Distance-Based Partial Least Squares Analysis

  • Anjali Krishnan
  • Nikolaus Kriegeskorte
  • Hervé Abdi
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 56)


Distances matrices are traditionally analyzed with statistical methods that represent distances as maps such as Metric Multidimensional Scaling (mds), Generalized Procrustes Analysis (gpa), Individual Differences Scaling (indscal), and distatis. Mds analyzes only one distance matrix at a time while gpa, indscal and distatis extract similarities between several distance matrices. However, none of these methods is predictive. Partial Least Squares Regression (plsr) predicts one matrix from another, but does not analyze distance matrices. We introduce a new statistical method called Distance-based Partial Least Squares Regression (displsr), which predicts one distance matrix from another. We illustrate displsr with data obtained from a neuroimaging experiment, which explored semantic categorization.

Key words

Partial least squares Regression Correlation Distance Mds Distatis 


  1. [1]
    H. Abdi, “Metric multidimensional scaling,” in Encyclopedia of Measurement and Statistics, N. J. Salkind, ed., pp. 598–605, Thousand Oaks (CA): Sage, 2007.Google Scholar
  2. [2]
    W. S. Torgerson, “Multidimensional scaling: I. Theory and method,” Psychometrika 17, pp. 401–419, 1952.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    J. Gower and G. Dijksterhuis, Procrustes Problems, New York: Oxford University Press, 2004.CrossRefMATHGoogle Scholar
  4. [4]
    J. D. Carroll and J.-J. Chang, “Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition,” Psychometrika 35, pp. 283–319, 1970.CrossRefMATHGoogle Scholar
  5. [5]
    H. Abdi, D. Valentin, A. J. O’Toole, and B. Edelman, “Distatis: The analysis of multiple distance matrices,” in Proceedings of the ieee Computer Society: International Conference on Computer Vision and Pattern Recognition, pp. 43–47, 2005.Google Scholar
  6. [6]
    H. Abdi, J.P. Dunlop, and L.A. Williams, “How to compute reliability estimates and display confidence and tolerance intervals for pattern classiffiers using the Bootstrap and 3-way multidimensional scaling Distatis,” in NeuroImage 17, pp. 89–95, 2009.CrossRefGoogle Scholar
  7. [7]
    H. Abdi, D. Valentin, S. Chollet, and C. Chrea, “ Analyzing assessors and products in sorting tasks: DISTATIS, theory and applications,” in Food Quality and Preference, 18, pp. 627–640, 2007.CrossRefGoogle Scholar
  8. [8]
    A. Krishnan, L. J. Williams, A. R. McIntosh, and H. Abdi, “Partial least squares (pls) methods for neuroimaging: A tutorial and review,” NeuroImage 56, pp. 455–475, 2011.CrossRefGoogle Scholar
  9. [9]
    L.R., Tucker, “An inter-battery method of factor analysis.” Psychometrika 23, pp. 111–136, 1958.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    F.L. Bookstein, P.L. Sampson, A.P. Streissguth, and H.M. Barr, “Exploting redundant measurements of dose and developmental outcome: New methods from the behavioral teratology of alcohol,” Developmental Psychology 32, pp. 404–415, 1996.CrossRefGoogle Scholar
  11. [11]
    P.D. Sampson, A.P. Streissguth, H.M. Barr, and F.S.Bookstein, “Neurobehavioral effect of prenatal alcohol: Part II, partial least square analysis,” Neurotoxicology and Teratology 11, pp. 477–491.Google Scholar
  12. [12]
    A. Tishler, D. Dvir, A. Shenhar, and S. Lipovetsky, “Identifying critical success factors in defense development projects: A multivariate analysis,” Technological Forecasting and Social Change 51, pp. 151–171, 1996.CrossRefGoogle Scholar
  13. [13]
    A. Tishler, and S. Lipovetsky, “Modelling and forecasting with robust canonical analysis: method and application,” Computers and Operations Research 27, pp. 217–232, 2000.CrossRefMATHGoogle Scholar
  14. [14]
    S. Dolédec, and D. Chessel, “Co-inertia analysis: an alernative methods for studying sepcies-environment relationships.” Fresehwater Biology 31, pp. 277–294.Google Scholar
  15. [15]
    H. Abdi, “Partial least squares regression and projection on latent structure regression (PLS Regression),” WIREs Computational Statistics 2, pp. 97–106, 2010.CrossRefGoogle Scholar
  16. [16]
    H. Wold, “Soft modelling: The basic design and some extensions,” in Systems under indirect observation: Causality-structure-prediction Part II, K. Jöreskog and H. Wold, eds., pp. 1–54, Amsterdam: North-Holland Publishing Company, 1982.Google Scholar
