We start with a formal introduction to Granger causality for the case of two times series, denoted as x=[x1,x2,…,xN] and y=[y1,y2,…,yN], with N being the length of the time series. In this work, y alludes to the NDVI anomaly time series at a given pixel, whereas x can represent the time series of any climatic variable at that pixel (e.g. temperature, precipitation, radiation). Granger causality can be interpreted as predictive causality, for which one attempts to forecast yt (at the specific timestamp t) given the values of x and y in previous timestamps. postulated that x causes y if the autoregressive forecast of y improves when information of x is taken into account. In order to make this definition more precise, it is important to introduce a performance measure to evaluate the forecast. Below, we will work with the coefficient of determination R2, which is here defined as follows: R2(y,y^)=1-RSSTSS=1-∑i=P+1N(yi-y^i)2∑i=P+1N(yi-y¯)2, where y represents the observed time series, y¯ is the mean of this time series, y^ is the predicted time series obtained from a given forecasting model, and P is the length of the lag-time moving window. Therefore, the R2 can be interpreted as the fraction of explained variance by the forecasting model, and it increases when the performance of the model increases, reaching the theoretical optimum of 1 for an error-free forecast and being negative when the predictions are less representative of the observations than the mean of the observations. Using R2, one can now define Granger causality in a more formal way.

Definition 1. We say that time series x Granger causes y if R2(y,y^) increases when xt-1,xt-2,…,xt-P are included in the prediction of yt, in contrast to considering yt-1,yt-2,…,yt-P only, where P is the lag-time moving window.

In climate sciences, linear vector autoregressive (VAR) models are often employed to make forecasts . A linear VAR model of order P boils down to the following representation: ytxt=β01β02+∑p=1Pβ11pβ12pβ21pβ22pyt-pxt-p+ϵ1ϵ2, with βij being parameters that need to be estimated and ϵ1 and ϵ2 referring to two white noise error terms. This model can be used to derive the predictions required to determine Granger causality. In that sense, time series x Granger causes time series y if at least one of the parameters β12p for any p significantly differs from 0. Specifically, and since we are focusing on the vegetation time series as the only target, the following two models are compared: yt=y^t+ϵ1=β01+∑p=1P(β11pyt-p+β12pxt-p)+ϵ1yt=y^t+ϵ1=β01+∑p=1Pβ11pyt-p+ϵ1. We will refer to Eq. () as the “full model” and to Eq. () as the “baseline model”, since the former incorporates all available information and the latter only information of y.

Comparing the above two models, x Granger causes y if the full model manifests a substantially better predictive performance in terms of R2 than the baseline model. To this end, statistical tests can be employed, for which one typically assumes that the errors in the model follow a Gaussian distribution . However, our above definition differs from the perspective in research papers that develop statistical tests for Granger causality , because we intend to move away from statistical hypothesis testing, since the assumptions behind such testing are typically violated when working with climate data where neither variables nor observational techniques are fully independent from each other in most cases, and errors are not normally distributed (see Sect.  for further discussion).

In climate studies, the Granger-causal relationship between two time series x and y has often been investigated in the bivariate setting . However, such an analysis might lead to incorrect conclusions, because additional (confounding) effects exerted by other climatic or environmental variables are not taken into account . This problem can be mitigated by considering time series of additional variables. For example, let us assume one has observed a third variable w, which might act as a confounder in deciding whether x Granger causes y. The above definition then naturally extends as follows.

Definition 2. We say that time series x Granger causes y conditioned on time series w if R2(y,y^) increases when xt-1,xt-2,…, xt-P are included in the prediction of yt, in contrast to considering yt-1,yt-2,…,yt-P and wt-1,wt-2,…,wt-P only, where P is the lag-time moving window.

Similarly as above, we refer to the two models as full and baseline model, respectively. Therefore, in the trivariate setting, Granger causality might be tested using the following linear VAR model: ytxtwt=β01β02β03+∑p=1Pβ11pβ12pβ13pβ21pβ22pβ23pβ31pβ32pβ33pyt-pxt-pwt-p+ϵ1ϵ2ϵ3, where a causal relationship between x and y exists if at least one β12p significantly differs from 0. As previously mentioned, the time series w might also have a causal effect on y and be correlated with x. For this reason, w should be included in both models (baseline and full), so that the method can cope with cross-correlations between predictors or, in our case, between the climatic drivers of vegetation anomalies. An extension of this definition for more than three time series is straightforward.

