Skip to main content
SpringerLink
Log in

On the Correspondence Between Procrustes Analysis and Bidimensional Regression

  • Published:
Journal of Classification Aims and scope Submit manuscript

Abstract

Procrustes analysis is defined as the problem of fitting a matrix of data to a target matrix as closely as possible (Gower and Dijksterhuis, 2004). The problem can take many forms, but the most common form, orthogonal Procrustes analysis, has as allowable transformations, a translation, a scaling, an orthogonal rotation, and a reflection. Procrustes analysis and other rotation methods have a long history in quantitative psychology, as well as in other fields, such as biology (Siegel and Benson, 1982) and shape analysis (Kendall, 1984). In the field of quantitative geography, the use of bidimensional regression (Tobler, 1965) has recently become popular. Tobler (1994) defines bidimensional regression as “an extension of ordinary regression to the case in which both the independent and dependent variables are two-dimensional.” In this paper, it is established that orthogonal Procrustes analysis (without reflection) and Euclidean bidimensional regression are the same. As such, both areas of development can borrow from the other, allowing for a richer landscape of possibilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • AHMED, N., and MILLER, H.J. (2007), “Time-Space Transformations of Geographic Space for Exploring, Analyzing and Visualizing Transportation Systems”, Journal of Transport Geography, 15, 2–17.

    Article  Google Scholar 

  • BAKDASH, J.Z., LINKENAUGER, S.A., and PROFFITT, D. (2008), “Comparing Decision-Making and Control for Learning a Virtual Environment: Backseat Drivers Learn Where They Are Going”, Proceedings of the Human Factors and Ergonomics Society, 52, 2117–2121.

    Article  Google Scholar 

  • BOOKSTEIN, F.L. (1991), Morphometric Tools for Landmark Data: Geometry and Biology, New York, NY: Cambridge University Press.

    MATH  Google Scholar 

  • BROWNE, M.W. (1967), “On Oblique Procrustes Rotation”, Psychometrika, 32, 125–132.

    Article  MathSciNet  Google Scholar 

  • CARBON, C.-C. (2013), BiDimRegression: Bidimensional Regression Modeling Using R, Journal of Statistical Software, 52, 1–11.

    Article  Google Scholar 

  • CARBON, C.-C., and LEDER, H. (2005), “The Wall Inside the Brain: Overestimation of Distances Crossing the Former Iron Curtain”, Psychonomic Bulletin and Review, 12, 746–750.

    Article  Google Scholar 

  • CLIFF, N. (1966), “Orthogonal Rotation to Congruence”, Psychometrika, 31, 33–42.

    Article  MathSciNet  Google Scholar 

  • CONSTANTINE, A.C., and GOWER, J.C. (1978), “ Some Properties and Applications of Simple Orthogonal Matrices”, IMA Journal of Applied Mathematics, 21, 445–454.

    Article  MathSciNet  MATH  Google Scholar 

  • CONSTANTINE, A.C., and GOWER, J.C. (1982), “Models for the Analysis of Inter-Regional Migration”, Environmental Planning A, 14, 477–497.

    Article  Google Scholar 

  • DUSEK, T. (2012), “Bidimensional Regression in Spatial Analysis”, Regional Statistics (English Edition), 52, 61–73.

    Article  Google Scholar 

  • FRIEDMAN, A., and KOHLER, B. (2003), “Bidimensional Regression: Assessing the Configural Similarity and Accuracy of Cognitive Maps and Other Two-Dimensional Data Sets”, Psychological Methods, 8, 468–491.

    Article  Google Scholar 

  • GIRAUDO, M.-D., GAYRAUD, D., and HABIB, M. (1997), “Visuospatial Ability of Parkinsonians and Elderly Adults in Location Memory Tasks”, Brain and Cognition, 34, 259–273.

    Article  Google Scholar 

  • GOODALL, C. (1991), “Procrustes Methods in the Statistical Analysis of Shape”, Journal of the Royal Stastistical Society. Series B (Methodological), 53, 285–339.

  • GOWER, J.C. (1975), “Generalized Procrustes Analysis”, Psychometrika, 40, 33–51.

    Article  MathSciNet  MATH  Google Scholar 

  • GOWER, J.C., and DIJKSTERHUIS, G.B. (2004), Procrustes Problems, NewYork, NY: Oxford University Press.

