC(A i) + LC(A j) Step 3We get a n-p fractional factorial design in which all main effects are linear trend free. Remark 1. These designs may not be minimum level change designs.. If the interactions, involving mapped factor is assigned to a new factor then LC for the new factor can not be obtained directly. ILLUSTRATIO 3 To construct 4-1 fractional factorial designs, consider the 3 design constructed in Illustration with mapped factor denoted by C. The table below gives the time counts of all two factor interactions along with number of level changes. We observe that the two factor interaction AB is linear trend free (TC = 0). Assigning D= AB, we get a 4-1 resolution III fractional factorial design with defining relation I= ABD, with all main effects linear trend free. Further, the total number of level changes(lc) for the design is 17, where LC(D)= LC(A)+ LC(B) = 6. 4 Dr. Veena Budhraja, Mrs. Puja Thapliyal
5 Table 3: 4-1 Resolution III fractional factorial design A B C D=AB AC BC LC TC Table 4 lists the n-p (4 n 9, 1 p 5) fractional factorial designs with total LC. The designs obtained may not be minimum level change designs but all main effects in the designs generated are having LTF property. Table 4: n-p (4 n 9, 1 p 4) Fractional factorial designs. Design Unmapped Factors/ Mapped Factor Assignment LC for Generated Factor A,B/C D=AB 6 A,B,C/D E=ABC 14 A,B,C,D/ E F=ABCD 30 A,B,C/ D E=ABC,F=BC 14, 6 A,B,C,D,E/ F G=ABCDE 56 A,B,C,D/E G=ABC,F=BCD 8, 14 A,B,C/D G=AB, F=BC, E=AC A,B,C,D,E,F/G H=ABCDEF 16 1, 6, 10 A,B,C,D,E/ F G=ABCD,H=BCDE 60, 30 A,B,C,D,F/E G=ABD, H=BCD, F=ACD 6, 14, A,B,C,D,E,F/ G H=CDEF, J=ABEF 30, 10 A,B,C,D,E/ F G=CDE, J=ABDE H=ABC, 14, 56,54 A,B,C,D/E F=ABC,G=BCD,H=A CD, J=ABD 8,14,,6 5 Dr. Veena Budhraja, Mrs. Puja Thapliyal
6 The resolution of the design depends on the assignment of the factors, which depends on the experimenter s requirement. 6. UIFORMITY In order to compare GRAY design with design of Coster& Cheng(1988) we calculated the measure of uniformity CD and WD. Using analytical formula given byhickernell(1998a,b) and Fang et.al.(00) for both the designs and are given in Table 5.Hickernell (1998b) gave analytical formula for the CD and WD as n k ( CD ( )) 1 n u j kj 05. ukj n u j i ki 0. 5 u ji 0. 5 uki u ji k n 1 k ( WD( )) n 3 u j i ki uji uki uji We compare our designs with the Coster and Cheng(1988) designs. The uniformity measures for both the designs are tabulated in Table 5. Table 5: Uniformity measures CD and WD values for the designs. Gray Designs CC Designs Design CD WD CD WD From Table 5 we observe that the uniformity measure for both Gray and CC designs are same. Hence under the assumption that errors are independent with zero mean and constant variance, both the designs are equally efficient in terms of uniformity. Conclusion An article provides a different approach to construct minimum level change factorial designs and trend free factorial designs, using gray codes. The proposed designs are proven to be equally efficient in terms of uniformity, with designs of Coster and Cheng (1988) designs as evident from Table 5. REFERECES 6 Dr. Veena Budhraja, Mrs. Puja Thapliyal
