Development of an Agreement Metric Based Upon the RAND Index for the Evaluation of Dimensionality Reduction Techniques, with Applications to Mapping Customer Data

We develop a metric i¾?, based upon the RAND index, for the comparison and evaluation of dimensionality reduction techniques. This metric is designed to test the preservation of neighborhood structure in derived lower dimensional configurations. We use a customer information data set to show how i¾?can be used to compare dimensionality reduction methods, tune method parameters, and choose solutions when methods have a local optimum problem. We show that i¾?is highly negatively correlated with an alienation coefficient K that is designed to test the recovery of relative distances. In general a method with a good value of i¾?also has a good value of K. However the monotonic regression used by Nonmetric MDS produces solutions with good values of i¾?, but poor values of K.

[1]  Paul E. Green,et al.  Marketing Applications of MDS: Assessment and Outlook , 1975 .

[2]  Russell S. Winer,et al.  A Panel-Data Based Method for Merging Joint Space and Market Response Function Estimation , 1987 .

[3]  Ajay K. Manrai,et al.  Mds Maps for Product Attributes and Market Response: An Application to Scanner Panel Data , 1999 .

[4]  Anil K. Jain,et al.  An Intrinsic Dimensionality Estimator from Near-Neighbor Information , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Ajay K. Manrai,et al.  A New Multidimensional Scaling Methodology for the Analysis of Asymmetric Proximity Data in Marketing Research , 1992 .

[6]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[7]  Joseph L. Zinnes,et al.  Theory and Methods of Scaling. , 1958 .

[8]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[9]  J. Berge,et al.  Perceptual Mapping Based on Idiosyncratic Sets of Attributes , 1994 .

[10]  Donna L. Hoffman,et al.  Constructing MDS Joint Spaces from Binary Choice Data: A Multidimensional Unfolding Threshold Model for Marketing Research , 1987 .

[11]  A. Householder,et al.  Discussion of a set of points in terms of their mutual distances , 1938 .

[12]  William M. Rand,et al.  Objective Criteria for the Evaluation of Clustering Methods , 1971 .

[13]  Young-Seuk Park,et al.  Self-Organizing Map , 2008 .

[14]  J. Douglas Carroll,et al.  PARAMAP vs. Isomap: A Comparison of Two Nonlinear Mapping Algorithms , 2006, J. Classif..

[15]  Michel Wedel,et al.  A Comparison of Multidimensional Scaling Methods for Perceptual Mapping , 1999 .

[16]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[17]  Ingwer Borg,et al.  Measuring the Similarity of MDS Configurations. , 1985, Multivariate behavioral research.

[18]  Anil K. Jain,et al.  Incremental nonlinear dimensionality reduction by manifold learning , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  I K Fodor,et al.  A Survey of Dimension Reduction Techniques , 2002 .

[20]  Juyoung Kim,et al.  A Gravity‐Based Multidimensional Scaling Model for Deriving Spatial Structures Underlying Consumer Preference/Choice Judgments , 2002 .

[21]  Ma Conway,et al.  Handbook of perception and cognition , 1996 .

[22]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[23]  Teuvo Kohonen,et al.  Self-Organizing Maps , 2010 .

[24]  Kilian Q. Weinberger,et al.  Unsupervised Learning of Image Manifolds by Semidefinite Programming , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[25]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[26]  Wagner A. Kamakura,et al.  Defection Detection: Measuring and Understanding the Predictive Accuracy of Customer Churn Models , 2006 .

[27]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[28]  Ann B. Lee,et al.  Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  A. Buja,et al.  Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing, and Proximity Analysis , 2009 .

[30]  Brigitte Le Roux,et al.  Fionn Murtagh, "Correspondence analysis and data coding with Java and R", Computer Science and Data Analysis Series, London, Chapmann and Hall/CRC, 2005 , 2008 .

[31]  W. Torgerson.  Multidimensional scaling: I. Theory and method , 1952 .

[32]  Andreas Buja,et al.  Visualization Methodology for Multidimensional Scaling , 2002, J. Classif..

[33]  Bernhard Schölkopf,et al.  A kernel view of the dimensionality reduction of manifolds , 2004, ICML.

[34]  J. Douglas Carroll,et al.  Psychometric Methods in Marketing Research: Part II, Multidimensional Scaling , 1997 .

[35]  Peter J. Bickel,et al.  Maximum Likelihood Estimation of Intrinsic Dimension , 2004, NIPS.

[36]  Lawrence K. Saul,et al.  Analysis and extension of spectral methods for nonlinear dimensionality reduction , 2005, ICML.

[37]  J. Carroll,et al.  Interpoint Distance Comparisons in Correspondence Analysis , 1986 .

[38]  P. Green.  Marketing Applications of MDS: Assessment and Outlook , 1975 .