    Book  MATH  Google Scholar 

  • GROENEN, P.J.F., GIAQUINTO, P., and KIERS, H.A.L. (2003), “An Improved Majorization Algorithm for Robust Procrustes Analysis”, in New Developments in Classification and Data Analysis, eds. M. Vichi, P. Monari, S. Mignani, and A. Montanari, Heidelberg, Germany: Springer, pp. 151–158.

  • GRUVAEUS, G.T. (1970), “A General Approach to Procrustes Pattern Rotation”, Psychometrika, 35, 493–505.

    Article  Google Scholar 

  • HURLEY, J.R., and CATTELL, R.B. (1962), “The Procrustes Program: Producing Direct Rotation to Test a Hypothesized Factor Structure”, Behavioral Science, 7, 258–262.

    Article  Google Scholar 

  • ISHIKAWA, T. (2013), “Retention of Memory for Large-Scale Spaces”, Memory, 21(7), 807–817.

    Article  Google Scholar 

  • KARE, S., SAMAL, A., and MARX, D. (2010), “Using Bidimensional Regression to Assess Face Similarity”, Machine Vision and Applications, 21, 261–274.

    Article  Google Scholar 

  • KENDALL, D.G. (1981), “The Statistics of Shape”, in Interpreting Multivariate Data, ed. V. Barnett, New York, NY: Wiley, pp. 75–80.

  • KENDALL, D.G. (1984), “Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces”, Bulletin of the London Mathematical Society, 16, 81–121.

    Article  MathSciNet  MATH  Google Scholar 

  • KITCHIN, R.M. (1996), “Methodological Convergence in Cognitive Mapping Research: Investigating Configurational Knowledge”, Journal of Environmental Psychology, 16, 163–185.

    Article  Google Scholar 

  • KORTH, B., and TUCKER, L.R. (1976), “Procrustes Matching by Congruence Coefficients”, Psychometrika, 41, 531–535.

    Article  MATH  Google Scholar 

  • KOSCHAT, M.A., and SWAYNE, D.F. (1991), “A Weighted Procrustes Criterion”, Psychometrika, 56, 229–239.

    Article  Google Scholar 

  • LEE, H. (2012), “Bayesian Bidimensional Regression and Its Extension”, Master’s thesis, Washington University, St. Louis, MO.

  • MARDIA, K.V., and DRYDEN, I.L. (1989a), “Shape Distributions for Landmark Data”, Advances in Applied Probability, 21, 742–755.

  • MARDIA, K.V., and DRYDEN, I.L. (1989b), “The StatisticalAnalysis of Data”, Biometrika, 76, 271–281.

  • MOOIJAART, A., and COMMANDEUR, J.J.F. (1990), “A General Solution of theWeighted Orthonormal Procrustes Problem”, Psychometrika, 55, 657–663.

    Article  MATH  Google Scholar 

  • NAKAYA, T. (1997), “Statistical Inferences in Bidimensional Regression Models”, Geographical Analysis, 29, 169–186.

    Article  Google Scholar 

  • OHUCHI, M., IWAYA, Y., SUZUKI, Y., and MUNEKATA, T. (2006), “Cognitive-Map Formation of Blind Persons in a Virtual Sound Environment”, Proceedings of the 12th International Conference on Auditory Display, pp. 1–7.

  • PEAY, E.R. (1988), “Multidimensional Rotation and Scaling of Configurations to Optimal Agreement”, Psychometrika, 53, 199–208.

    Article  MathSciNet  MATH  Google Scholar 

  • ROHLF, F.J., and SLICE, D. (1990), “Extensions of the Procrustes Method for the Optimal Superimposition of Landmarks”, Systematic Zoology, 39, 40–59.

    Article  Google Scholar 

  • SCHINAZI, V.R., NADI, D., NEWCOMBE, N.S., SHIPLEY, T.F., and EPSTEIN, R.A. (2013), “Hippocampal Size Predicts Rapid Learning of a Cognitive Map in Humans”, Hippocampus, 23, 515–528.

    Article  Google Scholar 

  • SCHMID, K.K., MARX, D.B., and SAMAL, A. (2011), “Weighted Bidimensional Regression”, Geographical Analysis, 43, 1–13.

    Article  Google Scholar 

  • SCHMID, K.K., MARX, D.B., and SAMAL, A. (2012), “Tridimensional Regression for Comparing and Mapping 3D Anatomical Structures”, Anatomy Research International, 1–9. f

  • SCHÖNEMANN, P.H. (1966), “A Generalized Solution of the Orthogonal Procrustes Problem”, Psychometrika, 31, 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  • SCHÖNEMANN, P.H., and CARROLL, R.M. (1970), “Fitting One Matrix to Another Under Choice of a Central Dilation and a Rigid Motion”, Psychometrika, 35, 245–256.

