Journal topic
Biogeosciences, 17, 1033–1061, 2020
https://doi.org/10.5194/bg-17-1033-2020
Biogeosciences, 17, 1033–1061, 2020
https://doi.org/10.5194/bg-17-1033-2020

Research article 26 Feb 2020

Research article | 26 Feb 2020

# Estimating causal networks in biosphere–atmosphere interaction with the PCMCI approach

Estimating causal networks in biosphere–atmosphere interaction with the PCMCI approach
Christopher Krich1,2, Jakob Runge3, Diego G. Miralles2, Mirco Migliavacca1, Oscar Perez-Priego1, Tarek El-Madany1, Arnaud Carrara4, and Miguel D. Mahecha1,5 Christopher Krich et al.
• 1Max Planck Institute for Biogeochemistry, 07745 Jena, Germany
• 2Laboratory of Hydrology and Water Management, Ghent University, Ghent 9000, Belgium
• 3German Aerospace Center, Institute of Data Science, 07745 Jena, Germany
• 4Fundación Centro de Estudios Ambientales del Mediterráneo (CEAM), 46980 Paterna, Spain
• 5German Centre for Integrative Biodiversity Research (iDiv), Deutscher Platz 5e, 04103 Leipzig, Germany

Correspondence: Christopher Krich (ckrich@bgc-jena.mpg.de)

Abstract

The dynamics of biochemical processes in terrestrial ecosystems are tightly coupled to local meteorological conditions. Understanding these interactions is an essential prerequisite for predicting, e.g. the response of the terrestrial carbon cycle to climate change. However, many empirical studies in this field rely on correlative approaches and only very few studies apply causal discovery methods. Here we explore the potential for a recently proposed causal graph discovery algorithm to reconstruct the causal dependency structure underlying biosphere–atmosphere interactions. Using artificial time series with known dependencies that mimic real-world biosphere–atmosphere interactions we address the influence of non-stationarities, i.e. periodicity and heteroscedasticity, on the estimation of causal networks. We then investigate the interpretability of the method in two case studies. Firstly, we analyse three replicated eddy covariance datasets from a Mediterranean ecosystem. Secondly, we explore global Normalised Difference Vegetation Index time series (GIMMS 3g), along with gridded climate data to study large-scale climatic drivers of vegetation greenness. We compare the retrieved causal graphs to simple cross-correlation-based approaches to test whether causal graphs are considerably more informative. Overall, the results confirm the capacity of the causal discovery method to extract time-lagged linear dependencies under realistic settings. For example, we find a complete decoupling of the net ecosystem exchange from meteorological variability during summer in the Mediterranean ecosystem. However, cautious interpretations are needed, as the violation of the method's assumptions due to non-stationarities increases the likelihood to detect false links. Overall, estimating directed biosphere–atmosphere networks helps unravel complex multidirectional process interactions. Other than classical correlative approaches, our findings are constrained to a few meaningful sets of relations, which can be powerful insights for the evaluation of terrestrial ecosystem models.

1 Introduction

The terrestrial biosphere responds to atmospheric drivers such as radiation intensity, temperature, vapour pressure deficit, and composition of trace gases. On the other hand, the biosphere influences the atmosphere via partitioning incoming net radiation into sensible, latent, and ground heat fluxes, as well as via controlling the exchange of trace gases and volatile organic compounds. Over the past few decades, many of these processes have been identified and their physical, chemical, and biological effects have been investigated (see, e.g. Monson and Baldocchi2014; McPherson2007, for overviews). However, there are still substantial unknowns regarding the exact causal dependencies among the different processes , which leads to large uncertainties when predicting, e.g. ecosystem responses to drought conditions .

Multiple ecological monitoring systems have been set up to monitor ecosystem dynamics. Networks of eddy covariance towers continuously monitor carbon, water, and energy fluxes in high temporal resolution . Satellite remote sensing data complement this picture and can be used in tandem . They typically only monitor vegetation states at multi-day resolutions and some products offer nearly complete global coverage . The actual and future satellite missions are leading to rapid development in the field with ever higher spatial, temporal, and spectral measurements .

The study of biosphere–atmosphere interactions using observations typically relies on correlative approaches, or is based on model data, i.e. requires a priori knowledge. In recent years, a new branch in statistics aiming for causal inference from empirical data has experienced substantial progress. The idea of causal inference had already emerged in the early 20th century (Wright1921). Later, suggested one of the first applicable formalisms; since then, several efforts in ecology and climate science have concentrated on the bivariate form of Granger causality . From an information theory perspective, transfer entropy evolved as a frequently used measure to infer directionality and amount of information flow . For instance, used transfer entropy to estimate networks of information flow. These networks constructed for an agricultural site under drought and non-drought conditions showed substantial differences in connectivity, especially between subsystems comprised of variables of land and atmospheric conditions. Those changes in connectivity are attributed to changes in the feedback patterns between the subsystems for drought and normal conditions. The original forms of both Granger causality and transfer entropy are bivariate and converge for the case of vector auto-regressive models. While Granger causality is typically limited to linear relationships, transfer entropy also captures non-linear interactions but requires very large data quantities for the estimation of the probability density function.

Aiming to mitigate some of the limitations of the traditional Granger causality, used a conditional spectral Granger causality framework that allowed them to disentangle system inherent periodic couplings from external forcing. The disentanglement is enabled via decomposition into the frequency domain using wavelet theory. This method led them to the finding that soil respiration in a pine and hardwood forested ecosystem in winter is not influenced by canopy assimilation but only by temperature, a result that would not be detectable via lagged correlation or bivariate Granger causality. A time frequency representation of Granger causality was presented by , which allowed them to identify anomalous events in marine and ecological time series. used a similar approach to to investigate biosphere–atmosphere feedback loops. It was found that they can explain up to 30 % of variance in radiation and precipitation in certain regions. Recently, applied a non-linear multivariate conditional Granger causality framework to study climatic drivers of vegetation at the global scale. This approach revealed that water availability dominates plant productivity, as 61 % of the vegetated surface appeared water-limited rather than controlled by radiation or temperature . In cases of transfer entropy, developed a redundancy measure, which allows them to distinguish unique, synergistic, and redundant information transfer of a bivariate or (potentially) multivariate system to a target variable. This modification enables a stronger multivariate interpretation of process networks constructed with transfer entropy. Changes in connectivity then potentially point to different ecosystem response strategies to disturbances . These examples highlight that unexpected interaction patterns can in principle be identified from data only and may challenge theoretical assumptions. In fact, in the last few years the science of causal inference has developed a strong theoretical foundation and several algorithms have been proposed . However, only a few studies test the suitability of this latest generation of methods for understanding ecosystem dynamics (see, e.g. Shadaydeh et al.2018, 2019; Christiansen and Peters2020).

Ecological and climate data are often time-ordered. This property can be exploited to construct time series graphs . Recently, introduced an algorithm to estimate such graphs, called PCMCI, a combination of the PC algorithm (named after its inventors Peter and Clark; Spirtes and Glymour1991) and the Momentary Conditional Independence (MCI) test. PCMCI has been successfully applied to artificial and climatological case studies . Hence, this method could be potentially of high relevance for learning the causal dependency structure underlying biosphere–atmosphere interactions.

In this study, we explore the potential of PCMCI for disentangling and quantifying interactions and feedbacks between terrestrial biosphere state and fluxes and meteorological variables. The study is structured as follows. In Sect. 2, we justify and introduce the method from an ecological perspective, and we also describe artificial and real-world datasets explored in this study. The results in Sect. 3 describe the performance of the method on artificial time series data with known dependencies that mimic some basic properties of observed land surface fluxes such as heteroscedasticity. We then report on the exploration of three replicated eddy covariance measurement towers in a Mediterranean ecosystem and explore how the identified interdependencies of carbon and energy fluxes and micro-meteorological observations vary over time. Further, we present the analysis of global satellite data of vegetation greenness to understand the lagged dependency of ecosystems with respect to climatic drivers. Based on these results, Sect. 4 discusses the potentials and limitations of PCMCI for other applications in land–atmosphere studies and give recommendations for further methodological developments.

