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Max-linear models on directed acyclic graphs. (English) Zbl 1419.62138

Summary: 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.

MSC:

62H12 Estimation in multivariate analysis
05C90 Applications of graph theory

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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. · Zbl 0247.05128
[2] Bollen, K.A. (1989). Structural Equations with Latent Variables. Wiley Series in Probability and Mathematical Statistics : Applied Probability and Statistics . New York: Wiley. · Zbl 0731.62159
[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. · Zbl 1309.62063
[4] Butkovič, P. (2010). Max-linear Systems : Theory and Algorithms. Springer Monographs in Mathematics . London: Springer. · Zbl 1202.15032
[5] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory : An Introduction. Springer Series in Operations Research and Financial Engineering . New York: Springer. · Zbl 1101.62002
[6] Diestel, R. (2010). Graph Theory , 4th ed. Graduate Texts in Mathematics 173 . Heidelberg: Springer. · Zbl 1204.05001
[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 . · Zbl 1408.62077
[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,. · Zbl 0907.62001
[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. · Zbl 0743.05065
[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. · Zbl 1188.68291
[13] Pourret, O., Naim, P. and Marcot, B., eds. (2008) Bayesian Networks : A Practical Guide to Applications. Statistics in Practice . Chichester: Wiley. · Zbl 1275.62010
[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. · Zbl 0633.60001
[15] Resnick, S.I. (2007). Heavy-Tail Phenomena : Probabilistic and Statistical modeling. Springer Series in Operations Research and Financial Engineering . New York: Springer. · Zbl 1152.62029
[16] Rote, G. (1985). A systolic array algorithm for the algebraic path problem (shortest paths; matrix inversion). Computing 34 191-219. · Zbl 0562.68056
[17] Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation , Prediction , and Search , 2nd ed. Adaptive Computation and Machine Learning . Cambridge, MA: MIT Press. · Zbl 0806.62001
[18] Wang, Y. and Stoev, S.A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Adv. in Appl. Probab. 43 461-483. · Zbl 1225.60085
[19] Wright, S. · Zbl 0010.31305
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