1、BSI Standards Publication PD ISO/TR 13195:2015 Selected illustrations of response surface method Central composite designPD ISO/TR 13195:2015 PUBLISHED DOCUMENT National foreword This Published Document is the UK implementation of ISO/TR 13195:2015. The UK participation in its preparation was entrus
2、ted to Technical Committee MS/6, Methodologies for business process improvement using statistical methods. A list of organizations represented on this committee can be obtained on request to its secretary. This publication does not purport to include all the necessary provisions of a contract. Users
3、 are responsible for its correct application. The British Standards Institution 2015. Published by BSI Standards Limited 2015 ISBN 978 0 580 74473 0 ICS 03.120.30 Compliance with a British Standard cannot confer immunity from legal obligations. This Published Document was published under the authori
4、ty of the Standards Policy and Strategy Committee on 31 December 2015. Amendments issued since publication Date Text affectedPD ISO/TR 13195:2015 ISO 2015 Selected illustrations of response surface method Central composite design Illustrations choisies de mthodologie surface de rponse Plans composit
5、es centrs TECHNICAL REPORT ISO/TR 13195 Reference number ISO/TR 13195:2015(E) First edition 2015-12-15PD ISO/TR 13195:2015ISO/TR 13195:2015(E)ii ISO 2015 All rights reserved COPYRIGHT PROTECTED DOCUMENT ISO 2015, Published in Switzerland All rights reserved. Unless otherwise specified, no part of th
6、is publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below or ISOs member body in t
7、he country of the requester. ISO copyright office Ch. de Blandonnet 8 CP 401 CH-1214 Vernier, Geneva, Switzerland Tel. +41 22 749 01 11 Fax +41 22 749 09 47 copyrightiso.org www.iso.orgPD ISO/TR 13195:2015ISO/TR 13195:2015(E)Foreword iv Introduction v 1 Scope . 1 2 T erms and definitions . 1 3 Symbo
8、ls and abbreviated terms . 6 3.1 Symbols . 6 3.2 Abbreviated terms . 6 4 Generic descriptions of central composite designs . 7 4.1 Overview of the structure of the examples in Annexes A to D . 7 4.2 Overall objective(s) of a response surface experiment 7 4.3 Description of the response variable(s) 8
9、 4.4 Identification of measurement systems 8 4.5 Identification of factors affecting the response(s) . 8 4.6 Selection of levels for each factor 8 4.6.1 Factorial runs 9 4.6.2 Star runs 9 4.6.3 Centre run 9 4.7 Layout plan of the CCD with randomization principle 10 4.8 Analyse the results Numerical
10、summaries and graphical displays .10 4.9 Present the results 11 4.10 Perform confirmation run 12 5 Description of Annexes A through D .12 5.1 Comparing and contrasting the examples 12 5.2 Experiment summaries 13 Annex A (informative) Effects of fertilizer ingredients on the yield of a crop 14 Annex
11、B (informative) Optimization of the button tactility using central composite design 28 Annex C (informative) Semiconductor die deposition process optimization .41 Annex D (informative) Process yield-optimization of a palladium-copper catalysed C-C- bond formation .52 Annex E (informative) Background
12、 on response surface designs 70 Bibliography .80 ISO 2015 All rights reserved iii Contents PagePD ISO/TR 13195:2015ISO/TR 13195:2015(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing
13、International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaiso
14、n with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Direc
15、tives, Part 1. In particular the different approval criteria needed for the different types of ISO documents should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives). Attention is drawn to the possibility that so
16、me of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be in the Introduction and/or on the ISO list of patent declar
17、ations received (see www.iso.org/patents). Any trade name used in this document is information given for the convenience of users and does not constitute an endorsement. For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment, as well as information a
18、bout ISOs adherence to the WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information The committee responsible for this document is ISO/TC 69, Applications of statistical methods, Subcommittee SC 7, Applications of statistical and related tec
19、hniques for the implementation of Six Sigma.iv ISO 2015 All rights reservedPD ISO/TR 13195:2015ISO/TR 13195:2015(E) Introduction The present Technical Report takes one specific statistical tool (Central Composite Designs in Response Surface Methodology) and develops the topic somewhat generically (i