2 Method and data

## 2.1 From bivariate to multivariate measures of causality

Monitoring an ecosystem with continuous observations of net ecosystem exchange (NEE), the underlying gross primary production (GPP), and ecosystem respiration (Reco) together with the relevant drivers, i.e. global radiation (Rg), surface air temperature (T), and soil moisture (SM), allows us to study the dynamics of the carbon cycle in terrestrial ecosystems. To foster its understanding, a fundamental question is how these variables causally depend on each other. This requires the identification of directional dependencies such as the well-known effects of SM  GPP and GPP Reco and their differentiation from physically implausible links such as Reco GPP. Graphical causal models provide a framework to represent and identify causal relations based on conditional independence relations in data streams of this kind. In the case of an ecological monitoring site as described here, we can exploit the temporal information of the observations for identifying a time series graph as a type of graphical model (Runge2018). Formally this can be stated as follows. The variable ${X}_{t}^{i}$ is part of a multivariate stochastic process X (where i is the variable index, in the example $i\in \mathit{\left\{}{R}_{\mathrm{g}},T,\mathrm{GPP},{R}_{\mathrm{eco}},\mathrm{SM}\mathit{\right\}}$, and t is the time index). A time series graph 𝒢 visualises how the individual variables XiX depend on each other at specific time lags τ, i.e. ${X}_{t-\mathit{\tau }}^{i}$ with $\mathit{\tau }\in \mathit{\left\{}\mathrm{1},\mathrm{\dots },{\mathit{\tau }}_{max}\mathit{\right\}}$ (see Runge2018, for definitions). In the following, we refer to a variable ${X}_{t-\mathit{\tau }}^{i}$ that is causally affecting a variable ${X}_{t}^{j}$ as “parent” or “driver” and the latter as “receiver” or “target”. To come to a causal interpretation, it is important to exclude dependencies between two variables that are due to common drivers (${X}_{t-{\mathit{\tau }}_{\mathrm{1}}}^{i}←{X}_{t-{\mathit{\tau }}_{\mathrm{2}}}^{s}\to {X}_{t}^{j}$) or indirect paths (${X}_{t-{\mathit{\tau }}_{\mathrm{2}}}^{i}\to {X}_{t-{\mathit{\tau }}_{\mathrm{1}}}^{s}\to {X}_{t}^{j}$). For instance, when estimating the effects of GPP on Reco and Rg on Reco using a bivariate measure, one likely obtains implausible results like links that are too strong or even unexpected links because T, respectively, as the common driver and mediator (indirect path), is not accounted for. To exclude dependencies due to common drivers or indirect paths, conditional independence tests are used, denoted as $\mathrm{CI}\left({X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j}|\mathcal{S}\right)$, with some conditioning set 𝒮. If any variable (or their combination) in 𝒮 explains the dependence between ${X}_{t-\mathit{\tau }}^{i}$ and ${X}_{t}^{j}$, then the CI statistic is 0.

Two prominent methods that aim for directional dependencies are Granger causality and transfer entropy . Granger causality is typically estimated as a vector auto-regressive model and thus captures only linear links. Transfer entropy, based on information theory, also captures non-linear dependencies. It can be shown that for multivariate Gaussians, transfer entropy is equivalent to Granger causality . Both can be phrased as testing for conditional independence . In their original bivariate form, neither of these two methods accounts for third variables, but both can also be extended to deal with multivariate time series as required here . There are even non-linear and spectral modifications of Granger causality which have been applied to study biosphere–atmosphere interactions . However, the estimation of multivariate transfer entropy is challenging due to the “curse of dimensionality” , and multivariate Granger causality also exhibits low link detection power for larger number of variables (higher dimensions) and limited sample size, as is the case in our application . The strong decrease in detection power happens when using the whole past ${\mathbf{X}}_{t}^{-}=\left({\mathbf{X}}_{t-\mathrm{1}},{\mathbf{X}}_{t-\mathrm{2}},\mathrm{\dots }\right)$ of ${X}_{t}^{j}$, truncated at a maximum lag τmax, as a conditioning set 𝒮. The problem is that this set can contain a high number of conditions that are irrelevant. For example, when assessing the effect of Rg at a specific time lag τ on GPP using multivariate Granger causality one would create a vector auto-regressive model comprised of all variables, i.e. Rg, T, SM, GPP, and Reco, at each available lag, but Reco, dominated by heterotrophic respiration, is not expected to affect gross primary productivity and could be removed to decrease the dimensionality. However, manually selecting conditions is not desirable when the underlying dependence structure is unknown, which is why ideally the conditioning set is identified automatically.

PCMCI addresses this issue by reducing the set of conditions 𝒮 prior to quantifying the dependence between two variables. The two-step approach utilises a variant of the PC algorithm and the momentary conditional independence measure (MCI) . More detailed descriptions are given in Sect. 2.4 and 2.5, respectively (a full description of PCMCI including proofs and quantitative comparisons with other methods are provided in ). A schematic of the PCMCI approach is given in Scheme A1 in Appendix A. PCMCI belongs to the family of causal graphical models and follows the assumptions listed in Sect. 2.2. In the limit of infinite time series length, PCMCI converges to the true graph of dependencies, which is why we use the term “causal”. As we deal with finite sample length and partially unfulfilled assumptions, spurious links can still appear (beyond the expected false positive rate) and therefore each detected link has to be interpreted with caution.

## 2.2 Assumptions

PCMCI assesses the causal structure of a multivariate dataset or process X by estimating its time series graph. To draw causal conclusions from observational data, any causal method must adopt a number of assumptions . For the time series case, here we assume time order, the causal Markov condition, faithfulness, causal sufficiency, causal stationarity, and no contemporaneous causal effects . PCMCI is applied in combination with the ParCorr linear independence test based on partial correlations (see Sect. 2.3). This application additionally requires stationarity in the mean and variance and linear dependencies. In the following, we briefly discuss these assumptions (further details in Runge2018; Runge et al.2019b).

The time order within the time series allows us to orient directed links which are only pointing forward in time. This accounts for causal information propagating forward in time only, i.e. the cause shall precede the effect. Therefore, a directed causal link ${X}_{{t}_{\mathrm{1}}}^{i}\to {X}_{{t}_{\mathrm{2}}}^{j}$ can only exist between two nodes ${X}_{{t}_{\mathrm{1}}}^{i},{X}_{{t}_{\mathrm{2}}}^{j}$ if t1<t2. When a contemporaneous link is found, i.e. t1=t2, it is considered to be undirected. In ecological language it means that in order to claim that Rg is driving GPP, any change in Rg that is affecting GPP must be measured at a time step before the change in GPP occurs. The causal Markov and faithfulness assumptions relate the underlying physical causal mechanisms to statistical relationships manifest in the data. The causal Markov condition states that if two processes are not directly connected by some physical mechanism, then they should be statistically independent conditional on their direct drivers, like Rg and Reco conditional on T. The faithfulness assumption concerns the other direction: if two processes are statistically independent, then there cannot be a direct physical mechanism. The causal sufficiency assumption implies that every common cause of two or more variables XiX is included in X. If this is not the case, detected links may be indirect or due to an unobserved common driver. However, the absence of a link in the detected graph still implies that no direct link is present (as this only requires the assumption of faithfulness). For example, Rg is expected to influence Reco via T, the indirect path. However, a link between Rg and Reco might be detected if T is not included in the analysis. However, a missing link between T and Reco might indicate conditions inhibiting respiratory processes, i.e. very cold temperatures with frozen surfaces or very dry conditions with dead vegetation coverage. Causal stationarity refers to the existence of links over time. In a deciduous forest, for example, the ecosystem's CO2 exchange is not causally stationary as the link Rg GPP is given in summer but not in winter. Formally, a process X with graph 𝒢 is called causally stationary over a time index 𝒯 if and only if for all links ${X}_{t-\mathit{\tau }}^{i}\to {X}_{t}^{j}$ in the graph the condition ${X}_{t-\mathit{\tau }}^{i}\colorbox[rgb]{}{}{X}_{t}^{j}\phantom{\rule{0.25em}{0ex}}|\phantom{\rule{0.25em}{0ex}}{\mathbf{X}}_{t}^{-}\\mathit{\left\{}{X}_{t-\mathit{\tau }}^{i}\mathit{\right\}}$ holds for all t∈𝒯.

