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.
Abstract. It has been argued persuasively that, in order to evaluate climate models, the probability distributions of model output need to be compared to the corresponding empirical distributions of observed data. Distance measures between probability distributions, also called divergence functions, can be used for this purpose. We contend that divergence functions ought to be proper, in the sense that acting on modelers' true beliefs is an optimal strategy. The score divergences introduced in this paper derive from proper scoring rules and, thus, they are proper with the integrated quadratic distance and the Kullback-Leibler divergence being particularly attractive choices. Other commonly used divergences fail to be proper. In an illustration, we evaluate and rank simulations from 15 climate models for temperature extremes in a comparison to reanalysis data. 1. Introduction. Traditionally, climate models have been assessed by comparing summary statistics or point estimates that derive from the simulated model output to the corresponding observed quantities [1,2]. However, there is a growing consensus that in order to evaluate climate models the probability distribution of model output needs to be compared to the corresponding distribution of observed data. Guttorp [3, p. 820] argues powerfully the following:Climate models are difficult to compare to data. Often climatologists compute some summary statistic, such as global annual mean temperature, and compare climate models using observed (or rather estimated) forcings to the observed (or rather estimated) temperatures. However, it seems more appropriate to compare the distribution (over time and space) of climate model output to the corresponding distribution of observed data, as opposed to point estimates with or without confidence intervals.
We address the identifiablity and estimation of recursive max-linear structural equation models represented by an edge weighted directed acyclic graph (DAG). Such models are generally unidentifiable and we identify the whole class of DAGs and edge weights corresponding to a given observational distribution. For estimation, standard likelihood theory cannot be applied because the corresponding families of distributions are not dominated. Given the underlying DAG, we present an estimator for the class of edge weights and show that it can be considered a generalized maximum likelihood estimator. In addition, we develop a simple method for identifying the structure of the DAG. With probability tending to one at an exponential rate with the number of observations, this method correctly identifies the class of DAGs and, similarly, exactly identifies the possible edge weights.MSC 2010 subject classifications: Primary 60E15, 62H12; secondary 62G05, 60G70, 62-09
Reliable projections of extremes by climate models are becoming increasingly important in the context of climate change and associated societal impacts. Extremes are by definition rare events, characterized by a small sample associated with large uncertainties. The evaluation of extreme events in model simulations thus requires performance measures that compare full distributions rather than simple summaries. This paper proposes the use of the integrated quadratic distance (IQD) for this purpose. The IQD is applied to evaluate CMIP5 and CMIP6 simulations of monthly maximum and minimum near-surface air temperature over Europe and North America against both observation-based data and reanalyses. Several climate models perform well to the extent that these models’ performance is competitive with the performance of another data product in simulating the evaluation set. While the model rankings vary with region, season and index, the model evaluation is robust against changes in the grid resolution considered in the analysis. When the model simulations are ranked based on their similarity with the ERA5 reanalysis, more CMIP6 than CMIP5 models appear at the top of the ranking. When evaluated against the HadEX2 data product, the overall performance of the two model ensembles is similar.
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