7 [1] Coster, D.C., and Cheng, C.S Minimum cost trend free run orders of fractional Factorial design. The Annals of Statistics, 16, [] Cox, D.R Some systematic experimental designs. Biometrika, 38, [3] Cheng, C.S Run orders of factorial designs: Proceedings of the Berkeley conference in Honor of Jerzy eyman and Jack Kiefer,Vol.II, pp Lucien M.Le.Cam and R.A. Olishen eds. Wadsworth, Inc. [4] Cheng, C.S. and Steinberg, D.M Trend robust two level factorial designs. Biometrika, 78, [5] Dickinson, A.W Some run orders requiring a minimum numbers of Factor level changes for and main effect plans. Technometrics, 16, [6] Daniel, C. and Wilcoxon, F Factorial p-q plans robust against linear and quadratic trends. Technometrics, 8, [7] Fang, K.T, Ma. C.X., and Mukerjee, R. 00. Uniformity in fractional factorial.in:monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, eds. By Fang KT, Hickernall FJ, iederreiter H, Springer-Verlag Berlin. [8] Hickernell, F.J A generalized discrepancy and Quadrature error bound. Math. comput., 67, 1, [9] Hill, H.H Experimental Designs to adjust for time trends. Technometrics,, [10] Gilbert, E Gray Codes and Paths on the n-cube. Bell System Tech. J 37, [11] Golay, M.J.E otes on Digital Coding. Proceedings IEEE, 37, 657. [1] Joiner, B.L. and Campbell, C Designing experiments when run order is important. Technometrics, 18, [13] Kiefer, J Construction and optimality of generalized youden design. In J..Srivastava, Ed, A survey of statistical design and linear models, orth Holland, Amsterdam. 7 Dr. Veena Budhraja, Mrs. Puja Thapliyal
proposed. This method can easily be used to construct the trend free orthogonal arrays of higher level and higher strength.
International Journal of Scientific & Engineering Research, Volume 5, Issue 7, July-2014 1512 Trend Free Orthogonal rrays using some Linear Codes Poonam Singh 1, Veena Budhraja 2, Puja Thapliyal 3 * bstract
More informationA note on optimal foldover design
Statistics & Probability Letters 62 (2003) 245 250 A note on optimal foldover design Kai-Tai Fang a;, Dennis K.J. Lin b, HongQin c;a a Department of Mathematics, Hong Kong Baptist University, Kowloon Tong,
More informationA technique to construct linear trend free fractional factorial design using some linear codes
International Journal of Statistics and Mathematics Vol. 3(1), pp. 073-081, February, 2016. www.premierpublishers.org, ISSN: 2375-0499x IJSM Research Article A technique to construct linear trend free
More informationOptimal Selection of Blocked Two-Level. Fractional Factorial Designs
Applied Mathematical Sciences, Vol. 1, 2007, no. 22, 1069-1082 Optimal Selection of Blocked Two-Level Fractional Factorial Designs Weiming Ke Department of Mathematics and Statistics South Dakota State
More informationMixture Designs Based On Hadamard Matrices
Statistics and Applications {ISSN 2452-7395 (online)} Volume 16 Nos. 2, 2018 (New Series), pp 77-87 Mixture Designs Based On Hadamard Matrices Poonam Singh 1, Vandana Sarin 2 and Rashmi Goel 2 1 Department
More informationA General Criterion for Factorial Designs Under Model Uncertainty
A General Criterion for Factorial Designs Under Model Uncertainty Steven Gilmour Queen Mary University of London http://www.maths.qmul.ac.uk/ sgg and Pi-Wen Tsai National Taiwan Normal University Fall
More informationMaximal Rank - Minimum Aberration Regular Two-Level Split-Plot Fractional Factorial Designs
Sankhyā : The Indian Journal of Statistics 2007, Volume 69, Part 2, pp. 344-357 c 2007, Indian Statistical Institute Maximal Rank - Minimum Aberration Regular Two-Level Split-Plot Fractional Factorial
More informationE(s 2 )-OPTIMALITY AND MINIMUM DISCREPANCY IN 2-LEVEL SUPERSATURATED DESIGNS
Statistica Sinica 12(2002), 931-939 E(s 2 )-OPTIMALITY AND MINIMUM DISCREPANCY IN 2-LEVEL SUPERSATURATED DESIGNS Min-Qian Liu and Fred J. Hickernell Tianjin University and Hong Kong Baptist University
More informationMoment Aberration Projection for Nonregular Fractional Factorial Designs
Moment Aberration Projection for Nonregular Fractional Factorial Designs Hongquan Xu Department of Statistics University of California Los Angeles, CA 90095-1554 (hqxu@stat.ucla.edu) Lih-Yuan Deng Department