## 2.3 Independence test

At the core of PCMCI there are conditional independence tests $\mathrm{CI}\left({X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j},\mathcal{S}\right)$ to evaluate whether ${X}_{t-\mathit{\tau }}^{i}\colorbox[rgb]{}{}{X}_{t}^{j}\phantom{\rule{0.25em}{0ex}}|\phantom{\rule{0.25em}{0ex}}\mathcal{S}$ given a conditioning set 𝒮. Within the PCMCI software package TIGRAMITE (Runge2017), several independence tests are implemented. Here, we focus on the linear independence test called ParCorr. The ParCorr conditional independence test is based on partial correlations and a t test. This assumes the following model:

$\begin{array}{}\text{(1)}& {X}^{i}=\mathcal{S}{\mathbit{\beta }}_{{X}^{i}}+{\mathit{ϵ}}_{{X}^{i}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{X}^{j}=\mathcal{S}{\mathbit{\beta }}_{{X}^{j}}+{\mathit{ϵ}}_{{X}^{j}},\end{array}$

with coefficients β and Gaussian noise ϵ. This leads to the following residuals:

$\begin{array}{}\text{(2)}& {r}^{{X}^{i}}={X}^{i}-\mathcal{S}{\stackrel{\mathrm{^}}{\mathbit{\beta }}}_{{X}^{i}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{r}^{{X}^{j}}={X}^{j}-\mathcal{S}{\stackrel{\mathrm{^}}{\mathbit{\beta }}}_{{X}^{j}},\end{array}$

with estimated $\stackrel{\mathrm{^}}{\mathbit{\beta }}$. ParCorr removes the influence of 𝒮 on Xi and Xj via ordinary least-squares regression and tests for independence of the residuals using the Pearson correlation with a t test. The independence test returns a p value and test statistic value I, i.e. the correlation coefficient in case of ParCorr. Thus, to identify the effect of GPP on Reco that does account for their common driver T∈𝒮, ParCorr will perform a linear regression of T on both GPP and Reco accounting for time lags. The p value of the residuals' partial correlation test can be used to assess whether the two variables are dependent.

## 2.4 PC algorithm

To efficiently estimate $\mathrm{CI}\left({X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j}|\mathcal{S}\right)$ the conditioning set 𝒮 should be as small as possible, which means that it should only contain relevant conditions, which allows us to isolate the unique influence of ${X}_{t-\mathit{\tau }}^{i}$ on ${X}_{t}^{j}$. For an estimation of $\mathrm{CI}\left({{R}_{\mathrm{g}}}_{t-\mathit{\tau }},{\mathrm{GPP}}_{t}|\mathcal{S}\right)$, for example, 𝒮 should contain T and SM (at certain lags), as they influence the ability of an ecosystem to perform photosynthesis. Likewise, when estimating $\mathrm{CI}\left({T}_{t-\mathit{\tau }},{\mathrm{GPP}}_{t}|\mathcal{S}\right)$, 𝒮 should include Rg and SM for the same reasons. A sufficient set of relevant conditions includes the drivers and parents of the variable ${X}_{t}^{j}$. Consequently, the aim of the PC step is to identify an as small as possible superset of the parents of each variable included in the process. The algorithm uses a variant of the PC algorithm ; a comprehensive pseudo-code of this procedure is given in the Supplement of . In the limit of infinite sample size, the relevant conditions indeed converge to the true causal parents, although, practically, an estimate that contains a few irrelevant conditions, like Reco, is sufficient as well.

The PC step starts by initialising the whole past of a process: $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)={\mathbf{X}}_{t}^{-}=\mathit{\left\{}{X}_{t-\mathit{\tau }}^{i}:i=\mathrm{1},\mathrm{\dots },N,\mathit{\tau }=\mathrm{1},\mathrm{\dots },{\mathit{\tau }}_{max}\mathit{\right\}}$. Next, by evaluating $\mathrm{CI}\left({X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j},\mathcal{S}\right)$, conditions ${X}_{t-\mathit{\tau }}^{i}$ are removed from $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)$ that are independent of ${X}_{t}^{j}$ conditionally on a subset $\mathcal{S}\in \stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)\\left\{{X}_{t-\mathit{\tau }}^{i}\right\}$. 𝒮 starts as the empty set and is iteratively increased. For instance, let ${X}_{t-\mathit{\tau }}^{i}$ be Reco (at a specific lag) and ${X}_{t}^{j}$ be GPP. The conditional independence between GPP and Reco will be estimated first by using no conditions. If GPP and Reco appear related, one variable will be included in the conditioning set. If the residuals are still dependent, a second variable is included and so on. When T is part of the conditioning set, the residuals of GPP and Reco might not be dependent anymore and Reco is removed from the estimated set of parents of GPP. The PC algorithm adopted in PCMCI efficiently selects those conditioning sets to limit the number of tests conducted.

Every conditional independence test is evaluated at a significance threshold αpc, which is usually set to a liberal value between 0.1 and 0.4. Alternatively, in TIGRAMITE one can let αpc be unspecified. PCMCI then evaluates the best choice of αpc $\in \mathit{\left\{}\mathrm{0.1},\mathrm{0.2},\mathrm{0.3},\mathrm{0.4}\mathit{\right\}}$ based on the Akaike information criterion (AIC), which is further explained in .

## 2.5 MCI tests

MCI is the actual causal discovery step that ascribes a p value and strength to each possible link. MCI iterates through all pairs $\left({X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j}\right),i=\mathrm{1},\mathrm{\dots },N,\phantom{\rule{0.33em}{0ex}}\mathit{\tau }=\mathrm{0},\mathrm{\dots },{\mathit{\tau }}_{max}$, and calculates $\mathrm{CI}\left({X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j},\mathcal{S}\right)$, where 𝒮 consists of the two (super-)sets of parents $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)$ and $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t-\mathit{\tau }}^{i}\right)$ obtained in the PC step. $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t-\mathit{\tau }}^{i}\right)$ is constructed by shifting the time series of $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{i}\right)$ by τ. In cases where ${X}_{t-\mathit{\tau }}^{i}\in \stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)$, ${X}_{t-\mathit{\tau }}^{i}$ has to be removed from $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)$. If τ=0, conditional dependence is estimated for contemporaneous nodes ${X}_{t}^{j}$ and ${X}_{t}^{i}$. Due to missing time order, a dependence would be left undirected. Further, as the parents $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{i}\right)$ and $\stackrel{\mathrm{̃}}{\mathcal{P}}\left({X}_{t}^{j}\right)$ used in each conditional dependence test are defined to lie in the past of ${X}_{t}^{i}$ and ${X}_{t}^{j}$, links, both contemporaneous and lagged, can be spurious due to contemporaneous common drivers or contemporaneous indirect paths. The absence of a link, though, means that a physical (contemporaneous) link is unlikely (assuming faithfulness; see ). For simplicity, the previously given examples were omitting the time lag. Thus, if Reco responds instantaneously (considering the sampling temporal resolution) to changes in T but T responds with a time lag to Rg, both variables will likely appear contemporaneously coupled to Rg.

