Topic

# Scaling

About: Scaling is a research topic. Over the lifetime, 21588 publications have been published within this topic receiving 534528 citations.

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TL;DR: A method of scaling ratios using the principal eigenvector of a positive pairwise comparison matrix is investigated, showing that λmax = n is a necessary and sufficient condition for consistency.

Abstract: The purpose of this paper is to investigate a method of scaling ratios using the principal eigenvector of a positive pairwise comparison matrix. Consistency of the matrix data is defined and measured by an expression involving the average of the nonprincipal eigenvalues. We show that λmax = n is a necessary and sufficient condition for consistency. We also show that twice this measure is the variance in judgmental errors. A scale of numbers from 1 to 9 is introduced together with a discussion of how it compares with other scales. To illustrate the theory, it is then applied to some examples for which the answer is known, offering the opportunity for validating the approach. The discussion is then extended to multiple criterion decision making by formally introducing the notion of a hierarchy, investigating some properties of hierarchies, and applying the eigenvalue approach to scaling complex problems structured hierarchically to obtain a unidimensional composite vector for scaling the elements falling in any single level of the hierarchy. A brief discussion is also included regarding how the hierarchy serves as a useful tool for decomposing a large-scale problem, in order to make measurement possible despite the now-classical observation that the mind is limited to 7 ± 2 factors for simultaneous comparison.

8,117 citations

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Bell Labs

^{1}TL;DR: The numerical methods required in the approach to multi-dimensional scaling are described and the rationale of this approach has appeared previously.

Abstract: We describe the numerical methods required in our approach to multi-dimensional scaling. The rationale of this approach has appeared previously.

4,561 citations

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TL;DR: The various physical factors affecting measured diffraction intensities are discussed, as are the scaling models which may be used to put the data on a consistent scale and algorithms used by the CCP4 scaling program SCALA.

Abstract: The various physical factors affecting measured diffraction intensities are discussed, as are the scaling models which may be used to put the data on a consistent scale. After scaling, the intensities can be analysed to set the real resolution of the data set, to detect bad regions (e.g. bad images), to analyse radiation damage and to assess the overall quality of the data set. The significance of any anomalous signal may be assessed by probability and correlation analysis. The algorithms used by the CCP4 scaling program SCALA are described. A requirement for the scaling and merging of intensities is knowledge of the Laue group and point-group symmetries: the possible symmetry of the diffraction pattern may be determined from scores such as correlation coefficients between observations which might be symmetry-related. These scoring functions are implemented in a new program POINTLESS.

4,211 citations

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13 Mar 1991

TL;DR: In this paper, the authors present a directory of Symbols and Definitions for PCA, as well as some classic examples of PCA applications, such as: linear models, regression PCA of predictor variables, and analysis of variance PCA for Response Variables.

Abstract: Preface.Introduction.1. Getting Started.2. PCA with More Than Two Variables.3. Scaling of Data.4. Inferential Procedures.5. Putting It All Together-Hearing Loss I.6. Operations with Group Data.7. Vector Interpretation I : Simplifications and Inferential Techniques.8. Vector Interpretation II: Rotation.9. A Case History-Hearing Loss II.10. Singular Value Decomposition: Multidimensional Scaling I.11. Distance Models: Multidimensional Scaling II.12. Linear Models I : Regression PCA of Predictor Variables.13. Linear Models II: Analysis of Variance PCA of Response Variables.14. Other Applications of PCA.15. Flatland: Special Procedures for Two Dimensions.16. Odds and Ends.17. What is Factor Analysis Anyhow?18. Other Competitors.Conclusion.Appendix A. Matrix Properties.Appendix B. Matrix Algebra Associated with Principal Component Analysis.Appendix C. Computational Methods.Appendix D. A Directory of Symbols and Definitions for PCA.Appendix E. Some Classic Examples.Appendix F. Data Sets Used in This Book.Appendix G. Tables.Bibliography.Author Index.Subject Index.

3,534 citations