20、n the spirit of International Standards) but then illustrates it through the use of four detailed and distinct applications. The generic description focuses on the Central Composite Designs. The annexes containing the four illustrations follow the basic framework but also identify the nuances and pe
21、culiarities in the specific applications. Each example offers at least one “wrinkle” to the problem, which is generally the case for real applications. It is hoped that practitioners can identify with at least one of the four examples, if only to remind them of the basic material on response surface
22、 method that was encountered during their training. Each of the four examples is developed and analysed using statistical software of current vintage. The explanations throughout are devoid of mathematical detailsuch material can be readily obtained from the many design and analysis of experiments t
23、extbooks (such as those given in References 1 to 7). ISO 2015 All rights reserved vPD ISO/TR 13195:2015PD ISO/TR 13195:2015Selected illustrations of response surface method Central composite design 1 Scope This Technical Report describes the steps necessary to understand the scope of Response Surfac
24、e Methodology (RSM) and the method to analyse data collected using Central Composite Designs (CCD) through illustration with four distinct applications of this methodology. Response surface methodology (RSM) is used in order to investigate a relation between the response and the set of quantitative
25、predictor variables or factors. Especially after specifying the vital few controllable factors, RSM is used in order to find the factor setting which optimizes the response. 2 T erms a nd definiti ons For the purposes of this document, the following terms and definitions apply. 2.1 experiment purpos
26、ive investigation of a system through selective adjustment of controllable conditions and allocation of resources Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.1. (The notes are not reproduced here.) 2.2 response variable variable representing the outcome of an experiment (2.1) Note
27、1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.3. (Except for NOTE 3 the notes are not reproduced here.) Note 2 to entry: A common synonym is “output variable”. Note 3 to entry: The response variable is likely to be influenced by one or more predictor variables (2.3), the nature of which c
28、an be useful in controlling or optimizing the response variable. 2.3 predictor variable variable that can contribute to the explanation of the outcome of an experiment (2.1) Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.4. (The notes are not reproduced here.) Note 2 to entry: Natural
29、 predictor variables are expressed in natural units of measurement such as degrees Celcius (C) or grams per liter, for example. In RSM work, it is convenient to transform the natural variables to coded variables which are dimensionless variables, symmetric around zero and all with the same spread. 2
30、.4 modelformalized representation of outcomes of an experiment (2.1) Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.2. (The notes and examples are not reproduced here except for NOTE 2 which is NOTE 1 in ISO 3534-3.) TECHNICAL REPORT ISO/TR 13195:2015(E) ISO 2015 All rights reserved 1
31、PD ISO/TR 13195:2015ISO/TR 13195:2015(E) Note 2 to entry: The model consists of three parts. The first part is the response variable (2.2) that is being modelled. The second part is the deterministic or the systematic part of the model that includes predictor variable(s) (2.3). Finally, the third pa
32、rt is the residual error (2.12) that can involve pure random error (2.13) and misspecification error (2.14). The model applies for the experiment as a whole and for separate outcomes denoted with subscripts. The model is a mathematical description that relates the response variable to predictor vari
33、ables and includes associated assumptions. Outcomes refer to recorded or measured observations of the response variable. Note 3 to entry: In some areas the term transfer function is used for the systematic part of the model. EXAMPLE In the models considered in response surface methodology the determ
34、inistic or systematic part are polynomials in the predictor variables. A second order model with two predictor variables is written as where is the random error. The associated assumptions on the random error could be either that individual random errors are uncorrelated with constant variance or in