The link strength in the PCMCI framework can be given by the effect size of the conditional independence test statistic measure CI used in combination with MCI. In case of ParCorr, the effect size is given by the partial correlation value, which is between −1 and 1. Assuming a linear Gaussian model the partial correlation value is shown to directly depend on the receiver's and driver's variance as well as the coupling coefficient :

$\begin{array}{}\text{(3)}& {\mathit{\rho }}_{{X}_{t-\mathit{\tau }}^{i}\to {X}_{t}^{j}}^{\mathrm{MCI}}=\frac{c{\mathit{\sigma }}_{{X}_{t-\mathit{\tau }}^{i}}}{\sqrt{{\mathit{\sigma }}_{{X}_{t}^{j}}^{\mathrm{2}}+{c}^{\mathrm{2}}{\mathit{\sigma }}_{{X}_{t-\mathit{\tau }}^{i}}^{\mathrm{2}}}},\end{array}$

where ${\mathit{\sigma }}_{{X}_{t-\mathit{\tau }}^{i},{X}_{t}^{j}}$ are the variances of the noise terms driving ${X}_{t-\mathit{\tau }}^{i}$ and ${X}_{t}^{j}$, respectively, and c is their coupling coefficient. In practice, non-linear links can often also be well detected with ParCorr in so far as they are linearly approximated. In case the linear part is even “stronger” than the non-linear part, ParCorr might also have a better detection power than a non-linear independence test .

## 2.6 Data

### 2.6.1 Artificial dataset – test model

We tested the algorithm on artificial datasets prior to its application to real-world data. The artificial dataset was created using a test model that takes time series of measured global radiation (g) and creates three artificial time series that conceptually represent temperature (𝒯), gross primary production (𝒢𝒫𝒫), and ecosystem respiration (eco). Note that this test model is not intended to accurately represent observed land–atmosphere fluxes but only serves to test the procedure. The model incorporates one linear auto-dependence ${\mathcal{T}}_{t-{\mathit{\tau }}_{\mathrm{1}}}\to {\mathcal{T}}_{t}$, one linear additive cross-dependence ${{\mathcal{R}}_{\mathrm{g}}}_{t-{\mathit{\tau }}_{\mathrm{2}}}\to {\mathcal{T}}_{t}$, and two non-linear dependencies, multiplicative ${{\mathcal{R}}_{\mathrm{g}}}_{t-{\mathit{\tau }}_{\mathrm{3}}}\cdot {\mathcal{T}}_{t-{\mathit{\tau }}_{\mathrm{4}}}\to \mathcal{G}\mathcal{P}{\mathcal{P}}_{t}$ and multiplicative exponential $\mathcal{G}\mathcal{P}{\mathcal{P}}_{t-{\mathit{\tau }}_{\mathrm{5}}}\cdot {c}^{{\mathcal{T}}_{t-{\mathit{\tau }}_{\mathrm{6}}}}\to {{\mathcal{R}}_{\mathrm{eco}}}_{t}$ (see Fig. 1), according to the following equations:

$\begin{array}{}\text{(4)}& {\mathcal{R}}_{{\mathrm{g}}_{\mathrm{mo}}}& ={\mathcal{R}}_{{\mathrm{g}}_{\mathrm{obs}}},\text{(5)}& {\mathcal{T}}_{\mathrm{mo}}\left(t\right)& ={c}_{\mathrm{1}}\phantom{\rule{0.33em}{0ex}}{\mathcal{T}}_{\mathrm{mo}}\left(t-{\mathit{\tau }}_{\mathrm{1}}\right)\phantom{\rule{0.33em}{0ex}}+{c}_{\mathrm{2}}\phantom{\rule{0.33em}{0ex}}{\mathcal{R}}_{{\mathrm{g}}_{\mathrm{mo}}}\left(t-{\mathit{\tau }}_{\mathrm{2}}\right)\phantom{\rule{0.33em}{0ex}}+\phantom{\rule{0.33em}{0ex}}{\mathit{\xi }}_{T},\text{(6)}& \mathcal{G}\mathcal{P}{\mathcal{P}}_{\mathrm{mo}}\left(t\right)& ={c}_{\mathrm{3}}\phantom{\rule{0.33em}{0ex}}{\mathcal{R}}_{{\mathrm{g}}_{\mathrm{mo}}}\left(t-{\mathit{\tau }}_{\mathrm{3}}\right)\phantom{\rule{0.33em}{0ex}}\cdot {\mathcal{T}}_{\mathrm{mo}}\left(t-{\mathit{\tau }}_{\mathrm{4}}\right)\phantom{\rule{0.33em}{0ex}}+\phantom{\rule{0.33em}{0ex}}{\mathit{\xi }}_{\mathrm{GPP}},\text{(7)}& {\mathcal{R}}_{{\mathrm{eco}}_{\mathrm{mo}}}\left(t\right)& \phantom{\rule{0.25em}{0ex}}={c}_{\mathrm{4}}\phantom{\rule{0.33em}{0ex}}\mathcal{G}\mathcal{P}{\mathcal{P}}_{\mathrm{mo}}\left(t-{\mathit{\tau }}_{\mathrm{5}}\right)\phantom{\rule{0.33em}{0ex}}\cdot {c}_{\mathrm{5}}^{\frac{{\mathcal{T}}_{\mathrm{mo}}\left(t-{\mathit{\tau }}_{\mathrm{6}}\right)-{T}_{\mathrm{ref}}}{\mathrm{10}}}\phantom{\rule{0.33em}{0ex}}+\phantom{\rule{0.33em}{0ex}}{\mathit{\xi }}_{{\mathcal{R}}_{\mathrm{eco}}}.\end{array}$

The parameters c1, c2, …, c5 are referred to as coupling coefficients, and the time lags are noted as τ1, τ2, …, τ5. The subscripts mo and obs abbreviate model and observation, respectively. Tref is set to 15 C. The term ξ, termed “intrinsic” or “dynamical noise”, here represents values from uncorrelated, normally distributed noise. Having dynamic noise is essential for a method utilising conditional independence tests. It is based on the assumption that a process or state is never fully described by its deterministic part because there are unresolved intrinsic processes, summarised as ξ.

The model was fitted to real observational data (using radiation, temperature, and land–atmosphere fluxes) of daily time resolution, measured by the eddy-covariance method from FLUXNET, by minimising the sum of squared residuals using the gradient descent implemented in the Optim.jl package . We fitted the model to 72 sites listed in Table A2 given in Appendix A. The value range for the coupling coefficients c1 to c4 and c5 were set to [0.2,1] and [1,2.5], respectively. The lags were limited to integer values in the range [0,25]. The distributions of obtained lags and coupling coefficients are given in Figs. A2 to A5 in Appendix A. The fitting process thus generated 72 sets of parameters, containing coupling coefficients and lag values, which were used for the time series generation.

