Max-linear models on directed acyclic graphs

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 a unique 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 a minimum number of node and noise variables. Primary 60G70, 05C20; secondary 05C75.

Right Singular Vector Projection Graphs: Fast High Dimensional Covariance Matrix Estimation under Latent Confounding

“…As an additional test of robustness, we consider, in a second set of experiments, replacing the linear structural equation x = w +Γ h (1) with a max‐linear model (Gissibl and Klüppelberg, )our goal is as before to recover Σ=cov( w ). We present in Fig.…”

“…For example, it would be interesting to explore whether there are other estimators of the form (4) that depend on the spectrum of such that the scale of Σ is not lost, but in a sufficiently smooth way as not to have high variance even in the challenging scenarios that were mentioned above. Another interesting problem is that of controlling for latent confounding when the influence of the confounding is not linear, such as the max‐linear settings (Gissibl and Klüppelberg, ).…”

“…As an additional test of robustness, we consider, in a second set of experiments, replacing the linear structural equation x = w + Γh (1) with a max-linear model (Gissibl and Klüppelberg, 2018) x j = max{w j , .Γh/ j };…”

Identifiability and estimation of recursive max‐linear models

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Bayesian Networks for Max-Linear Models

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