    Article  Google Scholar 

  • SHI, J., SAMAL, A., and MARX, D. (2005), “Face Recognition Using Landmark-Based Bidimensional Regression”, Proceedings of the Fifth IEEE International Conference on Data Mining, pp. 1–4.

  • SIEGEL, A.F., and BENSON, R.H. (1982), “A Robust Comparison of Biological Shapes”, Biometrics, 38, 341–350.

    Article  MATH  Google Scholar 

  • SMITH, D.R., CRESPI, B.J., and BOOKSTEIN, F.L. (1997), “Fluctuating Asymmetry in the Honey Bee, Apis Mellifera: Effects of Ploidy and Hybridization”, Journal of Evolutionary Biology, 10, 551–574.

  • SYMINGTON, A., CHARLTON, M. E., and BRUNSDON, C.F. (2002), “Using Bidimensional Regression to Explore Map Lineage”, Computers, Environment and Urban Systems, 26, 201–218.

    Article  Google Scholar 

  • TEN BERGE, J.M.F. (1977), “Orthogonal Procrustes Rotation for Two or More Matrices”, Psychometrika, 42, 267–276.

    Article  MathSciNet  MATH  Google Scholar 

  • TEN BERGE, J.M.F. (1979), “On the Equivalence of Two Oblique Congruence Rotation Methods, and Orthogonal Approximations”, Psychometrika, 44, 359–364.

    Article  MathSciNet  MATH  Google Scholar 

  • TEN BERGE, J.M.F., and BEKKER, P.A. (1993), “The Isotropic Scaling Problem in Generalized Procrustes Analysis”, Computational Statistics and Data Analysis, 16, 201–204.

    Article  MATH  Google Scholar 

  • TEN BERGE, J.M.F., and KNOL, D.L. (1984), “Orthogonal Rotations to Maximal Agreement for Two or More Matrices of Different Column Orders”, Psychometrika, 49, 49–55.

    Article  Google Scholar 

  • TEN BERGE, J.M.F., and NEVELS, K. (1977), “A General Solution to Mosier’s Oblique Procrustes Problem”, Psychometrika, 42, 593–600.

    Article  MathSciNet  MATH  Google Scholar 

  • TOBLER, W.R. (1965), “Computation of the Correspondence of Geographical Patterns”, Papers of the Regional Science Association, 15, 131–139.

    Article  Google Scholar 

  • TOBLER,W.R. (1966), “Medieval Distortions: The Projections of Ancient Maps”, Annals of the Association of American Geographers, 56, 351–360.

    Article  Google Scholar 

  • TOBLER, W.R. (1978), “Comparing Figures by Regression”, Computer Graphics, 12, 193–195.

    Article  Google Scholar 

  • TOBLER,W.R. (1994), “Bidimensional Regression”, Geographical Analysis, 26, 187–212.

    Article  Google Scholar 

  • TRENDAFILOV, N.T. (2003), “On the ℓ1 Procrustes Problem”, Future Generation Computer Systems, 19, 1177–1186.

    Article  Google Scholar 

  • TRENDAFILOV, N.T., and WATSON, G.A. (2004), “The ℓ1 Oblique Procrustes Problem”, Statistics and Computing, 14, 39–51.

    Article  MathSciNet  Google Scholar 

  • VERBOON, P., and HEISER, W.J. (1992), “Resistant Orthogonal Procrustes Analysis”, Journal of Classification, 9, 237–256.

    Article  MATH  Google Scholar 

  • WALLER, D., and HAUN, D.B.M. (2003), “Scaling Techniques for Modeling Directional Knowledge”, Behavior Research Methods, Instruments, and Computers, 35, 285–293.

    Article  Google Scholar 

  • WATERMAN, S., and GORDON, D. (1984), “A Quantitative-Comparative Approach to Analysis of Distortion in Mental Maps”, Professional Geographer, 36, 326–337.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Psychology, University of Illinois at Urbana-Champaign, 603 East Daniel Street, Champaign, Illinois, 61820, USA

    Justin L. Kern

Authors
  1. Justin L. Kern
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Justin L. Kern.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kern, J.L. On the Correspondence Between Procrustes Analysis and Bidimensional Regression. J Classif 34, 35–48 (2017). https://doi.org/10.1007/s00357-017-9224-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00357-017-9224-z

Keywords

Search

Navigation