From each of the 72 sets of parameters we generated four sets of time series, each having a length of 500 years. The time series generation was initiated using two types of data: first, uncorrelated, normally distributed noise, and second, unprocessed radiation data as used during the fitting (the available radiation data were repeated to 500 years). The resulting datasets are called baseline dataset and seasonality dataset, respectively. In both cases, the model was run twice, once with homoscedastic (constant variance of ξ), once with heteroscedastic dynamical noise ξ. To induce heteroscedasticity, ξ was multiplied with a mean daily variance that was extracted for each variable at each FLUXNET site. In Fig. A6, a 5-year time series excerpt from the Hainich site is shown. A third dataset is generated by anomalisation (subtraction of smoothed seasonal mean) of the seasonality dataset.

### 2.6.2 Eddy covariance data – Majadas de Tiètar experimental site

Data from three towers located in Majadas de Tiètar, (ES-LMa, ES-LM1, ES-LM2), a Mediterranean Savanna in central Spain, are used (coordinates of the central tower: 395625′′ N, 54629′′ W). Measurements include the exchange of CO2 between the land surface and atmosphere at half-hourly resolution using the eddy covariance method. The three tower footprints received different fertilisation treatments in spring 2015 . We consider data from before the fertilisation from April 2014 to March 2015 of shortwave downward radiation (Rg), air temperature (T), net ecosystem exchange (NEE), vapour pressure deficit (VPD), sensible heat (H) and latent heat (LE). The average temperature within this period was 17.3 C, with a total precipitation of 765 mm. Most precipitation fell between October and April.

We expect the causal imprints in the data to vary between seasons and during the course of the day. To satisfy causal stationarity, we estimate networks separately for each month and consider only samples for which the potential radiation was above four-fifths of the potential daily maximum, which corresponds to midday samples. We used a mask type that limits only the receiver variable to the respective month and day time values (see Table A1 for PCMCI parameter settings). This setting causes time series lengths ranging from 239 datapoints in December to 372 datapoints in July. Minimal and maximal lags were set to 0 and 8, respectively. We left the data unprocessed, i.e. we did not subtract a seasonal mean for anomalisation. Constraining the samples to separate month and midday values reduces the effect of seasonality as a common driver that would lead to spurious links. Furthermore, to correct for multiple testing we applied the Benjamini–Hochberg false discovery rate correction . Thereby, the p values for the whole graph obtained from the MCI step are adjusted to control the number of false discoveries . We chose a two-sided significance level of 0.01.

### 2.6.3 Gridded global dataset

The second observational case study was performed on a global dataset. We used data with 0.5 spatial and monthly temporal resolution from 1982 to 2008. The dataset is composed of three climatic variables, global radiation (Rg), temperature (T) and precipitation (P), and one vegetation state index, the Normalised Difference Vegetation Index (NDVI). Both temperature and precipitation datasets were taken from the Climate Research Unit (CRU), version TS3.10 . The radiation data stems from the Climate Research Unit and National Centers for Environmental Prediction dataset (CRUNCEP, Viovy2016). The used NDVI data stems from the Global Inventory Monitoring and Modeling Systems (GIMMS) in version 3g_v1 .

To examine the influence of radiation, temperature, and precipitation on NDVI by means of PCMCI, we used the following settings. We compute the anomalies by subtracting a smoothed seasonal mean. A maximal time lag of 3 months was chosen based on the largest lag with significant partial correlation among all pairs of variables, partialling out only the autocorrelation of each variable. The receiver variable was limited to the growing season defined by T>0 and NDVI>0.2, which allows good comparison to . The significance level (αpc) in the condition selection phase (see Sect. 2.4) was chosen based on the AIC selection criterion. A concise list of PCMCI parameters that were altered from default settings is given in Table A1.

3 Results

## 3.1 Test model

As an example, in Fig. 1 we show PCMCI and lagged correlation networks in the form of process graphs. It is clearly visible that many more spurious links pass the significance threshold of 0.01 using lagged correlation as compared to using PCMCI. Those spurious links can complicate the analysis or lead to false conclusions and misleading hypotheses. We examined four cases of different time series lengths: 91 [$\mathrm{183}\le \mathrm{doy}\le \mathrm{274}$] and 120 [$\mathrm{153}\le \mathrm{doy}\le \mathrm{274}$] d and 1 and 5 years for daily data (doy: day of the year). For each time series length and each parameter set, the causal network structure was estimated for 100 realisations of the model (each based on a realisation of intrinsic noise), which allowed the estimation of false positive (FPR) and true positive (TPR) detection rates. The detection rates are calculated for each tower, FPR in general and TPR link-wise. The TPR for each link is its sum of detections among 100 realisations divided by 100. The FPR is the number of falsely detected links divided by the number of all possible false links and 100. The summary of the experiments, i.e. the overall false positive rate (FPR) and the distributions of the link's true positive rate (TPR) across sites are given in Figs. 2 and 3, respectively. The blue violin plots always report the case of normal distributed (non-heteroscedastic) intrinsic noise and the corresponding orange violin plots summarise the case of heteroscedastic noise. The effect of heteroscedasticity and seasonality is then assessed by comparing the distributions obtained from the baseline dataset to the results of the seasonality dataset.

Figure 1The artificial datasets are generated with a prescribed interaction structure (true network), which is obtained by fitting the test model to the FLUXNET sites. Here we show for four time series lengths the process graphs estimated via both lagged correlation and PCMCI. The data used stems from the homoscedastic realisation of the seasonality dataset of the Hainich site. The significance level was set to 0.01. The number of time lag labels were limited to five in the correlation networks. However, for the longest time series typically the whole range of lags (0–25) was significant.

Figure 2The distribution of false positive detection rates estimated for the baseline dataset, the seasonality dataset and the anomalised seasonality dataset (mean seasonal cycle subtracted). The distributions are given for different time series length (number of datapoints). Additionally, the distributions are split to show the impact of heteroscedastic noise (orange) compared to normal distributed noise (blue). The significance level of 0.01 is given by a blue horizontal line.

Figure 3The distribution of the true positive detection rates for each link in the test model estimated for the baseline dataset, the seasonality dataset, and the anomalised seasonality dataset. The distributions are given for different time series lengths (number of datapoints). Additionally, the distributions are split to show the impact of heteroscedastic noise (orange) compared to normally distributed noise (blue).

The FPR of homoscedastic time series in the baseline dataset is in the expected range of 0.01, the chosen significance level, indicating a well-calibrated test due to fulfilled assumptions. The assumption of stationarity is violated as soon as heteroscedasticity or seasonality is present. The effect on the FPR is an increase above 0.01 for time series length of 1 and 5 years, with a much stronger increase due to seasonality (factors of 4 and 8, respectively) than for heteroscedasticity (factor of 2).

The effect of non-stationarities on the TPR differs among the links. The detection of linear links (g→𝒯 and 𝒯→𝒯) is not affected by seasonality and slightly improves for heteroscedastic dynamical noise. The detection of non-linear links is improved by seasonality, with the strongest effects in the link 𝒯→𝒢𝒫𝒫. The link 𝒯→ℛeco has a stronger non-linearity, and therefore the detection rate shows a weaker effect on seasonality. Furthermore, the coupling coefficient c5, the base of T, can be close to or be exactly 1 (see Fig. A3). This would actually cause, on the one hand, 𝒯→ℛeco to vanish, rendering a detection impossible and, on the other hand, result in a linear dependence of 𝒢𝒫𝒫 on eco, which improves its detection. Heteroscedasticity seems to have a slight negative effect on non-linear links. In general, the TPRs in the seasonality dataset are quite high, even for non-linear links, and predominantly above 80 % and often reaching 100 %.

