Bayesian Analysis

Separable covariance arrays via the Tucker product, with applications to multivariate relational data

Peter D. Hoff

Full-text: Open access

PDF File (299 KB)

Abstract

Modern datasets are often in the form of matrices or arrays, potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we discuss an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We show how a particular array-matrix product can be used to generate the class of array normal distributions having separable covariance structure. We derive some properties of these covariance structures and the corresponding array normal distributions, and show how the array-matrix product can be used to define a semi-conjugate prior distribution and calculate the corresponding posterior distribution. We illustrate the methodology in an analysis of multivariate longitudinal network data which take the form of a four-way array.

Article information

Source
Bayesian Anal., Volume 6, Number 2 (2011), 179-196.

Dates
First available in Project Euclid: 13 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1339612040

Digital Object Identifier
doi:10.1214/11-BA606

Mathematical Reviews number (MathSciNet)
MR2806238

Zentralblatt MATH identifier
1330.62132

Subjects
Primary: 62F15: Bayesian inference
Secondary: 60E05: Distributions: general theory 62H05: Characterization and structure theory 62H12: Estimation 62J10: Analysis of variance and covariance 62P20: Applications to economics [See also 91Bxx]

Keywords
Gaussian matrix normal multiway data network tensor Tucker decomposition

Citation

Hoff, Peter D. Separable covariance arrays via the Tucker product, with applications to multivariate relational data. Bayesian Anal. 6 (2011), no. 2, 179--196. doi:10.1214/11-BA606. https://projecteuclid.org/euclid.ba/1339612040


Export citation

References

  • Browne, M. W. and Shapiro, A. (1991). "Invariance of covariance structures under groups of transformations." Metrika, 38(6): 345–355.
    Mathematical Reviews (MathSciNet): MR1144977
    Digital Object Identifier: doi:10.1007/BF02613631
    Zentralblatt MATH: 0757.62004
  • Carvalho, C. M. and West, M. (2007). "Dynamic Matrix-variate Graphical Models." Bayesian Analysis, 2(1): 69–98.
    Mathematical Reviews (MathSciNet): MR2289924
    Digital Object Identifier: doi:10.1214/07-BA204
  • Dawid, A. P. (1981). "Some matrix-variate distribution theory: notational considerations and a Bayesian application." Biometrika, 68(1): 265–274.
    Mathematical Reviews (MathSciNet): MR614963
    Zentralblatt MATH: 0464.62039
    Digital Object Identifier: doi:10.1093/biomet/68.1.265
  • De Lathauwer, L., De Moor, B., and Vandewalle, J. (2000). "A multilinear singular value decomposition"." SIAM Journal on Matrix Analysis and Applications, 21(4): 1253–1278.
    Mathematical Reviews (MathSciNet): MR1780272
    Zentralblatt MATH: 0962.15005
    Digital Object Identifier: doi:10.1137/S0895479896305696
  • Dutilleul, P. (1999). "The MLE algorithm for the matrix normal distribution." Journal of Statistical Computation and Simulation, 64(2): 105–123.
    Zentralblatt MATH: 0960.62056
  • Galecki, A. (1994). "General class of covariance structures for two or more repeated factors in longitudinal data analysis." Communications in Statistics-Theory and Methods, 23(11): 3105–3119.
    Zentralblatt MATH: 0875.62274
  • Kass, R. E. and Wasserman, L. (1995). "A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion." J. Amer. Statist. Assoc., 90(431): 928–934.
    Mathematical Reviews (MathSciNet): MR1354008
    Zentralblatt MATH: 0851.62020
  • Kolda, T. G. (2006). "Multilinear operators for higher-order decompositions." Technical Report SAND2006-2081, Sandia National Laboratories, Albuquerque, NM and Livermore, CA.
  • Kroonenberg, P. M. (2008). Applied multiway data analysis. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley-Interscience [John Wiley & Sons]. With a foreword by Willem J. Heiser and Jarqueline Meulman.
    Mathematical Reviews (MathSciNet): MR2378349
    Zentralblatt MATH: 1160.62002
  • Lu, N. and Zimmerman, D. L. (2005). "The likelihood ratio test for a separable covariance matrix." Statist. Probab. Lett., 73(4): 449–457.
    Mathematical Reviews (MathSciNet): MR2187860
  • McCullagh, P. (1987). Tensor methods in statistics. Monographs on Statistics and Applied Probability. London: Chapman & Hall.
    Mathematical Reviews (MathSciNet): MR907286
    Zentralblatt MATH: 0732.62003
  • Mitchell, M. W., Genton, M. G., and Gumpertz, M. L. (2006). "A likelihood ratio test for separability of covariances." J. Multivariate Anal., 97(5): 1025–1043.
    Mathematical Reviews (MathSciNet): MR2276147
    Zentralblatt MATH: 1089.62063
    Digital Object Identifier: doi:10.1016/j.jmva.2005.07.005
  • Quintana, J. M. and West, M. (1988). "Time Series Analysis of Compositional Data." In Bernardo, J. M., DeGroot, M. H., Lindley, D. V., and Smith, A. F. M. (eds.), Bayesian Statistics 3, 747–756. Clarendon Press [Oxford University Press].
  • Roy, A. and Khattree, R. (2005). "On implementation of a test for Kronecker product covariance structure for multivariate repeated measures data." Stat. Methodol., 2(4): 297–306.
    Mathematical Reviews (MathSciNet): MR2205602
    Digital Object Identifier: doi:10.1016/j.stamet.2005.07.003
  • Rubin, D. B. (1984). "Bayesianly justifiable and relevant frequency calculations for the applied statistician." Ann. Statist., 12(4): 1151–1172.
    Mathematical Reviews (MathSciNet): MR760681
    Zentralblatt MATH: 0555.62010
    Digital Object Identifier: doi:10.1214/aos/1176346785
    Project Euclid: euclid.aos/1176346785
  • Tseng, P. (2001). "Convergence of a block coordinate descent method for nondifferentiable minimization." J. Optim. Theory Appl., 109(3): 475–494.
    Mathematical Reviews (MathSciNet): MR1835069
    Zentralblatt MATH: 1006.65062
    Digital Object Identifier: doi:10.1023/A:1017501703105
  • Tucker, L. (1966). "Some mathematical notes on three-mode factor analysis"." Psychometrika, 31(3): 279–311.
    Mathematical Reviews (MathSciNet): MR205395
    Digital Object Identifier: doi:10.1007/BF02289464
  • Tucker, L. R. (1964). "The extension of factor analysis to three-dimensional matrices"." Contributions to mathematical psychology, 109–127.
  • Wang, H. and West, M. (2009). "Bayesian analysis of matrix normal graphical models." Biometrika, 96(4): 821–834.
    Mathematical Reviews (MathSciNet): MR2564493
    Zentralblatt MATH: 1179.62042
    Digital Object Identifier: doi:10.1093/biomet/asp049
  • West, M. and Harrison, J. (1997). Bayesian forecasting and dynamic models. Springer Series in Statistics. New York: Springer-Verlag, second edition.
    Mathematical Reviews (MathSciNet): MR1482232

See also

  • Related item: Genevera I. Allen. Comment on article by Hoff. Bayesian Anal., Vol. 6, Iss. 2 (2011), 197-201.
    Project Euclid: euclid.ba/1339612041
  • Related item: Hedibert Freitas Lopes. Comment on article by Hoff. Bayesian Anal., Vol. 6, Iss. 2 (2011), 203-204.
    Project Euclid: euclid.ba/1339612042

International Society for Bayesian Analysis

Bayesian Analysis

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more