It is well known in the statistical literature that predictions made on in-sample data, i.e. the same data that were used to fit the statistical model, tend to be optimistic. This process is often referred to as overfitting; i.e. by definition, the fitting process leads to parameter values that cause the model to mimic the observed data as closely as possible . In the context of Granger-causality analysis, overfitting will occur more prominently in the multivariate case, when the number of considered time series increases. The results in Sect.  are based on multivariate analysis; thus, they are vulnerable to overfitting; the situation further aggravates when switching from linear to non-linear models, because then the number of parameters typically increases to allow for a more flexible functional model form.

To prevent overfitting, out-of-sample data should be used in evaluating the predictive performance in Granger-causality studies . The most straightforward procedure for creating out-of-sample data is to separate the time frame into two parts, a training set and a test set, which typically constitute the first and last halves of the time frame. A few authors have adopted this approach for climatic attribution ; however, satellite Earth observation time series are usually too short to allow for train-test splitting in that fashion. An alternative approach, which uses the available data in an efficient manner, is cross-validation. To this end, the time frame is divided into a number of short intervals, typically a few years of data, in which one interval serves as a test set, while all remaining data are used for parameter fitting. This procedure is repeated until all intervals have served once as a test set, and the prediction errors obtained in each round are aggregated so that one global performance measure can be computed. We direct the reader to and for further discussion.

The inclusion of a regularization term in the fitting process of over-parameterized linear models will avoid overfitting. Typical regularizers that shrink the parameter vectors of linear models towards 0 are L2 norms (as in ridge regression), L1 norms (as in least absolute shrinkage and selection operator (LASSO) models), or a combination of the two norms (as in elastic nets) . Translated to VAR models, this implies that one should impose restrictions on the parameter matrix of Eq. (), as done in the recent theoretical paper of . In this work, we want to identify causal relationships between a vegetation time series and various climatic time series. Hence, there is only one target variable of interest, and a simpler approach can be adopted. Denoting the vegetation time series by y, one can mimic in the trivariate setting a VAR model by means of three autoregressive ridge regression models: yt=y^t+ϵ1=β01+∑p=1P(β11pyt-p+β12pxt-p+β13pwt-p)+ϵ1xt=x^t+ϵ2=β02+∑p=1P(β21pyt-p+β22pxt-p+β23pwt-p)+ϵ2wt=w^t+ϵ3=β03+∑p=1P(β31pyt-p+β32pxt-p+β33pwt-p)+ϵ3. In this article, we aim to detect the climate drivers of vegetation and not the feedback of vegetation on climate (see, e.g. ). Therefore, it suffices to retain Eq. () in our analysis as is stated above for the trivariate case (Eq. ). Concatenating all parameters of this model into a vector β=[β01,β11p,…,β13p], one fits the parameters in ridge regression by solving the following optimization problem: min⁡β∑P+1N(yt-y^t)2+λ||β||2, with λ being a regularization parameter, that is tuned using a validation set or nested cross-validation, and ||β||2 being a penalty term, i.e. the squared L2 norm of the coefficient vector. The sum only starts at P+1 because a moving window of P lags is considered. For simplicity, we describe the above approach for the trivariate setting, even though the total number of variables used in our study is a lot larger (see Sect. ); nonetheless, extensions to the multivariate setting are straightforward.

The methodology that we develop in this paper is closely connected to the methods explained in the previous section. However, as we hypothesize that the relationships between climate and vegetation can be highly non-linear , we also replace the linear VAR models in the Granger-causality framework with non-linear machine learning models. In other fields, such as neuroscience, kernel methods or other non-linear models have been used for the investigation of non-linear Granger-causality relationships between time series . In our analysis, we use simple non-linear methods that are applicable to large data sets. More sophisticated approaches typically do not scale well enough in global climate–vegetation data sets. Therefore, in our work, the machine learning algorithm we choose is random forests due to its excellent computational scalability . Random forests is a well-known method that has shown its merits in diverse application domains and has successfully been applied to Earth observations in both classification and regression problems . Briefly summarized, the random forest algorithm forms a combination of multiple decision trees, where each tree contributes a single vote to the final output, which is the most frequent class (for classification problems) or the average (for regression problems).