Comparing the TPRs of the non-linear links shows some disparity. The links 𝒯→𝒢𝒫𝒫 and 𝒯→ℛeco experience zero detection in the baseline dataset but partially considerable rates in the seasonality dataset with a strong dependence on the time series length. On the contrary, the median of the TPR of the links g→𝒢𝒫𝒫 and 𝒢𝒫𝒫→ℛeco is above 95 % in the seasonality dataset, even for time series length as short as 91 d, but remains high in the baseline dataset. The removal of the seasonal cycle keeps the TPRs largely unaffected but reduces the FPR. Nevertheless, it still remains above the significance level by factors of 4 and 2 for 5-year and 1-year time series lengths, respectively.

In summary, the seasonality dataset exhibits high TPR even for non-linear links. Compared to stationary time series, the detection of non-linear links actually benefits from seasonality. The high detection, though, comes at the cost of a high false positive rate for time series length of and above 1 year. To a certain degree, the increase in FPR can be counteracted by anomalisation.

## 3.2 Majadas de Tiètar dataset

First, we look at the link consistency by comparing networks that were obtained for each tower within a month. The comparison is done for 2 months with strongly differing climate conditions: April and August. In Fig. 4 we compare the estimated link strengths (effect size estimated via partial correlation) as long as the corresponding links are significant in at least one network. The confidence intervals are overlapping for the majority of links, suggesting that the uncertainty of the fluxes is much smaller than the observed effects . Exceptions are found for only a few links (${R}_{\mathrm{g}}\stackrel{\mathrm{0}}{⟶}T$, ${R}_{\mathrm{g}}\stackrel{\mathrm{1}}{⟶}LE$, $\mathrm{VPD}\stackrel{\mathrm{1}}{⟶}\mathrm{VPD}$, $H\stackrel{\mathrm{2}}{⟶}H$, $\mathrm{NEE}\stackrel{\mathrm{0}}{⟶}LE$, $H\stackrel{\mathrm{2}}{⟶}\mathrm{NEE}$, $\mathrm{NEE}\stackrel{\mathrm{4}}{⟶}\mathrm{NEE}$; the number above the arrow indicates the lag) where the detection rates do not or barely overlap. Cross-links (a link from one variable to another) with two or more significant appearances are predominantly at zero lag. Approximately half of the links with lag 1 are auto-dependent links (a link from the past of a variable to its present). Comparing the links between the months of April and August, distinct differences can be noticed. First, August has slightly fewer significant links compared to April. Second, the only links remaining that are significant in two or three towers are between atmospheric variables. Third, the remaining link strengths tend to be weaker in August than in April.

Figure 4Comparison of the networks of three eddy covariance measurement stations (LMa, LM1, LM2) located in Majadas de Tiètar (Spain). Links that are found to be significant in one of the three networks are included. For each link, the calculated strength of all three networks is plotted with its 90 % confidence interval. The colours blue, orange, and green correspond to the towers LMa, LM1, and LM2, respectively. The significance threshold is 0.01. If a link does not pass the significance, it is marked by a black dot. The links are grouped into lag 0 (top), lag 1 (middle), and all lags greater than 1 (bottom). Negative NEE is associated with carbon uptake by the ecosystem. Links at lag 0 are left undirected (–), yet as Rg is set as main driver, links incorporating Rg at lag 0 are directed ().

The difference among the seasons is further investigated in Fig. 5, which shows process graphs for each month of the year. We combine the networks of the three towers to one process graph by plotting every link that is significant in at least one tower. The process graphs in Fig. 5 visualise clearly gradual changes within the interaction structure of the biosphere–atmosphere system during the course of a year. During the main growing season from February to May, NEE is coupled strongly to the energy fluxes latent (LE) and sensible heat (H). These connections weaken, disappear, or even switch sign during the dry season. Less regularly, NEE also shows connections to radiation (Rg) and temperature (T). Between the atmospheric variables, a basic network between VPD, T, Rg, and H remains intact and relatively constant in strength. The dominance of contemporaneous links is found as well, as already seen in Fig. 4. Aside from the decoupling of NEE from any variable in the dry period, there are additional interesting patterns. For example, the positive reappearance of the link between NEE and LE in September. Here, the onset of precipitation events (see Fig. A10) occurred, which led to strong respiration peaks . Creating such a network via lagged correlation would result in much more significant links (causing the network to be not interpretable as opposed to PCMCI) and NEE does not decouple from the atmosphere in August (see Fig. A7).

Figure 5To visualise the gradual changes in interaction structure the networks of the three towers are combined for each month. The number of significant occurrences of a link is given by its width. The link strength, given by the link colour, is calculated by averaging the significant links of the towers. The link's lag is shown in the centre of each arrow, sorted in descending order of link strength. The resulting graphs are shown for April 2014 until March 2015. The significance threshold is 0.01. The networks of April and August, illustrated in Fig. 4, are highlighted by a box.

The above results demonstrate that PCMCI is sensitive enough to capture seasonal differences and certain physiological reasonable biosphere behaviour. Moreover, PCMCI yields a better interpretable network structure than pure correlation approaches.

## 3.3 Gridded global dataset

The significant lags and MCI values of each climatic variable on NDVI were subject to inspection. In Fig. 6 the maximal MCI value and the corresponding lag are plotted for the links ${X}_{t-\mathit{\tau }}\to \mathrm{NDVI}:\mathit{\tau }\in \mathit{\left\{}\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\mathit{\right\}}$, with X being one of the climatic drivers: radiation, temperature, or precipitation. The chosen significance threshold was set to 0.05. Figure 7 shows the climatic driver with the largest MCI per grid point. PCMCI detects a regionally varying influence of climatic drivers. As expected, the boreal regions are strongly driven by temperature instantaneously, while (semi-)arid regions, which correspond predominantly to grass- or prairie-dominated areas, respond strongest to precipitation at a time lag of 1 month. Radiation is found to have a comparatively low spatial effect, with hotspots in southern and eastern China, central Russia, and eastern Canada.

Figure 6Influence of climatic drivers on NDVI as calculated by PCMCI. The first column shows the estimated causal influence given as maximal absolute MCI value of climatic drivers on NDVI. The second column gives the time lag at which the maximal absolute MCI value occurs (in months).

Figure 7Map of the strongest climatic driver (largest absolute MCI value) per grid point.

The dominant lags are found to be 0 and 1. Only a very small fraction of the total area shows a maximal MCI value at a higher lag of two or three months. The lags are also not equally distributed among the climatic drivers. Radiation and temperature are predominantly strongest at lag 0, while precipitation has a much larger fraction of area showing the strongest response at lag 1. Regions where the impact of Rg on NDVI is strongest at lag 1 tend to respond negatively to Rg but positively to precipitation at lag 1. On the other hand, a large part of the regions with the strongest impact of precipitation at lag 0 respond negatively to it but positively to radiation.

In summary, PCMCI estimates coherent interaction patterns that match well with anticipated behaviour based on vegetation type and prevailing climatic conditions.

4 Discussion

Causal discovery methods promise an improved understanding and can help to come up with new hypotheses about the interaction between biosphere and atmosphere . However, the underlying assumptions need to be properly taken into account. The coupled biosphere–atmosphere system possesses several challenges that potentially violate the underlying assumptions of causal discovery in general and the employed method's assumptions in particular. Here, we investigate the effect of a violation of assumptions on PCMCI network estimates.

With regard to expected non-linearities in biosphere–atmosphere interactions, using a linear independence test within the PCMCI framework may not be adequate. We justify our choice with the following arguments: first, non-linearities are often approximated linearly. Second, a linear regression based test has a much higher power for detecting linear links than a non-parametric test (Runge2018) and can hence already detect links at smaller sample sizes. Third, linear partial correlation is easily interpretable, for example, positive and negative MCI values. This motivation is supported on the one hand by the results of the test model and on the other hand by additionally performed analyses on the observational datasets using Gaussian regression and distance correlation as an independence test. These results (see Figs. A8, A9, A11) show similar patterns, but due to the low sample sizes exhibit lower statistical significance. In general, the derived results show a high detection power with a strong consistency in calculated effect strengths on eddy covariance data and global, regularly gridded reanalysis data that leads to easily interpretable patterns. Observed drawbacks are a high FPR in cases of violated assumptions, especially strong periodicity, as well as the appearance of contemporaneous lags in measurement datasets.