  17. [17]
    S. Wold, M. Sjöström, and L. Eriksson, “Pls-regression: A basic tool of chemometrics,” Chemometrics and Intelligent Laboratory Systems 58, pp. 109–130, 2001.CrossRefGoogle Scholar
  18. [18]
    H. Abdi, “Singular value decomposition (svd) and generalized singular value decomposition (gsvd),” in Encyclopedia of Measurement and Statistics, N. Salkind, ed., pp. 907–912, Thousand Oaks (CA): Sage, 2007.Google Scholar
  19. [19]
    M. Greenacre, Theory and Applications of Correspondence Analysis, Academic Press, London, 1984.MATHGoogle Scholar
  20. [20]
    H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition, New York, Springer, 2011.CrossRefMATHGoogle Scholar
  21. [21]
    B. L. Bush and R. B. Nachbar, Jr., “Sample-distance partial least squares: pls optimized for many variables, with application to comfa,” Journal of Computer-aided Molecular Design 7, pp. 587–619, 1993.CrossRefGoogle Scholar
  22. [22]
    Y. C. Martin, C. T. Lin, C. Hetti, and J. DeLazzer, “Pls analysis of distance matrices to detect nonlinear relationships between biological potency and molecular properties,” Journal of Medicinal Chemistry 38, pp. 3009–3015, 1995.CrossRefGoogle Scholar
  23. [23]
    P. Legendre and M. J. Anderson, “Distance-based redundancy analysis: Testing multispecies responses in multifactorial ecological experiments,” Ecological Monographs 69, pp. 1–24, 1999.CrossRefGoogle Scholar
  24. [24]
    M. A. Zapala and N. J. Schork, “Multivariate regression analysis of distance matrices for testing associations between gene expression patterns and related variables,” Proceedings of the National Academy of Sciences 103, pp. 19430–19435, 2006.CrossRefGoogle Scholar
  25. [25]
    S. Rännar, F. Lindgren, P. Geladi, and S. Wold, “A pls kernel algorithm for data sets with many variables and fewer objects. Part I: Theory and algorithm,” Journal of Chemometrics 8, pp. 111–125, 1994.CrossRefGoogle Scholar
  26. [26]
    A. Höskuldsson, “Pls regression methods,” Journal of Chemometrics 2, pp. 211–228, 1988.CrossRefGoogle Scholar
  27. [27]
    H. Abdi, “Congruence: Congruence coefficient, R V coefficient and Mantel coefficient,” in Encyclopedia of Research Design, N. Salkind, D. D.M., and B. Frey, eds., pp. 222–229, Thousand Oaks (CA): Sage, 2010.Google Scholar
  28. [28]
    H. Abdi, “R V coefficient and congruence coefficient,” in Encyclopedia of Measurement and Statistics, N. Salkind, ed., pp. 849–853, Thousand Oaks (CA): Sage, 2007.Google Scholar
  29. [29]
    E. J. Dietz, “Permutation tests for association between two distance matrices,” Systematic Zoology 32, pp. 21–26, 1983.CrossRefGoogle Scholar
  30. [30]
    J. Josse, J. Pagès, and F. Husson, “Testing the significance of the R V coefficient,” Computational Statistics & Data Analysis 53, pp. 82–91, 2008.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    N. Kriegeskorte, M. Mur, and P. Bandettini, “Representational similarity analysis connecting the branches of systems neuroscience,” Frontiers in Systems Neuroscience 2, p. doi:10.3389/neuro.06.004.2008, 2008.Google Scholar
  32. [32]
    R. Kiani, H. Esteky, K. Mipour, and K. Tanaka, “Object category structure in response patterns of neuronal population in monkey inferior temporal cortex,” Journal of Neurophysiology 97, pp. 4296–4309, 2007.CrossRefGoogle Scholar
  33. [33]
    J. Daugman, “How iris recognition works,” I eee Transactions on Circuits and Systems for Video Technology 14, pp. 21–30, 2004.CrossRefGoogle Scholar
  34. [34]
    G. Orban, D. van Essen, and W. Vanduffel, “Comparative mapping of higher visual areas in monkeys and humans,” Trends in Cognitive Sciences 8, pp. 315–324, 2004.CrossRefGoogle Scholar
  35. [35]
    B. Efron and R. Tibshirani, “Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy,” Statistical Science 1, pp. 54–77, 1986.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Anjali Krishnan
    • 1
  • Nikolaus Kriegeskorte
    • 2
  • Hervé Abdi
    • 3
  1. 1.Institute of Cognitive ScienceUniversity of Colorado BoulderBoulderUSA
  2. 2.MRC Cognition and Brain Sciences UnitUniversity of CambridgeCambridgeUK
  3. 3.School of Behavioral and Brain SciencesThe University of Texas at DallasRichardsonUSA

Personalised recommendations