Compared to most application domains where random forests are applied, we employ the algorithm in a slightly different way as an autoregressive non-linear method for time series forecasting. In practice, this means that we replace the full and baseline linear model of Sect.  by a random forest model. At each pixel, the vegetation time series is still considered as a response variable, and the various climate time series serve as predictor variables (see Sect.  for an overview of our database). For a given value of the NDVI time series y at timestamp t, we investigate properties of the different predictor time series – temperature, radiation, etc. – by considering a moving window including a number of previous months (Fig.). In this way, the definition of Granger causality in Sect.  is adopted. Any climatic time series x Granger causes vegetation time series y if the predictive performance in terms of R2 improves when the moving window xt-1,xt-2,…,xt-P is incorporated in the random forests, in contrast to considering yt-1,yt-2,…,yt-P and wt-1,wt-2,…,wt-P only. Analogous to the linear case, we will speak of a full random forest model when all variables are taken into account and of a baseline random forest model when only the moving window yt-1,yt-2,…,yt-P of y is considered as a predictor. In Fig. , this principle is extended to four time series. The baseline random forest predictions of NDVI at t1 are based on the observations from the green moving window only, whereas the full random forest model includes the three red moving windows as well.

In our experiments, we treat each continental pixel as a separate problem and use the Scikit-learn library for the random forest regressor implementation, with the number of trees equal to 100 and the maximum number of predictor variables per node equal to the square root of the total number of predictor variables. Changes in these parameters or in the randomness of the algorithm do not cause substantial changes in the results (not shown). Model performance is assessed by means of 5-fold cross-validation. The window length is fixed to 12 months because initial experimental results revealed that longer time windows did not lead to improvements in the predictions (results omitted). Finally, we also experimented with techniques that exploit spatial correlations to improve the predictive performance of the model (see Sect. ).

Generally, the null hypothesis (H0) of Granger causality is that the baseline model has equal prediction error as the full model. Alternatively, if the full model predicts the target variable y significantly better than the baseline model, H0 is rejected. In some applications, inference is drawn in VAR by testing for significance of individual model parameters. Other studies have used likelihood-ratio tests, in which the full and baseline models are nested models . However, in both cases, the models are trained and evaluated on the same in-sample data. As it has been discussed above, the performance of any Granger-causal model should be validated on out-of-sample data to avoid overfitting (see Sect. ). Therefore, the null hypothesis of non-causality in the formulation stated above should be tested for by comparing out-of-sample prediction errors. To this end, statistical tests have been proposed and applied both in the econometric literature as well as in Granger-causality studies in the context of climate science. These kinds of tests, which compare out-of-sample prediction errors, are available for models for which parameter estimation is done through ordinary least squares or maximum likelihood estimation . Moreover, the asymptotic and finite-sample properties of a battery of tests for comparing forecasting accuracies of different models have been studied and, more recently, further tests aiming specifically at nested models have been proposed .

Unfortunately, all the tests mentioned above were designed to compare the out-of-sample prediction errors of linear parametric models . In climate, relations between variables are highly non-linear and tend to become even more non-linear as the temporal resolution of the data becomes finer . Therefore, it would be convenient to have at our disposal a statistical test to assess the significance of any quantitative evidence of climate (Granger) causing vegetation anomalies. Ideally, the test would be model independent so that any non-linear model could be used. One well-known model-independent test to compare the accuracy of two forecasts is the Diebold–Mariano test (DM test) . Although its application to Granger causality is promising, the test does not hold for nested models, because under H0 the prediction errors from two nested models are exactly the same and perfectly correlated . An alternative approach for comparing the predictive performance of different models is to use resampling methods such as the bootstrap or schemes such as 5×2 cross-validation . Methods based on the bootstrap have been used before in Granger-causality studies with climate data . However, these results need to be interpreted with care because, by increasing the number of bootstrap samples, the power of any paired test (such as the Wilcoxon signed rank test) to detect significant differences between the error distributions of both models (full and baseline) increases as well. For these reasons, we conclude that developing a statistical test that is able to handle non-stationary time series and non-linear models is not a trivial task. To the best of our knowledge, no such test exists in the current literature. In this paper, we focus on expressing Granger causality in a quantitative instead of a qualitative way and stress the gained improvement with the use of a non-linear model.