## 4.1 Lessons learned from the test model

The probability of detecting a link with PCMCI depends strongly on a link's MCI effect size, which is larger for strong variance in the driver and a low variance in the receiver (see Sect. 2.5). Several results can be explained by this observation. First, the variance of three out of five drivers of cross dependencies in the test model are either directly or indirectly (via 𝒢𝒫𝒫) influenced by g, which has the highest variance of all variables. Consequently, the detection power of the three links is large, almost 100 %. In comparison, the other variables' variances are weaker, since they are influenced by 𝒯, which results in a lower detection power. This is the origin of the disparity in detection rates of the non-linear links. Second, the partially strong increase in TPR of non-linear links (influenced in a multiplicative way by 𝒯) from the baseline dataset to the seasonality dataset can also be explained by this increase in variance. A multiplicative link is actually not generally expected to be found by ParCorr (Runge2018), but the value of the multiplicative factor is dominated by the seasonal value, and not the dynamical noise, which might cause a scaling of the dynamical noise terms rather than a random distortion. Third, the dependence on the variance ratio can also explain the difference in TPR between homoscedastic (equal error variance) and heteroscedastic (error variance changing over time) time series, i.e. the variance of g and 𝒢𝒫𝒫 exhibits a strong seasonality, with its peak in summer, while the variance of 𝒯 is rather constant. This explains, for example, the strong decrease in TPR for the link 𝒯→𝒢𝒫𝒫 at 91 d time series length when comparing homoscedasticity to heteroscedasticity. The decrease in TPR is less pronounced when another season, implying a different variance, is chosen for this comparison. As links with weak driver variance and strong response variance are more likely to be missed, one may ask what effect this will have on the detection of feedback loops where one variable has low and the other high variance. Here lies a limitation of the test model where no feedback loops were implemented.

Summarising the results of the test model, the different detection rates, disparity among non-linear links, and the detection of multiplicative links are largely explainable via the effect of the variance on the link detection. Yet, the discussion revealed the need for further research in several aspects. On the one hand, feedback loops are not included in the test model yet are an important aspect in natural systems. On the other hand, removing non-stationarities is essential to keep the false positive rate in the expected range, but standard procedures of subtracting the mean seasonal cycle are not sufficient. Further, the effect of non-stationarity on the causal network structure needs to be investigated.

## 4.2 Causal interpretation of estimated networks from observational data

Nevertheless, robust patterns were identified in our studies that are also consistent with other studies. Furthermore, a causal analysis has the advantage of an enhanced interpretability compared to correlative approaches. First of all, we could show that the networks' estimated link strengths are consistent for observational data, even though measurement error affects the data. The dataset used was suitable for this analysis, as the measurement stations are located in a reasonably homogeneous ecosystem that shows only little spatial variation . Thus, also the interaction between biosphere and atmosphere is expected to change only marginally across space within this ecosystem. Second, the gradual changes in plant activity that are taking place in the ecosystem of Majadas de Tiètar throughout the year do emerge very well in the coupling strength of daytime NEE to the atmospheric variables. The observed decoupling during the dry season is in accordance with the one of a soybean field during drought conditions observed by . The gradual changes in ecosystem activity are not visible in a pure (lagged) correlation analysis or are only visible in colour or density changes but the large number of significant links prevents any detailed interpretation on the physical mechanisms and changes thereof. The large number of significant links compared to the PCMCI networks stems solely from the absence of conditioning in common drivers or mediating variables, which often leads to an overestimation of the link strength in correlation networks. As a result, processes, such as the decoupling of NEE during the dry period, stay hidden. To reduce the effect of confounding, analyses often utilise partial correlation (see, e.g. Buermann et al.2018). However, a partial correlation can introduce new dependencies (as opposed to removing them) if one conditions on causal effects of the variables under consideration (the “marrying parents” effect). This issue is avoided in PCMCI by only conditioning on past variables. Additionally, PCMCI chooses only relevant variables as conditions by applying the PC condition selection step, which is especially valuable in high-dimensional study cases and improves detection power and computation time (Runge2018).

The global study of climatic drivers of vegetation shows a general pattern of lags and dependence strengths of vegetation on climatic variables that is easily interpretable. The boreal regions appear energy limited and especially driven by temperature (see Fig. 6c), while the strongest dependence of (semi-)arid regions on precipitation reflects their limitation in water supply. Two recent studies performed a similar analysis. Both and investigated lagged effects and dependence strengths of NDVI on precipitation, temperature, and radiation. estimated the lags of the strongest effects via an univariate regression of the climatic drivers on NDVI and subsequently used those lags to fit a multivariate regression model of the climatic drivers on NDVI and determined their relative effects. applied a non-linear Granger causality framework utilising a random forest predictive model; the method was presented by . We recognise that similar patterns are observed in , but the lags at the maximal MCI value are usually lower than the one found in , which stems from the methodical differences. Besides having used anomaly values, PCMCI regresses both NDVI and the climatic drivers on their parents before calculating the MCI value (see Sect. 2.5). This especially removes the influence of autocorrelation. shows how autocorrelation affects the correlative lag causing it to be larger for stronger autocorrelation; thereby, the correlative lag may become larger than the causal lag. Therefore, according to Fig. 6b, the causal information embedded in monthly resolution is predominantly received within 1 month. Finding the strongest causal links at a time lag up to 1 month appear in agreement with . Additionally, the spatial distribution of the strongest climatic influences compares well, but there are certain noteworthy differences that do not necessarily stem from masking differences, i.e. that we took only values belonging to the growing season while took the whole time series. First, there is little significant Granger causality of water availability found in boreal regions while there are significant negative causal dependencies detected via PCMCI. Second, NDVI in arid regions is not (or is only partially) Granger caused by radiation and temperature but in part shows a negative PCMCI value for those variables. There might be physiological reasons that can explain the PCMCI patterns, i.e. waterlogging or temperatures that are too high. To explain the differences though, we could identify two possible reasons. First, masked out negative influences of radiation, arguing that radiation is not negatively affecting NDVI. They found that negative influences of Rg are usually a consequence of poor conditioning on other variables. Second, a precipitation event in boreal regions coincides with a reduction in radiation and temperature. Boreal regions usually do not suffer from water shortages. Thus, they respond more strongly to the reduction of radiation and temperature than precipitation. As precipitation is coupled negatively to radiation and temperature at lag 0, the effect of precipitation on NDVI is found to be negative. Thus, the link $P\stackrel{-}{\to }\mathrm{NDVI}$ might be an effect of the contemporaneous common driver scheme $P\stackrel{-}{\to }{R}_{\mathrm{g}}\stackrel{+}{\to }\mathrm{NDVI}$ and therefore would not be causal. In fact, a similar argumentation can be given for the negative impact of temperature and radiation on NDVI in arid regions.

In summary, we pointed out the need for careful interpretations in applying causal discovery methods and especially highlighted the challenges linked to the study of biosphere–atmosphere interaction via PCMCI. We demonstrated that the network structures estimated from observational data are explainable with respect to plant physiology and climatic effects. Finally, our study shows that causal methods can deliver better interpretability and a much improved process understanding in comparison to correlation and bivariate Granger causality analyses that are ambiguous to interpret since they do not account for common drivers.