More informationUNIFORM FRACTIONAL FACTORIAL DESIGNS
The Annals of Statistics 2012, Vol. 40, No. 2, 81 07 DOI: 10.1214/12-AOS87 Institute of Mathematical Statistics, 2012 UNIFORM FRACTIONAL FACTORIAL DESIGNS BY YU TANG 1,HONGQUAN XU 2 AND DENNIS K. J. LIN
More informationQUASI-ORTHOGONAL ARRAYS AND OPTIMAL FRACTIONAL FACTORIAL PLANS
Statistica Sinica 12(2002), 905-916 QUASI-ORTHOGONAL ARRAYS AND OPTIMAL FRACTIONAL FACTORIAL PLANS Kashinath Chatterjee, Ashish Das and Aloke Dey Asutosh College, Calcutta and Indian Statistical Institute,
More informationMATH602: APPLIED STATISTICS
MATH602: APPLIED STATISTICS Dr. Srinivas R. Chakravarthy Department of Science and Mathematics KETTERING UNIVERSITY Flint, MI 48504-4898 Lecture 10 1 FRACTIONAL FACTORIAL DESIGNS Complete factorial designs
More informationConstruction of optimal Two- Level Supersaturated Designs
RASHI 1 (2) :41-50 (2016) Construction of optimal Two- Level Supersaturated Designs Basudev Kole and Gourav Kumar Rai Department of Statistics, Mathematics & Computer Application Bihar Agricultural University,
More informationStrategy of Experimentation III
LECTURE 3 Strategy of Experimentation III Comments: Homework 1. Design Resolution A design is of resolution R if no p factor effect is confounded with any other effect containing less than R p factors.
More informationFRACTIONAL FACTORIAL
FRACTIONAL FACTORIAL NURNABI MEHERUL ALAM M.Sc. (Agricultural Statistics), Roll No. 443 I.A.S.R.I, Library Avenue, New Delhi- Chairperson: Dr. P.K. Batra Abstract: Fractional replication can be defined
More informationDefinitive Screening Designs
Definitive Screening Designs Bradley Jones September 2011 Copyright 2008, SAS Institute Inc. All rights reserved. Joint work with Chris Nachtsheim Outline 1. Motivation 2. Design Structure 3. Design Construction
More informationProjection properties of certain three level orthogonal arrays
Metrika (2005) 62: 241 257 DOI 10.1007/s00184-005-0409-9 ORIGINAL ARTICLE H. Evangelaras C. Koukouvinos A. M. Dean C. A. Dingus Projection properties of certain three level orthogonal arrays Springer-Verlag
More informationOptimal Fractional Factorial Plans for Asymmetric Factorials
Optimal Fractional Factorial Plans for Asymmetric Factorials Aloke Dey Chung-yi Suen and Ashish Das April 15, 2002 isid/ms/2002/04 Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi 110
More informationSome optimal criteria of model-robustness for two-level non-regular fractional factorial designs
Some optimal criteria of model-robustness for two-level non-regular fractional factorial designs arxiv:0907.052v stat.me 3 Jul 2009 Satoshi Aoki July, 2009 Abstract We present some optimal criteria to
More informationSome Nonregular Designs From the Nordstrom and Robinson Code and Their Statistical Properties
Some Nonregular Designs From the Nordstrom and Robinson Code and Their Statistical Properties HONGQUAN XU Department of Statistics, University of California, Los Angeles, CA 90095-1554, U.S.A. (hqxu@stat.ucla.edu)
More informationInteraction balance in symmetrical factorial designs with generalized minimum aberration
Interaction balance in symmetrical factorial designs with generalized minimum aberration Mingyao Ai and Shuyuan He LMAM, School of Mathematical Sciences, Peing University, Beijing 100871, P. R. China Abstract:
More informationMixture Designs Based on Hadamard Matrices
Statistics and Applications {ISSN 2452-7395 (online)} Volume 16 Nos. 2, 2018 (New Series), pp 77-87 Mixture Designs Based on Hadamard Matrices Poonam Singh 1, Vandana Sarin 2 and Rashmi Goel 2 1 Department