Our non-linear Granger-causality framework is used to disentangle the effect of past climate variability on global vegetation dynamics. To this end, climate data sets of observational nature – mostly based on satellite and in situ observations – have been assembled to construct time series (see Sect. ) that are then used to predict NDVI anomalies following the linear and non-linear causality frameworks described in Sect. . Data sets have been selected from the current pool of satellite and in situ observations on the basis of meeting a series of spatiotemporal requirements: (a) expected relevance of the variable for driving vegetation dynamics, (b) multidecadal record and global coverage available, and (c) adequate spatial and temporal resolution. The selected data sets can be classified into three different categories: water availability (including precipitation, snow water equivalent, and soil moisture data sets), temperature (both for the land surface and the near-surface atmosphere), and radiation (considering different radiative fluxes independently). Rather than using a single data set for each variable, we have collected all data sets meeting the above requirements. This has led to a total of 21 different data sets which are listed in Table . They span the study period 1981–2010 at the global scale and have been converted to a common monthly temporal resolution and 1∘×1∘ latitude–longitude spatial resolution. To do so, we have used averages to resample original data sets found at finer native resolution and linear interpolation to resample coarser-resolution ones.

For temperature, we consider seven different products based on in situ and satellite data: Climate Research Unit (CRU-HR) , University of Delaware (UDel) , NASA Goddard Institute for Space Studies (GISS) , merged land-ocean surface temperature (MLOST) , International Satellite Cloud Climatology Project (ISCCP) , and global land surface temperature data (CFSR-Land) . We also included one reanalysis data set, the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim . In the case of precipitation, eight products have been collected. Four of them result from the merging of in situ data only: Climate Research Unit (CRU-HR) , University of Delaware (UDel) , Climate Prediction Center Unified analysis (CPC-U) , and the Global Precipitation Climatology Centre (GPCC) . The rest result from a combination of in situ and satellite data, and may include reanalysis: CPC Merged Analysis of Precipitation (CMAP) , ERA-Interim , Global Precipitation Climatology Project (GPCP) , and Multi-Source Weighted-Ensemble Precipitation (MSWEP) . For radiation, two different products have been collected (considering incoming short-wave/long-wave and surface net radiation as different time series): the first is the NASA Global Energy and Water cycle Exchanges (GEWEX) surface radiation budget (SRB) based on satellite data, and the second is the ERA-Interim reanalysis . For soil moisture, we use the Global Land Evaporation Amsterdam Model (GLEAM) and the Climate Change Initiative (CCI) product . Two different soil moisture products by CCI are considered: the passive microwave data set and the combined active/passive product (). Moreover, snow water equivalent data come from the GlobSnow project .

To conclude, as a proxy for the state and activity of vegetation, we use the third-generation (3G) Global Inventory Modeling and Mapping Studies (GIMMS) satellite-based NDVI , a commonly used long-term global record of NDVI . We note that this data set is used to derive the response variable in our approach (seasonal NDVI anomalies; see Sect. ), while all other data sets are converted to predictor variables. The length of the NDVI record (1981–2010) sets the study period to an interval of 30 years.

In climate studies, Granger causality has already been applied on time series of seasonal anomalies . The latter may be obtained in a two-step decomposition procedure by first subtracting the seasonal cycle and then the long-term trend from the raw time series. Several competing decomposition methods have been proposed in the literature, including additive models, multiplicative models, and more sophisticated methods based on break points (see, e.g. ). In our framework, we used the following approach: in a first step, at each given pixel, the “raw” time series of the target variable yt and the climate predictors (xt, wt,…) are detrended linearly based on a simple linear regression with the timestamp t as a predictor variable applied to the entire study period. For the case of the target variable, this can be denoted as follows: yt≈ytTr=α0+α1t, with α0 and α1 being the intersect and the slope of the linear regression, respectively. We obtain in this way the detrended time series ytD=yt-ytTr. This detrending is needed to remove non-stationary signals in climatic time series, and allows us to draw the emphasis to the shorter-term multi-month dynamics. By detrending, one can assure that the mean of the probability distribution does not change over time; however, other moments of the probability distribution, such as the variance, might still be time dependent. As classical statistical procedures for testing Granger causality (i.e. autoregressive model, statistical tests) are developed for stationary time series, those methods are in fact not applicable to non-stationary climate data. In a second step, after subtracting the trend from the raw time series, the seasonal cycle ytS is calculated. When the assumption is made that the seasonal cycle is annual and constant over time, one can simply estimate it as the monthly expectation. To this end, the multi-year average for each of the 12 months of the year is calculated. Finally, the anomalies ytR can then be computed by subtracting the corresponding monthly expectation from the detrended time series: ytR=ytD-ytS. This procedure is schematically represented in Fig. .