## 4.3 Outlook

The preceding discussion has shed light on the merits of PCMCI, as well as the challenges of applying causal discovery methods. discuss further challenges and methods and give an outlook of how multiple methods can be combined to alleviate limitations.

5 Conclusions

Here we tested PCMCI, an algorithm that estimates causal graphs from empirical time series. We specifically explored two types of datasets that are highly relevant in biogeosciences: eddy covariance measurements of land–atmosphere fluxes and global satellite remote sensing of vegetation greenness. The causal graphs estimated from the eddy covariance data collected in a Mediterranean site confirm patterns we would expect in these ecosystems: during the dry season's plants senescence, for instance, the ecosystem's carbon cycle (NEE) decouples from meteorological variability. On the contrary, during the main growing season, with warm and humid conditions, strong links between NEE, LE and H characterise the graph. Using the causal framework, not only the strongly contrasting states emerge in the graph structure but also the gradual transitions that relate to minor changes like the connectivity of sensible heat to temperature with progressing dryness. A purely correlative analysis, instead, is not able to resolve these patterns. PCMCI allows us to identify and focus on fewer (only those that are highly relevant) dependencies. Applying the approach to three replicated eddy covariance systems shows the robustness of the method to random errors in the fluxes measurements and confirms one of the assumptions of eddy covariance: above a relatively homogeneous terrain the fluxes measured should be spatially invariant as should the underlying causal relationship between climate and fluxes. The global analysis of NDVI in relation to climatic drivers confirms the known patterns of dependence strengths of vegetation on climatic variables: boreal regions are energy limited and especially driven by temperature and secondarily by radiation, while in semi-arid regions vegetation dynamics are strongly dependent on water supply. However, obtained response times of vegetation to climatic variations are lower using PCMCI than correlation which can be attributed to a better treatment of the autocorrelation in the time series and cross-relations among climate variables. Compared to merely correlative approaches, this leads to a interpretable pattern of driver–response relationships. In short, the new developments achieved in causal inference allow us to gain well-constrained insights on processes that would otherwise be drowning in the correlation chaos. Therefore, we hope that this study fosters usage of causal inference in analysing interactions and feedbacks of the biosphere–atmosphere system and furthermore exhibits our demand of further developments.

Appendix A

Table A1PCMCI parameters that were used differently from default settings.

Table A2List of FLUXNET sites used for the generation of artificial datasets and the time period used.

Scheme A1Schematic overview of the aim and approach of PCMCI using time series graphs as a visualisation tool.

Figure A1Distribution of coupling coefficients obtained after fitting the test model to the Fluxnet sites. Shown here are the distributions used for generation of heteroscedastic time series.

Figure A2Distribution of coupling coefficients obtained after fitting the test model to the Fluxnet sites. Shown here are the distributions used for generation of homoscedastic time series.

Figure A3Distribution of time lags obtained after fitting the test model to the Fluxnet sites. Shown here are the distributions used for generation of heteroscedastic time series.

Figure A4Distribution of time lags obtained after fitting the test model to the Fluxnet sites. Shown here are the distributions used for generation of homoscedastic time series.

Figure A5Observed (blue) and test model (orange) time series for the Hainich Fluxnet site. The model data were produced with heteroscedastic noise.

Figure A6The same as Fig. 5 but using simple correlation analysis to estimate the graph structures. The number of significant occurrences of a link is given by its width. The link strength, given by the link colour, is calculated by averaging the significant links of the towers. Link labels indicating the lag were removed to improve link visibility. They typically ranged from 1 to 8 (full range of possible lags). The resulting graphs are shown for April 2014 until March 2015. The significance threshold is 0.01.

Figure A7The same as Fig. 4, but the analysis was performed using a non-linear independence test. Comparison of the networks of three eddy covariance measurement stations (LMa, LM1, LM2) located in Majadas de Tiètar (Spain). Links that are found to be significant in one of the three networks are included. For each link, the calculated strength of all three networks is plotted with its 90 % confidence interval. The colours blue, orange, and green correspond to the towers LMa, LM1, and LM2, respectively. The significance threshold is 0.01. If a link does not pass the significance, it is marked by a black dot. The links are grouped into lag 0 (a, b), lag 1 (c, d) and all lags greater than 1 (e, f). Links at lag 0 are left undirected (), yet as Rg is set as main driver, links incorporating Rg at lag 0 are directed (). Note that Gaussian process regression and distance correlation (GPDC) only yields positive link strengths. Further, the strength values estimated with GPDC are rather weak due to the low number of datapoints and the larger sensitivity of that method to the sample size.

Figure A8The same as Fig. 5, but the analysis was performed using a non-linear independence test. The number of significant occurrences of a link is given by its width. The link strength, given by the link colour, is calculated by averaging the significant links of the towers. The link's lag is shown in the centre of each arrow, sorted in descending order of link strength. The resulting graphs are shown for April 2014 till March 2015. The significance threshold is 0.01. Note that GPDC only yields positive link strengths. Further, the strength values estimated with GPDC are rather weak due to the low number of datapoints and the larger sensitivity of that method to the sample size

Figure A9Daily aggregated precipitation in Majadas de Tiètar measured at the three tower sites from April 2014 to March 2015. Missing values are plotted as gaps.

Figure A10Influence of climatic drivers on NDVI as calculated by PCMCI in conjunction with the non-linear independence test GPDC (similar to Fig. 6). The first and second columns show the estimated causal influences of climatic drivers on NDVI at lag 0 and 1, respectively.

Code and data availability
Code and data availability.

The eddy covariance data of the FLUXNET sites can be downloaded from the official web page (https://fluxnet.fluxdata.org/, last access: 14 June 2018). For further information, please see Table A2.

CRU temperature and precipitation data are available at http://catalogue.ceda.ac.uk/uuid/3f8944800cc48e1cbc29a5ee12d8542d ().

The NDVI dataset is available at https://ecocast.arc.nasa.gov/data/pub/gimms/3g.v1/ ().

The TIGRAMITE software package that includes PCMCI can be found on github https://github.com/jakobrunge/tigramite/ (). All other code will be made available upon request.

Author contributions
Author contributions.

CK and MDM designed the study with contributions from JR and DGM. CK conducted the analysis and wrote the manuscript. All authors helped to improve the manuscript. AC, MM, TEM, and OPP conducted field experiments in Majadas de Tiètar, processed and provided its data, and helped with their interpretation.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We thank Maha Shadaydeh, Rune Christiansen, Jonas Peters, Milan Flach and Markus Reichstein for useful comments. Christopher Krich thanks the Max Planck Research School for global Biogeochemical Cycles for supporting his PhD project. The authors affiliated with the Max Planck Institute for Biogeochemistry thank the European Space Agency for funding the “Earth System Data Lab” project.

This work used eddy covariance data acquired and shared by the FLUXNET community, including these networks: AmeriFlux, AfriFlux, AsiaFlux, CarboAfrica, CarboEuropeIP, CarboItaly, CarboMont, ChinaFlux, Fluxnet-Canada, GreenGrass, ICOS, KoFlux, LBA, NECC, OzFlux-TERN, TCOS-Siberia, and USCCC. The ERA-Interim reanalysis data are provided by ECMWF and processed by LSCE. The FLUXNET eddy covariance data processing and harmonisation was carried out by the European Fluxes Database Cluster, AmeriFlux Management Project, and Fluxdata project of FLUXNET, with the support of CDIAC; ICOS Ecosystem Thematic Center; and the OzFlux, ChinaFlux, and AsiaFlux offices.

Financial support
Financial support.

The article processing charges for this open-access publication were covered by the Max Planck Society.

Review statement
Review statement.

This paper was edited by Ivonne Trebs and reviewed by Benjamin L. Ruddell and two anonymous referees.

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