More informationOptimal Two-Level Regular Fractional Factorial Block and. Split-Plot Designs
Optimal Two-Level Regular Fractional Factorial Block and Split-Plot Designs BY CHING-SHUI CHENG Department of Statistics, University of California, Berkeley, California 94720, U.S.A. cheng@stat.berkeley.edu
More informationCS 147: Computer Systems Performance Analysis
CS 147: Computer Systems Performance Analysis Fractional Factorial Designs CS 147: Computer Systems Performance Analysis Fractional Factorial Designs 1 / 26 Overview Overview Overview Example Preparing
More informationConstruction and analysis of Es 2 efficient supersaturated designs
Construction and analysis of Es 2 efficient supersaturated designs Yufeng Liu a Shiling Ruan b Angela M. Dean b, a Department of Statistics and Operations Research, Carolina Center for Genome Sciences,
More informationTWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS
STAT 512 2-Level Factorial Experiments: Irregular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS A major practical weakness of regular fractional factorial designs is that N must be a
More informationFlorida State University Libraries
Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2006 Efficient Mixed-Level Fractional Factorial Designs: Evaluation, Augmentation and Application Yong
More informationComparison of Re-sampling Methods to Generalized Linear Models and Transformations in Factorial and Fractional Factorial Designs
Journal of Modern Applied Statistical Methods Volume 11 Issue 1 Article 8 5-1-2012 Comparison of Re-sampling Methods to Generalized Linear Models and Transformations in Factorial and Fractional Factorial
More informationActa Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2015
Acta Mathematica Sinica, English Series Jul., 2015, Vol. 31, No. 7, pp. 1163 1170 Published online: June 15, 2015 DOI: 10.1007/s10114-015-3616-y Http://www.ActaMath.com Acta Mathematica Sinica, English
More informationUnit 2 Session - 6 Combinational Logic Circuits
Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums
More informationOptimizations and Tradeoffs. Combinational Logic Optimization
Optimizations and Tradeoffs Combinational Logic Optimization Optimization & Tradeoffs Up to this point, we haven t really considered how to optimize our designs. Optimization is the process of transforming
More informationConstruction of column-orthogonal designs for computer experiments
SCIENCE CHINA Mathematics. ARTICLES. December 2011 Vol. 54 No. 12: 2683 2692 doi: 10.1007/s11425-011-4284-8 Construction of column-orthogonal designs for computer experiments SUN FaSheng 1,2, PANG Fang
More informationReg. No. Question Paper Code : B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER Second Semester. Computer Science and Engineering
Sp 6 Reg. No. Question Paper Code : 27156 B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2015. Second Semester Computer Science and Engineering CS 6201 DIGITAL PRINCIPLES AND SYSTEM DESIGN (Common
More informationp and q are two different primes greater than 25. Pass on the least possible value of p + q.
A1 p and q are two different primes greater than 25. Pass on the least possible value of p + q. A3 A circle has an area of Tπ. Pass on the area of the largest square which can be drawn inside the circle.
More information23. Fractional factorials - introduction
173 3. Fractional factorials - introduction Consider a 5 factorial. Even without replicates, there are 5 = 3 obs ns required to estimate the effects - 5 main effects, 10 two factor interactions, 10 three
More informationComputer Aided Construction of Fractional Replicates from Large Factorials. Walter T. Federer Charles E. McCulloch. and. Steve C.