We do not limit our approach to considering raw and anomaly time series of the data sets in Table  as predictors but also take into consideration different lag times, past cumulative values, and extreme indices (see following text). These additional predictors, here referred to as “higher-level variables”, are calculated based on raw and anomaly time series. Our application of Granger causality can be interpreted as a way to identify patterns in climate during past moving windows (see Fig. ) that are predictive with respect to the anomalies of vegetation time series. Therefore, by feeding predictor variables from previous timestamps to a linear (or non-linear) predictive model, one can identify subsequences of interest in the moving window specified for timestamp t, a technique that is similar to so-called shapelets . In addition, vegetation dynamics may not necessarily reflect the climatic conditions from, e.g. 3 months ago, but the average of the, e.g. three antecedent months. This integrated response to antecedent environmental and climatic conditions is referred to here as a “cumulative” response. More formally, we construct a cumulative variable of k months as the sum of time series observations in the last k months: Cumul[xt-1,xt-2,…,xt-k]=∑p=1kxt-p. Note that, unlike in the case of lagged variables, cumulative variables always include the period up to time t. Figure  illustrates an example of a 4-month cumulative variable. In our analysis, we experimented with time lags covering a wide range of time-lag values and concluded that including lags of more than 6 months did not yield substantial predictive power.

Another type of higher-level predictor variable that can be constructed from the data sets in Table  are extreme indices. Over the last few years, several research studies have focused on defining and indexing climate extremes . As an example, the Expert Team on Climate Change Detection and Indices (ETCCDI) recommends the use of a range of extreme indices related to temperature and precipitation . Here, we calculate a variety of analogous indices for the whole set of the collected climatic variables, based on both the raw data sets as well as on the seasonal anomalies (see Table ). In addition, we derived lagged and cumulative predictor variables from these extremes' indices to incorporate the potential impact of climatic extremes occurring, e.g. 3 months ago, or during the previous, e.g. 3 months, respectively. All these resulting time series appear as additional predictor variables in our non-linear Granger-causality framework (see Sect. ).

Combining the different climate and environmental predictor variables described above, we obtain a database of 4571 predictor variables per 1∘ pixel, covering 30 years at a monthly temporal resolution.

In a first experiment, we evaluate the extent to which climate variability Granger causes the anomalies in vegetation using a standard Granger-causality approach, in which only linear relationships between climate (predictors) and vegetation (target variable) are considered. To this end, ridge regression is used as a linear VAR model in the Granger-causality approach (note that this ridge regression will be substituted by the non-linear random forest approach in Sect. ). In the application of the ridge regression, we use all climatic and environmental predictor variables (Sect. ) and adopt a nested 5-fold cross-validation to properly tune the hyper parameter λ (see Eq. ). Figure a shows the predictive performance of the full ridge regression model. While the model explains more than 40 % of the variability in NDVI anomalies in some regions (R2>0.4), this is by itself not necessarily indicative of climate Granger causing the vegetation anomalies, as it may reflect simple correlations. In order to test the latter, we compare the results of the full model to a baseline model, i.e. an autoregressive ridge regression model that only uses previous values of NDVI to predict the NDVI at time t (see Sect. ). If climate Granger caused the variability of NDVI at a given pixel, the full ridge regression model (Fig. a) would show an increase in the predictive power over the predictions based on the baseline ridge regression model. However, the results unequivocally show that – when only linear relationships between vegetation and climate are considered – the areas for which vegetation anomalies are Granger caused by climate are very limited, involving mainly semiarid regions and central Europe (Fig. b).

