Additive Trees for Fitting Three-Way (Multiple Source) Proximity Data

  • Hans-Friedrich KöhnEmail author
  • Justin L. Kern
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 265)


Additive trees are graph-theoretic models that can be used for constructing network representations of pairwise proximity data observed on a set of N objects. Each object is represented as a terminal node in a connected graph; the length of the paths connecting the nodes reflects the inter-object proximities. Carroll, Clark, and DeSarbo (J Classif 1:25–74, 1984) developed the INDTREES algorithm for fitting additive trees to analyze individual differences of proximity data collected from multiple sources. INDTREES is a mathematical programming algorithm that uses a conjugate gradient strategy for minimizing a least-squares loss function augmented by a penalty term to account for violations of the constraints as imposed by the underlying tree model. This article presents an alternative method for fitting additive trees to three-way two-mode proximity data that does not rely on gradient-based optimization nor on penalty terms, but uses an iterative projection algorithm. A real-world data set consisting of 22 proximity matrices illustrated that the proposed method gave virtually identical results as the INDTREES method.


Additive trees Three-way data Individual differences Iterative projection 


  1. Barthélemy, J. P., & Guénoche, A. (1991). Tree and proximity representations. Chichester: Wiley.Google Scholar
  2. Boyle, J. P., & Dykstra, R. L. (1985). A method for finding projections onto the intersection of convex sets in Hilbert spaces. In R. L. Dykstra, R. Robertson, & F. T. Wright (Eds.), Advances in order restricted inference (Lecture Notes in Statistics, Vol 37) (pp. 28–47). Berlin: Springer.Google Scholar
  3. Carroll, J. D. (1976). Spatial, non-spatial and hybrid models for scaling. Psychometrika, 41, 439–463.CrossRefGoogle Scholar
  4. Carroll, J. D., & Pruzansky, S. (1980). Discrete and hybrid scaling models. In E. Lantermann & H. Feger (Eds.), Similarity and choice (pp. 108–139). Bern: Huber.Google Scholar
  5. Carroll, J. D., Clark, L. A., & DeSarbo, W. S. (1984). The representation of three-way proximities data by single and multiple tree structure models. Journal of Classification, 1, 25–74.CrossRefGoogle Scholar
  6. Corter, J. E. (1982). ADDTREE/P: A PASCAL program for fitting additive trees based on Sattath and Tversky’s ADDTREE algorithm. Behavior Research Methods and Instrumentation, 14, 353–354.CrossRefGoogle Scholar
  7. De Soete, G. (1983). A least-squares algorithm for fitting additive trees to proximity data. Psychometrika, 48, 621–626.CrossRefGoogle Scholar
  8. De Soete, G., & Carroll, J. D. (1989). Ultrametric tree representations of three-way three-mode data. In R. Coppi & S. Belasco (Eds.), Analysis of multiway data matrices (pp. 415–426). Amsterdam: North Holland.Google Scholar
  9. De Soete, G., & Carroll, J. D. (1996). Tree and other network models for representing proximity data. In P. Arabie, L. J. Hubert, & G. De Soete (Eds.), Clustering and classification (pp. 157–197). River Edge, NJ: World Scientific.Google Scholar
  10. Deutsch, F. (2001). Best approximation in inner product spaces. New York: Springer.Google Scholar
  11. Dykstra, R. L. (1983). An algorithm for restricted least-squares regression. Journal of the American Statistical Association, 78, 837–842.MathSciNetCrossRefGoogle Scholar
  12. Fletcher, R., & Reeves, C. M. (1964). Function minimization by conjugate gradients. Computer Journal, 7, 149–154.MathSciNetCrossRefGoogle Scholar
  13. Han, S. P. (1988). A successive projection method. Mathematical Programming, 40, 1–14.MathSciNetCrossRefGoogle Scholar
  14. Hubert, L. J., & Arabie, P. (1995). Iterative projection strategies for the least-squares fitting of tree structures to proximity data. British Journal of Mathematical and Statistical Psychology, 48, 281–317.CrossRefGoogle Scholar
  15. Hubert, L. J., Arabie, P., & Meulman, J. (2006). The structural representation of proximity matrices with MATLAB. Philadelphia: SIAM.Google Scholar
  16. Hornik, K. (2018). clue: Cluster Ensembles. R package version 0.3-56. Retrieved from the Comprehensive R Archive Network [CRAN] website
  17. Křivánek, M. (1986). On the computational complexity of clustering. In E. Diday, Y. Escouffier, L. Lebart, J. P. Pagés, Y. Schektman, & R. Tomassone (Eds.), Data analysis and information, IV (pp. 89–96). Amsterdam: North Holland.Google Scholar
  18. Sattath, S., & Tversky, A. (1977). Additive similarity trees. Psychometrika, 42, 319–345.CrossRefGoogle Scholar
  19. Semple, C., & Steel, M. (2003). Phylogenetics. Oxford, UK: Oxford University Press.Google Scholar
  20. Smith, T. J. (1998). A comparison of three additive tree algorithms that rely on a least-squares loss criterion. The British Journal of Mathematical and Statistical Psychology, 51, 269–288.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Educational PsychologyUniversity of Illinois at Urbana-ChampaignChampaignUSA

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