Computer Aided Construction of Fractional Replicates from Large Factorials by Walter T. Federer Charles E. McCulloch and Steve C. Wang Biometrics Unit and Statistics Center Cornell University Ithaca, NY
More informationAlgorithmic Construction of Efficient Fractional Factorial Designs With Large Run Sizes
Algorithmic Construction of Efficient Fractional Factorial Designs With Large Run Sizes Hongquan Xu Department of Statistics University of California Los Angeles, CA 90095-1554 (hqxu@stat.ucla.edu) February
More informationTotal Time = 90 Minutes, Total Marks = 50. Total /50 /10 /18
University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Midterm Examination Instructor: M. Sachdev October 23rd, 2007 Total Time = 90 Minutes, Total
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analysis of Variance and Design of Experiment-I MODULE IX LECTURE - 38 EXERCISES Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Example (Completely randomized
More informationExtended Binary Linear Codes from Legendre Sequences
Extended Binary Linear Codes from Legendre Sequences T. Aaron Gulliver and Matthew G. Parker Abstract A construction based on Legendre sequences is presented for a doubly-extended binary linear code of
More informationFractional Replications
Chapter 11 Fractional Replications Consider the set up of complete factorial experiment, say k. If there are four factors, then the total number of plots needed to conduct the experiment is 4 = 1. When
More informationDefinitive Screening Designs with Added Two-Level Categorical Factors *
Definitive Screening Designs with Added Two-Level Categorical Factors * BRADLEY JONES SAS Institute, Cary, NC 27513 CHRISTOPHER J NACHTSHEIM Carlson School of Management, University of Minnesota, Minneapolis,
More informationarxiv: v1 [stat.me] 10 Jul 2009
6th St.Petersburg Workshop on Simulation (2009) 1091-1096 Improvement of random LHD for high dimensions arxiv:0907.1823v1 [stat.me] 10 Jul 2009 Andrey Pepelyshev 1 Abstract Designs of experiments for multivariate
More informationOPTIMAL DESIGN INPUTS FOR EXPERIMENTAL CHAPTER 17. Organization of chapter in ISSO. Background. Linear models
CHAPTER 17 Slides for Introduction to Stochastic Search and Optimization (ISSO)by J. C. Spall OPTIMAL DESIGN FOR EXPERIMENTAL INPUTS Organization of chapter in ISSO Background Motivation Finite sample
More informationOn the construction of asymmetric orthogonal arrays
isid/ms/2015/03 March 05, 2015 http://wwwisidacin/ statmath/indexphp?module=preprint On the construction of asymmetric orthogonal arrays Tianfang Zhang and Aloke Dey Indian Statistical Institute, Delhi
More informationLogic Simplification. Boolean Simplification Example. Applying Boolean Identities F = A B C + A B C + A BC + ABC. Karnaugh Maps 2/10/2009 COMP370 1
Digital Logic COMP370 Introduction to Computer Architecture Logic Simplification It is frequently possible to simplify a logical expression. This makes it easier to understand and requires fewer gates
More informationCONSTRUCTION OF NESTED (NEARLY) ORTHOGONAL DESIGNS FOR COMPUTER EXPERIMENTS
Statistica Sinica 23 (2013), 451-466 doi:http://dx.doi.org/10.5705/ss.2011.092 CONSTRUCTION OF NESTED (NEARLY) ORTHOGONAL DESIGNS FOR COMPUTER EXPERIMENTS Jun Li and Peter Z. G. Qian Opera Solutions and
More informationConstruction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing
Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing Vladimir Brayman Webtrends October 19, 2012 Advantages of Conducting Designed Experiments in Digital Marketing Availability
More informationDyadic diaphony of digital sequences
Dyadic diaphony of digital sequences Friedrich Pillichshammer Abstract The dyadic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube In this paper we
More informationELC224C. Karnaugh Maps
KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to K-maps How to use K-maps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on
More informationUnit 3 Session - 9 Data-Processing Circuits
Objectives Unit 3 Session - 9 Data-Processing Design of multiplexer circuits Discuss multiplexer applications Realization of higher order multiplexers using lower orders (multiplexer trees) Introduction
More informationUniversity, Wuhan, China c College of Physical Science and Technology, Central China Normal. University, Wuhan, China Published online: 25 Apr 2014.
This article was downloaded by: [0.9.78.106] On: 0 April 01, At: 16:7 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 10795 Registered office: Mortimer House,
More informationA Coset Pattern Identity between a 2 n p Design and its Complement
A Coset Pattern Identity between a 2 n p Design and its Complement Peng Zeng, Hong Wan, and Yu Zhu Auburn University, Purdue University, and Purdue University February 8, 2010 Abstract: The coset pattern
More informationB. L. Raktoe and W. T. Federer Cornell University ABSTRACT
On Irregular 1_ Fractions of a 2m Factorial 2n B. L. Raktoe and W. T. Federer Cornell University ABSTRACT A rigorous definition of regular and irregular fractional replicates from an sm factorial is presented.
More informationFractional designs and blocking.
