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Graph-Theoretic Procedures for Dimension Identification

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  • Brito, María R.
  • Quiroz, Adolfo J.
  • Yukich, J. E.

Abstract

We consider the problem of identifying the dimension in which a sample of data points lives, when only their interpoint distances are known. We study as a random variable the average "reach" of vertices in the k-nearest-neighbors graph associated to the interpoint distance matrix, and we show how this variable can be used to accurately (from a probabilistic viewpoint) identify the unknown dimension at low computational cost. We discuss results that serve as the theoretical foundation for the methodology proposed. We illustrate how our method can help in dimension reduction procedures.

Suggested Citation

  • Brito, María R. & Quiroz, Adolfo J. & Yukich, J. E., 2002. "Graph-Theoretic Procedures for Dimension Identification," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 67-84, April.
    • Handle: RePEc:eee:jmvana:v:81:y:2002:i:1:p:67-84
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    References listed on IDEAS

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    1. Brito, M. R. & Chávez, E. L. & Quiroz, A. J. & Yukich, J. E., 1997. "Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection," Statistics & Probability Letters, Elsevier, vol. 35(1), pages 33-42, August.
    2. Wishik, S.M., 1978. "The use of incentives for fertility reduction," American Journal of Public Health, American Public Health Association, vol. 68(2), pages 113-114.
    3. J. Kruskal, 1964. "Nonmetric multidimensional scaling: A numerical method," Psychometrika, Springer;The Psychometric Society, vol. 29(2), pages 115-129, June.
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    Cited by:

    1. S. Camelo & M. González-Lima & A. Quiroz, 2015. "Nearest neighbors methods for support vector machines," Annals of Operations Research, Springer, vol. 235(1), pages 85-101, December.
    2. González-Barrios, José María & Quiroz, Adolfo J., 2003. "A clustering procedure based on the comparison between the k nearest neighbors graph and the minimal spanning tree," Statistics & Probability Letters, Elsevier, vol. 62(1), pages 23-34, March.
    3. Díaz, Mateo & Quiroz, Adolfo J. & Velasco, Mauricio, 2019. "Local angles and dimension estimation from data on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 229-247.
    4. Brito, M.R. & Quiroz, A.J. & Yukich, J.E., 2013. "Intrinsic dimension identification via graph-theoretic methods," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 263-277.

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