Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2693-2720.

Max-linear models on directed acyclic graphs

Nadine Gissibl and Claudia Klüppelberg

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a new recursive structural equation model where all variables can be written as max-linear function of their parental node variables and independent noise variables. The model is max-linear in terms of the noise variables, and its causal structure is represented by a directed acyclic graph. We detail the relation between the weights of the recursive structural equation model and the coefficients in its max-linear representation. In particular, we characterize all max-linear models which are generated by a recursive structural equation model, and show that its max-linear coefficient matrix is the solution of a fixed point equation. We also find the minimum directed acyclic graph representing the recursive structural equations of the variables. The model structure introduces a natural order between the node variables and the max-linear coefficients. This yields representations of the vector components, which are based on the minimum number of node and noise variables.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2693-2720.

Dates
Received: January 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051222

Digital Object Identifier
doi:10.3150/17-BEJ941

Mathematical Reviews number (MathSciNet)
MR3779699

Zentralblatt MATH identifier
06853262

Keywords
directed acyclic graph graphical model max-linear model minimal representation path analysis structural equation model

Citation

Gissibl, Nadine; Klüppelberg, Claudia. Max-linear models on directed acyclic graphs. Bernoulli 24 (2018), no. 4A, 2693--2720. doi:10.3150/17-BEJ941. https://projecteuclid.org/euclid.bj/1522051222


Export citation

References

  • [1] Aho, A.V., Garey, M.R. and Ullman, J.D. (1972). The transitive reduction of a directed graph. SIAM J. Comput. 1 131–137.
  • [2] Bollen, K.A. (1989). Structural Equations with Latent Variables. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [3] Bühlmann, P., Peters, J. and Ernest, J. (2014). CAM: Causal additive models, high-dimensional order search and penalized regression. Ann. Statist. 42 2526–2556.
  • [4] Butkovič, P. (2010). Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. London: Springer.
  • [5] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [6] Diestel, R. (2010). Graph Theory, 4th ed. Graduate Texts in Mathematics 173. Heidelberg: Springer.
  • [7] Ernest, J., Rothenhäusler, D. and Bühlmann, P. (2016). Causal inference in partially linear structural equation models: Identifiability and estimation. Preprint. Available at arXiv:1607.05980.
  • [8] Koller, D. and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. Adaptive Computation and Machine Learning. Cambridge, MA: MIT Press.
  • [9] Lauritzen, S.L. (1996). Graphical Models. Oxford Statistical Science Series 17. New York: Oxford Univ. Press,.
  • [10] Lauritzen, S.L., Dawid, A.P., Larsen, B.N. and Leimer, H.-G. (1990). Independence properties of directed Markov fields. Networks 20 491–505.
  • [11] Mahr, B. (1981). A bird’s-eye view to path problems. In Graph-Theoretic Concepts in Computer Science (Proc. Sixth Internat. Workshop, Bad Honnef, 1980). Lecture Notes in Computer Science 100 335–353. Springer, Berlin-New York.
  • [12] Pearl, J. (2009). Causality: Models, Reasoning, and Inference, 2nd ed. Cambridge: Cambridge Univ. Press.
  • [13] Pourret, O., Naim, P. and Marcot, B., eds. (2008) Bayesian Networks: A Practical Guide to Applications. Statistics in Practice. Chichester: Wiley.
  • [14] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. New York: Springer.
  • [15] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [16] Rote, G. (1985). A systolic array algorithm for the algebraic path problem (shortest paths; matrix inversion). Computing 34 191–219.
  • [17] Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. Adaptive Computation and Machine Learning. Cambridge, MA: MIT Press.
  • [18] Wang, Y. and Stoev, S.A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Adv. in Appl. Probab. 43 461–483.
  • [19] Wright, S. (1934). The method of path coefficients. Ann. Math. Stat. 5 161–215.