Fractional designs and blocking Petter Mostad mostad@chalmers.se Review of two-level factorial designs Goal of experiment: To find the effect on the response(s) of a set of factors each factor can be set
More informationIntroduction to Karnaugh Maps
Introduction to Karnaugh Maps Review So far, you (the students) have been introduced to truth tables, and how to derive a Boolean circuit from them. We will do an example. Consider the truth table for
More information18Ï È² 7( &: ÄuANOVAp.O`û5 571 Based on this ANOVA model representation, Sobol (1993) proposed global sensitivity index, S i1...i s = D i1...i s /D, w
A^VÇÚO 1 Êò 18Ï 2013c12 Chinese Journal of Applied Probability and Statistics Vol.29 No.6 Dec. 2013 Optimal Properties of Orthogonal Arrays Based on ANOVA High-Dimensional Model Representation Chen Xueping
More informationBmMT 2017 Individual Round Solutions November 19, 2017
1. It s currently 6:00 on a 12 hour clock. What time will be shown on the clock 100 hours from now? Express your answer in the form hh : mm. Answer: 10:00 Solution: We note that adding any multiple of
More informationThe Binary Self-Dual Codes of Length Up To 32: A Revised Enumeration*
The Binary Self-Dual Codes of Length Up To 32: A Revised Enumeration* J. H. Conway Mathematics Department Princeton University Princeton, New Jersey 08540 V. Pless** Mathematics Department University of
More informationThe Hamming Codes and Delsarte s Linear Programming Bound
The Hamming Codes and Delsarte s Linear Programming Bound by Sky McKinley Under the Astute Tutelage of Professor John S. Caughman, IV A thesis submitted in partial fulfillment of the requirements for the
More informationSession 3 Fractional Factorial Designs 4
Session 3 Fractional Factorial Designs 3 a Modification of a Bearing Example 3. Fractional Factorial Designs Two-level fractional factorial designs Confounding Blocking Two-Level Eight Run Orthogonal Array
More informationStat 890 Design of computer experiments
Stat 890 Design of computer experiments Will introduce design concepts for computer experiments Will look at more elaborate constructions next day Experiment design In computer experiments, as in many
More informationTWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS
STAT 512 2-Level Factorial Experiments: Regular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS Bottom Line: A regular fractional factorial design consists of the treatments
More informationUnit 6: Fractional Factorial Experiments at Three Levels
Unit 6: Fractional Factorial Experiments at Three Levels Larger-the-better and smaller-the-better problems. Basic concepts for 3 k full factorial designs. Analysis of 3 k designs using orthogonal components
More informationYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
American Society for Quality A Note on the Graphical Analysis of Multidimensional Contingency Tables Author(s): D. R. Cox and Elizabeth Lauh Source: Technometrics, Vol. 9, No. 3 (Aug., 1967), pp. 481-488
More information3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value.
3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value. One-way ANOVA Source DF SS MS F P Factor 3 36.15??? Error??? Total 19 196.04 Completed table is: One-way
More informationMinimax design criterion for fractional factorial designs
Ann Inst Stat Math 205 67:673 685 DOI 0.007/s0463-04-0470-0 Minimax design criterion for fractional factorial designs Yue Yin Julie Zhou Received: 2 November 203 / Revised: 5 March 204 / Published online:
More informationStatistica Sinica Preprint No: SS R2
Statistica Sinica Preprint No: SS-2016-0423.R2 Title Construction of Maximin Distance Designs via Level Permutation and Expansion Manuscript ID SS-2016-0423.R2 URL http://www.stat.sinica.edu.tw/statistica/
More informationResolvable partially pairwise balanced designs and their applications in computer experiments
Resolvable partially pairwise balanced designs and their applications in computer experiments Kai-Tai Fang Department of Mathematics, Hong Kong Baptist University Yu Tang, Jianxing Yin Department of Mathematics,
More informationCONSTRUCTION OF OPTIMAL MULTI-LEVEL SUPERSATURATED DESIGNS. Hongquan Xu 1 and C. F. J. Wu 2 University of California and University of Michigan
CONSTRUCTION OF OPTIMAL MULTI-LEVEL SUPERSATURATED DESIGNS Hongquan Xu 1 and C. F. J. Wu University of California and University of Michigan A supersaturated design is a design whose run size is not large
More informationSome Construction Methods of Optimum Chemical Balance Weighing Designs I
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(6): 778-783 Scholarlin Research Institute Journals, 3 (ISS: 4-76) jeteas.scholarlinresearch.org Journal of Emerging Trends in Engineering
More informationarxiv: v1 [math.co] 27 Jul 2015
Perfect Graeco-Latin balanced incomplete block designs and related designs arxiv:1507.07336v1 [math.co] 27 Jul 2015 Sunanda Bagchi Theoretical Statistics and Mathematics Unit Indian Statistical Institute
More informationCHAPTER 7. Solutions for Exercises
CHAPTER 7 Solutions for Exercises E7.1 (a) For the whole part we have: Quotient Remainders 23/2 11 1 11/2 5 1 5/2 2 1 2/2 1 0 1/2 0 1 Reading the remainders in reverse order we obtain: 23 10 = 10111 2
More informationSuppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks.