For further comparison, we analyse the predictive performance obtained when (linear) Pearson correlation coefficients are calculated on the training data sets, selecting the highest correlation to the target variable for any of the 4571 predictor variables at each pixel. Figure c shows that the explained variance is again rather low and, for most regions, substantially lower than the R2 of the baseline ridge regression model, here considered as the minimum to interpret this predictive power as Granger causal. These results indicate that, despite being routinely used as a standard tool in climate–biosphere studies (see, e.g. ), univariate correlation analyses are unable to extract the nuances of the relationships between climate and vegetation dynamics.

To analyse the effect of climate on vegetation more thoroughly, we substitute the linear ridge regression model (VAR) by the non-linear random forest model. Results in Fig.  highlight the differences. Compared to the results in Sect. , the predictive power substantially increases by considering non-linear relationships between vegetation and climate (Fig. a). This is the case for most land regions but is especially remarkable in semiarid regions of Australia, Africa, and Central and North America, which are frequently exposed to water limitations. In those regions, more that 40 % of the variance of NDVI anomalies can be explained by antecedent climate variability. These results are further investigated by , who highlight the crucial role of water supply for the anomalies in vegetation greenness in these and other regions. On the other hand, the variance of NDVI explained in other areas, such as the Eurasian taiga, tropical rainforests, or China, is again below 10 %. We hypothesize two potential reasons: (a) the uncertainty in the observations used as target and predictors are typically larger in these regions (especially in tropical forests and at higher latitudes), and (b) these are regions in which vegetation anomalies are not necessarily primarily controlled by climate but may be predominantly driven by phenological and biotic factors , occurrence of wildfires , limitations imposed by the availability of soil nutrients , or agricultural practices . Nonetheless, the explained variance shown in Fig. a is again not necessarily indicative of Granger causality. As we did in Fig. b, in order to test whether the climatic and environmental controls do, in fact, Granger cause the vegetation anomalies, we compare the results of our full random forest model to a baseline random forest model which only uses previous values of NDVI to predict the NDVI at time t. As seen in Fig. b, in this case, the improvement over the baseline is unambiguous. One can conclude that – while not considering all potential control variables in our analysis – climate dynamics indeed (Granger) cause vegetation anomalies in most of the continental land surface, with a larger impact on subtropical regions and midlatitudes. Moreover, a comparison between Figs. b and b unveils that these causal relationships are highly non-linear, as expected given the distinct resistance and resilience of different ecosystems, which are reflected by a progressive response and recovery of vegetation to these perturbations .

For a better understanding of the results obtained by the two models, we average the performance of each model regionally. More specifically, we use the International Geosphere-Biosphere Program (IGBP) land cover classification to stratify the mean and variance of R2 for both the baseline and the full model in Fig.  per IGBP land cover class. The bar plot in Fig. shows that the full model outperforms the baseline model in all IGBP land cover classes, i.e. that Granger causality exists for all these biomes. In the parentheses, we note the number of pixels per region. The error bars indicate that the variances of the two models are analogous; i.e. they are low or high in both models in the same land cover class. For the Closed Shrublands region, one can observe the highest difference between the two models, yet only 19 pixels belong to this biome type. In savannah regions, the performance of the full model is high in comparison with other regions (see Fig. ). On the other hand, the lowest performance improvement of the full model with respect to the baseline is observed for the regions of Deciduous Needleleaf Forests and Evergreen Broadleaf Forests. This shows that for these two regions climate is not identified as a major control over vegetation dynamics (see discussion in previous paragraph about tropical and boreal regions).

Environmental dynamics reveal their effect on vegetation at different timescales. Since the adaptation of vegetation to environmental changes requires some time, and because soil and atmosphere have a memory, a necessary aspect to investigate is the potential lag-time response of vegetation to climate dynamics which relates to the ecosystem resistance and resilience properties. The idea of exploring lag times was introduced by several studies in the past (see, e.g. ), and it has been adopted in various studies more recently . These studies indicate that lag times depend on both the specific climatic control variable and the characteristics of the ecosystem. As explained in Sect. , in our analysis shown in Figs. 4 and 5, we moved beyond traditional cross-correlations and incorporated higher-level variables in the form of cumulative and lagged responses to extreme climate. As mentioned in Sect. , our experiments indicated that lags of more than 6 months do not add extra predictive power (not shown), even though the effect of anomalies in water availability on vegetation can extend for several months ().

