The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether $X$ causes $Y$ or, alternatively, $Y$ causes $X$, given joint observations of two variables $X,Y$. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: methods based on Additive Noise Models (ANMs) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 data sets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the ground truth causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the method based on Additive Noise Models that has originally been proposed by Hoyer et al. (2009), which obtains an accuracy of 63 $\pm$ 10 % and an AUC of 0.74 $\pm$ 0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method.

The accurate estimation of carbon and heat fluxes at global scale is paramount for future policy decisions in the context of global climate change. This paper analyzes the relative relevance of potential remote sensing and meteorological drivers of global carbon and energy fluxes over land. The study is done in an indirect way via upscaling both Gross Primary Production (GPP) and latent energy (LE) using Gaussian Process regression (GPR). In summary, GPR is successfully compared to multivariate linear regression (RMSE gain of +4.17% in GPP and +7.63% in LE) and kernel ridge regression (+2.91% in GPP and +3.07% in LE). The best GP models are then studied in terms of explanatory power based on the analysis of the lengthscales of the anisotropic covariance function, sensitivity maps of the predictive mean, and the robustness to distortions in the input variables. It is concluded that GPP is predominantly mediated by several vegetation indices and land surface temperature (LST), while LE is mostly driven by LST, global radiation and vegetation indices.