The Annals of Applied Statistics

Multilinear tensor regression for longitudinal relational data

Peter D. Hoff

Full-text: Open access

Abstract

A fundamental aspect of relational data, such as from a social network, is the possibility of dependence among the relations. In particular, the relations between members of one pair of nodes may have an effect on the relations between members of another pair. This article develops a type of regression model to estimate such effects in the context of longitudinal and multivariate relational data, or other data that can be represented in the form of a tensor. The model is based on a general multilinear tensor regression model, a special case of which is a tensor autoregression model in which the tensor of relations at one time point are parsimoniously regressed on relations from previous time points. This is done via a separable, or Kronecker-structured, regression parameter along with a separable covariance model. In the context of an analysis of longitudinal multivariate relational data, it is shown how the multilinear tensor regression model can represent patterns that often appear in relational and network data, such as reciprocity and transitivity.

Article information

Source
Ann. Appl. Stat., Volume 9, Number 3 (2015), 1169-1193.

Dates
Received: November 2014
Revised: May 2015
First available in Project Euclid: 2 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1446488735

Digital Object Identifier
doi:10.1214/15-AOAS839

Mathematical Reviews number (MathSciNet)
MR3418719

Zentralblatt MATH identifier
06525982

Keywords
Array normal Bayesian inference event data international relations network Tucker product vector autoregression

Citation

Hoff, Peter D. Multilinear tensor regression for longitudinal relational data. Ann. Appl. Stat. 9 (2015), no. 3, 1169--1193. doi:10.1214/15-AOAS839. https://projecteuclid.org/euclid.aoas/1446488735


Export citation

References

  • Akdemir, D. and Gupta, A. K. (2011). Array variate random variables with multiway Kronecker delta covariance matrix structure. J. Algebr. Stat. 2 98–113.
  • Basu, S., Dunagan, J., Duh, K. and Muniswamy-Reddy, K.-K. (2012). Blr-d: Applying bilinear logistic regression to factored diagnosis problems. ACM SIGOPS Operating Systems Review 45 31–38.
  • De Lathauwer, L., De Moor, B. and Vandewalle, J. (2000). A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21 1253–1278 (electronic).
  • Durante, D. and Dunson, D. B. (2014). Nonparametric Bayes dynamic modelling of relational data. Biometrika 101 883–898.
  • Fu, W., Song, L. and Xing, E. P. (2009). Dynamic mixed membership blockmodel for evolving networks. In Proceedings of the 26th Annual International Conference on Machine Learning 329–336. ACM, New York.
  • Gabriel, K. R. (1998). Generalised bilinear regression. Biometrika 85 689–700.
  • Hanneke, S., Fu, W. and Xing, E. P. (2010). Discrete temporal models of social networks. Electron. J. Stat. 4 585–605.
  • Hoff, P. (2008). Modeling homophily and stochastic equivalence in symmetric relational data. In Advances in Neural Information Processing Systems (J. Platt, D. Koller, Y. Singer and S. Roweis, eds.) 20 657–664. MIT Press, Cambridge, MA.
  • Hoff, P. D. (2007). Extending the rank likelihood for semiparametric copula estimation. Ann. Appl. Stat. 1 265–283.
  • Hoff, P. D. (2011a). Hierarchical multilinear models for multiway data. Comput. Statist. Data Anal. 55 530–543.
  • Hoff, P. D. (2011b). Separable covariance arrays via the Tucker product, with applications to multivariate relational data. Bayesian Anal. 6 179–196.
  • Kolda, T. G. and Bader, B. W. (2009). Tensor decompositions and applications. SIAM Rev. 51 455–500.
  • Krivitsky, P. N. and Handcock, M. S. (2014). A separable model for dynamic networks. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 29–46.
  • Li, X., Zhou, H. and Li, L. (2013). Tucker tensor regression and neuroimaging analysis. Available at arXiv:1304.5637.
  • Luenberger, D. G. and Ye, Y. (2008). Linear and Nonlinear Programming, 3rd ed. International Series in Operations Research & Management Science 116. Springer, New York.
  • Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • Pettitt, A. N. (1982). Inference for the linear model using a likelihood based on ranks. J. R. Stat. Soc. Ser. B. Stat. Methodol. 44 234–243.
  • Potthoff, R. F. and Roy, S. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika 51 313–326.
  • Shi, J. V., Xu, Y. and Baraniuk, R. G. (2014) Sparse bilinear logistic regression. Available at arXiv:1404.4104.
  • Snijders, T. A. (2001). The statistical evaluation of social network dynamics. Sociological Methodology 31 361–395.
  • Snijders, T., Steglich, C. and Schweinberger, M. (2007). Modeling the coevolution of networks and behavior. In Longitudinal Models in the Behavioral and Related Sciences (K. van Montfort, J. Oud and A. Satorra, eds.) 41–71. Lawrence Erlbaum Associates, Mahwah, NJ.
  • Srivastava, M. S., von Rosen, T. and von Rosen, D. (2009). Estimation and testing in general multivariate linear models with Kronecker product covariance structure. Sankhyā 71 137–163.
  • Tucker, L. R. (1964). The extension of factor analysis to three-dimensional matrices. In Contributions to Mathematical Psychology (H. Gulliksen and N. Frederiksen, eds.) 110–127. Holt, Rinehart and Winston, New York.
  • Ward, M. D., Ahlquist, J. S. and Rozenas, A. (2013). Gravity’s rainbow: A dynamic latent space model for the world trade network. Network Science 1 95–118.
  • Ward, M. D. and Hoff, P. D. (2007). Persistent patterns of international commerce. Journal of Peace Research 44 157–175.
  • Westveld, A. H. and Hoff, P. D. (2011). A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict. Ann. Appl. Stat. 5 843–872.
  • White, H. (1981). Consequences and detection of misspecified nonlinear regression models. J. Amer. Statist. Assoc. 76 419–433.
  • White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50 1–25.
  • Xing, E. P., Fu, W. and Song, L. (2010). A state-space mixed membership blockmodel for dynamic network tomography. Ann. Appl. Stat. 4 535–566.