58 2. 2 factorials in 2 blocks Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks. Some more algebra: If two effects are confounded with
More informationCS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:
CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n
More informationCS 5014: Research Methods in Computer Science
Computer Science Clifford A. Shaffer Department of Computer Science Virginia Tech Blacksburg, Virginia Fall 2010 Copyright c 2010 by Clifford A. Shaffer Computer Science Fall 2010 1 / 254 Experimental
More informationProbability density of nonlinear phase noise
Keang-Po Ho Vol. 0, o. 9/September 003/J. Opt. Soc. Am. B 875 Probability density of nonlinear phase noise Keang-Po Ho StrataLight Communications, Campbell, California 95008, and Graduate Institute of
More informationBinary codes from rectangular lattice graphs and permutation decoding
Binary codes from rectangular lattice graphs and permutation decoding J. D. Key a,,1 P. Seneviratne a a Department of Mathematical Sciences, Clemson University, Clemson SC 29634, U.S.A. Abstract We examine
More informationSoo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Figure 12: Figure 13:
1.0 ial Experiment Design by Block... 3 1.1 ial Experiment in Incomplete Block... 3 1. ial Experiment with Two Blocks... 3 1.3 ial Experiment with Four Blocks... 5 Example 1... 6.0 Fractional ial Experiment....1
More informationE-optimal approximate block designs for treatment-control comparisons
E-optimal approximate block designs for treatment-control comparisons Samuel Rosa 1 1 Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Slovakia We study E-optimal block
More informationConstructions of digital nets using global function fields
ACTA ARITHMETICA 105.3 (2002) Constructions of digital nets using global function fields by Harald Niederreiter (Singapore) and Ferruh Özbudak (Ankara) 1. Introduction. The theory of (t, m, s)-nets and
More informationHigh School Mathematics Contest Spring 2006 Draft March 27, 2006
High School Mathematics Contest Spring 2006 Draft March 27, 2006 1. Going into the final exam, which will count as two tests, Courtney has test scores of 80, 81, 73, 65 and 91. What score does Courtney
More informationOn the Compounds of Hat Matrix for Six-Factor Central Composite Design with Fractional Replicates of the Factorial Portion
American Journal of Computational and Applied Mathematics 017, 7(4): 95-114 DOI: 10.593/j.ajcam.0170704.0 On the Compounds of Hat Matrix for Six-Factor Central Composite Design with Fractional Replicates
More information2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationHomework 02 Solution updated
CS3-Automata and Complexity Theory Homework 2 Solution updated Due On: 5hrs Wednesday, December 2, 25 Max Points: 25 Problem [5+5+5+ points] Give DFA for the following languages, over the alphabet {,}
More informationJEE/BITSAT LEVEL TEST
JEE/BITSAT LEVEL TEST Booklet Code A/B/C/D Test Code : 00 Matrices & Determinants Answer Key/Hints Q. i 0 A =, then A A is equal to 0 i (a.) I (b.) -ia (c.) -I (d.) ia i 0 i 0 0 Sol. We have AA I 0 i 0
More informationA Classes of Variational Inequality Problems Involving Multivalued Mappings
Science Journal of Applied Mathematics and Statistics 2018; 6(1): 43-48 http://www.sciencepublishinggroup.com/j/sjams doi: 10.11648/j.sjams.20180601.15 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online)
More informationEhrhart polynomial for lattice squares, cubes, and hypercubes
Ehrhart polynomial for lattice squares, cubes, and hypercubes Eugen J. Ionascu UWG, REU, July 10th, 2015 math@ejionascu.ro, www.ejionascu.ro 1 Abstract We are investigating the problem of constructing
More informationBoolean Algebra and Digital Logic 2009, University of Colombo School of Computing
IT 204 Section 3.0 Boolean Algebra and Digital Logic Boolean Algebra 2 Logic Equations to Truth Tables X = A. B + A. B + AB A B X 0 0 0 0 3 Sum of Products The OR operation performed on the products of
More informationExperimental design (DOE) - Design
Experimental design (DOE) - Design Menu: QCExpert Experimental Design Design Full Factorial Fract Factorial This module designs a two-level multifactorial orthogonal plan 2 n k and perform its analysis.