35、dependent and normally distributed. The deterministic part of the model is the second degree polynomial in the predictor variables x 1and x 2 which explains the mean (Ey) of the response variable as a function of the predictor variables. 2.5 factorfeature under examination as a potential cause of va
36、riation Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.5. (The notes are not reproduced here.) Note 2 to entry: Generally the symbol k is used to indicate the number of factors in the experiment. 2.6 factor level setting, value or assignment of a factor (2.5) Note 1 to entry: Adapted
37、from ISO 3534-3:2013, definition (3.1.12). (The notes are not reproduced here.) 2.7 coding of factor levelsone-to-one relabelling of factor levels Note 1 to entry: The coding of factor levels facilitates the identification of the design and the properties of the design. Note 2 to entry: In response
38、surface experiments the actual (or natural or operational) levels are relabelled such that the coded levels are numeric and symmetric around 0. Note 3 to entry: A two-level factor is usually coded to have coded levels 1 and +1. A factorial design where all factors are two-level factors can be coded
39、such that all runs are represented as factorial runs (2.9). Note 4 to entry: In central composite designs numeric (or continuous) factors with five levels are considered, except for the face-centred central composite deigns, where only three levels are needed, see note 6 to 2.7. If the actual (or na
40、tural or operational) levels are l 1 l 2 l 3 l 4 l 5then the middle level l 3shall be the average of the lowest level l 1and the highest level l 5 , and, furthermore, l 3shall be the average of the intermediate levels l 2and l 4 . The form of the coding operation can be expressed as where C is half
41、the distance from l 2to l 4 . With this coding of the factors each run (2.8) of a central composite design can be identified as either a factorial point (2.9), a centre point (2.10), or an star point (2.11). This is the coding used in textbooks for discussing central composite designs.2 ISO 2015 All
42、 rights reservedPD ISO/TR 13195:2015ISO/TR 13195:2015(E) Note 5 to entry: An alternative coding is sometimes applied in the computations in software programs. The form of the coding operation can be expressed as where M is half the distance from the lowest level l 1to the highest level l 5 . This co
43、ding will be referred to as software coding in this Technical Report. Note 6 to entry: In the face-centred CCD, only three levels of each factor are needed, so l 1= l 2specific settings of every factor (2.5) used on a particular experimental unit (2.15) Note 1 to entry: Ultimately, the impact of the
44、 factors will be captured through their representation in the predictor variables (2.3) and the extent to which the model matches the outcome of the experiment (2.1). EXAMPLE Consider a chemical process experiment (2.1) in which a high yield is the objective and the predictor variables are temperatu
45、re, duration, and concentration of a catalyst. A run could be a setting of temperature of 350 C, 30 min duration and 10 % concentration of the catalyst, assuming that all of these settings are possible and permissible. Note 2 to entry: Adapted from ISO 3534-3:2013, definition 3.1.13. 2.9 factorial p
46、oint factorial run cube point cube run vector of factor level settings of the form (a 1 , a 2 , ., a k ), where each a iequals 1 or +1 as a notation for the coded levels of the factors Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.37. (The notes are not reproduced here.) 2.10 centre
47、point centre run vector of factor level settings of the form (a 1 , a 2 , ., a k ), where all a iequal 0, as notation for the coded levels of the factors Note 1 to entry: Adapted from ISO 3534-3:2013, definition 3.1.38. (The notes are not reproduced here.) ISO 2015 All rights reserved 3PD ISO/TR 131
48、95:2015ISO/TR 13195:2015(E) 2.11 star point axial point star run axial run vector of factor level (2.7) settings of the form (a 1 , a 2 , , a k ), where one a iequals or and the other a i s equal 0, as notation for the coded levels of the factors (2.6) Note 1 to entry: For a k factor experiment, thi
49、s process yields 2k-star points of the form: (, 0, , 0), (0, , 0, , 0), , (0, 0, , ). Note 2 to entry: Star points are added to the design in order to estimate a quadratic response surface. Note 3 to entry: Special values of give a nice geometric structure. For a k factor experiment, if = k then the factorial points and the star po