To disentangle the response of vegetation to past cumulative climate anomalies and climatic extremes, Fig. a visualizes the predictive performance when cumulative variables and extreme indices are not included as predictive variables in the random forest model. As shown in Fig. b, in almost all regions of the world the predictive performance decreases substantially compared to the full random forest model approach, i.e. using the full repository of predictors (Fig. a), especially in regions such as the Sahel, the Horn of Africa, or North America. In those regions, 10–20 % of the variability in NDVI is explained by the occurrence of prolonged anomalies and/or extremes in climate, illustrating again the non-linear responses of vegetation. For more detailed results about lagged vegetation responses for specific climate drivers and the effect of climate extremes on vegetation, the reader is referred to .

Because of uncertainties in the observational records used in our study to represent climate and predict vegetation dynamics, and given that ecosystems and regional climate conditions usually extend over areas that exceed the spatial resolution of these records, one may expect that the predictive performance of our models becomes more robust when including climate information from neighbouring pixels. In addition, it is quite likely that neighbouring areas have similar climatic conditions which, in turn, affect vegetation dynamics in a similar manner. We therefore also consider an extension of our framework to exploit spatial autocorrelations, inspired by , who achieved spatial smoothness via an additional penalty term that punishes dissimilarity between coefficients for spatial neighbours. In our analysis, we incorporate spatial autocorrelations at a given pixel by extending the predictor variables of our models with the predictor variables of the eight neighbouring pixels. We provide such an extension both for the full and the baseline random forest model. As such, for the full random forest model, a vector of 41 139 (4571 × 9) predictor variables is formed for each pixel.

Figure c illustrates the performance of the full random forest model that includes the spatial information. As one can observe in Fig. d, the explained variance of NDVI anomalies remains similar to the original model that depicts the same approach without spatial autocorrelation (Fig. a). While in most areas the performance slightly increases, the explained variance never improves by more than 10 %; as a result, incorporating spatial autocorrelations in our framework does not seem to further improve the quantification of Granger causality and is not considered in further applications of the framework (see ). A possible explanation for this result is that the model without the spatial information cannot be outperformed because of the large dimensionality of the feature space, which may include redundant information, in combination with the low number of observations per pixel (Fig. a). Note that in this case the number of observations per pixel remains the same as in the original model (360 observations) while the number of predictor variables is 9 times larger.

In Sect. , we advocated that Granger-causality analysis should target NDVI anomalies, as opposed to raw NDVI values. There are several fundamental reasons for this. First, by applying a decomposition, one can subtract long-term trends from the NDVI time series, making the resulting time series more stationary. This is absolutely needed, as existing Granger-causality tests cannot be applied for non-stationary time series. Secondly, by subtracting the seasonal cycle from the time series, one is not only able to remove a confounding factor that may contribute predictive power without bearing causality but also able to remove a clear autoregressive component that can be well explained from the NDVI time series themselves. As vegetation has a strong seasonal cycle, it is not difficult to predict subsequent vegetation conditions by using the past observations of the seasonal cycle only. To corroborate this aspect, we repeat our analysis in Sect. , but this time the raw NDVI time series instead of the NDVI anomalies are considered as the target variable. We again compare the full and the baseline random forest models.

The results are visualized in Fig. a. As it can be observed, worldwide the R2 is close to the optimum of 1. However, due to the overwhelming domination of the seasonal cycle, it becomes very difficult, or even impossible, to unravel any potential Granger-causal relationships with climate time series in the Northern Hemisphere; see Fig. b. The predictability of NDVI based on the seasonal NDVI cycle itself is already so high that nothing can be gained by adding additional climatic predictor variables (see also the large amplitude of the seasonal cycle of NDVI at those latitudes compared to the NDVI anomalies, as illustrated in Fig. ). Therefore, a non-linear baseline autoregressive model is able to explain most of the variance in the time series. Moreover, as observed in Fig. , temperature and radiation also manifest strong seasonal cycles that often coincide with the NDVI cycle. For most regions on Earth, such a stationary seasonal cycle is less present for variables such as precipitation. This can potentially yield wrong conclusions, such as that temperature in the Northern Hemisphere is driving most NDVI variability, since the two seasonal cycles have the same pattern. However, based on the above discussion, it becomes clear that results of that kind should be treated with caution: for climate data, a Granger-causality analysis should be applied after decomposing time series into seasonal anomalies.