More informationKarnaugh Maps Objectives
Karnaugh Maps Objectives For Karnaugh Maps of up to 5 variables Plot a function from algebraic, minterm or maxterm form Obtain minimum Sum of Products and Product of Sums Understand the relationship between
More informationLecture 4: Linear Codes. Copyright G. Caire 88
Lecture 4: Linear Codes Copyright G. Caire 88 Linear codes over F q We let X = F q for some prime power q. Most important case: q =2(binary codes). Without loss of generality, we may represent the information
More informationSOME EQUATIONS WITH FEATURES OF DIGIT REVERSAL AND POWERS
SOME EQUATIONS WITH FEATURES OF DIGIT REVERSAL AND POWERS GEOFFREY B CAMPBELL AND ALEKSANDER ZUJEV Abstract. In this paper we consider integers in base 10 like abc, where a, b, c are digits of the integer,
More informationCSCI 220: Computer Architecture-I Instructor: Pranava K. Jha. BCD Codes
CSCI 220: Computer Architecture-I Instructor: Pranava K. Jha BCD Codes Q. Give representation of the decimal number 853 in each of the following codes. (a) 8421 code (c) 84(-2)(-1) code (b) Excess-three
More informationMinimization techniques
Pune Vidyarthi Griha s COLLEGE OF ENGINEERING, NSIK - 4 Minimization techniques By Prof. nand N. Gharu ssistant Professor Computer Department Combinational Logic Circuits Introduction Standard representation
More information2003 AIME Given that ((3!)!)! = k n!, where k and n are positive integers and n is as large as possible, find k + n.
003 AIME 1 Given that ((3!)!)! = k n!, where k and n are positive integers and n is as large 3! as possible, find k + n One hundred concentric circles with radii 1,, 3,, 100 are drawn in a plane The interior
More information2 k, 2 k r and 2 k-p Factorial Designs
2 k, 2 k r and 2 k-p Factorial Designs 1 Types of Experimental Designs! Full Factorial Design: " Uses all possible combinations of all levels of all factors. n=3*2*2=12 Too costly! 2 Types of Experimental
More informationCONSTRUCTION OF SLICED ORTHOGONAL LATIN HYPERCUBE DESIGNS
Statistica Sinica 23 (2013), 1117-1130 doi:http://dx.doi.org/10.5705/ss.2012.037 CONSTRUCTION OF SLICED ORTHOGONAL LATIN HYPERCUBE DESIGNS Jian-Feng Yang, C. Devon Lin, Peter Z. G. Qian and Dennis K. J.
More informationA-optimal diallel crosses for test versus control comparisons. Summary. 1. Introduction
A-otimal diallel crosses for test versus control comarisons By ASHISH DAS Indian Statistical Institute, New Delhi 110 016, India SUDHIR GUPTA Northern Illinois University, Dekal, IL 60115, USA and SANPEI
More informationIndividual Round CHMMC November 20, 2016
Individual Round CHMMC 20 November 20, 20 Problem. We say that d k d k d d 0 represents the number n in base 2 if each d i is either 0 or, and n d k ( 2) k + d k ( 2) k + + d ( 2) + d 0. For example, 0
More information