Introduction
Vegetation dynamics and the distribution of ecosystems are largely driven by
the availability of light, temperature, and water; thus, they are mostly
sensitive to climate conditions (;
). Meanwhile, vegetation also plays a crucial role in the
global climate system. Plant life alters the characteristics of the
atmosphere through the transfer of water vapour, exchange of carbon dioxide,
partition of surface net radiation (e.g. albedo), and impacts on wind speed
and direction (;
). Because of the strong two-way relationship between
terrestrial vegetation and climate variability, predictions of future climate
can be improved through a better understanding of the vegetation response to
past climate variability.
The current wealth of Earth observation data can be used for this purpose.
Nowadays, independent sensors on different platforms collect optical,
thermal, microwave, altimetry, and gravimetry information, and are used to
monitor vegetation, soils, oceans, and atmosphere (e.g.
). The longest composite records
of environmental and climatic variables already span up to 35 years, enabling
the study of multidecadal climate–biosphere interactions. Simple
correlation statistics and multilinear regressions using some of these data
sets have led to important steps forward in understanding the links between
vegetation and climate (e.g. ).
However, these methods in general are insufficient when it comes to assessing
causality, particularly in systems like the land–atmosphere continuum in
which complex feedback mechanisms are involved. A commonly used alternative
consists of Granger-causality modelling . Analyses of this
kind have been applied in climate attribution studies to investigate the
influence of one climatic variable on another, e.g. the Granger-causal
effect of CO2 on global temperature
, of
vegetation and snow coverage on temperature , of
sea surface temperatures on the North Atlantic Oscillation
, or of the El Niño–Southern Oscillation on the
Indian monsoon . Nonetheless, Granger causality
should not be interpreted as “real causality”; one assumes that a time
series A Granger causes a time series B if the past of A is helpful in
predicting the future of B (see Sect. 2 for a more formal definition).
However, the underlying statistical model that is commonly considered in such
a context is a linear vector autoregressive model, which is (again), by
definition, linear; see, e.g. .
In this article, we show new experimental evidence that advocates the need
non-linear methods to study climate–vegetation dynamics due to the
non-linear nature of these interactions
. To this end, we have assembled a
large, comprehensive database, comprising various global data sets of
temperature, radiation, and precipitation, originating from multiple online
resources. We use the Normalized Difference Vegetation Index (NDVI) to
characterize vegetation, which is commonly used as a proxy of plant
productivity . We followed an inclusive data
collection approach, aiming to consider all available data sets with a
worldwide coverage, and at least a 30-year time span and monthly temporal
resolution (Sect. ). Our novel non-linear Granger-causality
framework is used for finding climatic drivers of vegetation and consists of
several steps (Sect. ). In a first step, we apply time series
decomposition techniques to the vegetation and the various climatic time
series to isolate seasonal cycles, trends, and anomalies. Subsequently, we
explore various techniques for constructing more complex features from the
decomposed climatic time series. In a final step, we run a Granger-causality
analysis on the NDVI anomalies, while replacing traditional linear vector
autoregressive models with random forests. This framework allows for modelling
non-linear relationships and prevents overfitting. The results of the global
application of our framework are discussed in Sect. .
A Granger-causality framework for geosciences
Linear Granger causality revisited
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.
Overfitting and out-of-sample testing
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:
minβ∑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.
Non-linear Granger causality
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.
An illustrative example of the moving window approach considered in
the analysis of vegetation drivers at a given timestamp t1. Here, NDVI takes
the role of the time series y in Eq. (). In
addition, three climate predictor time series are shown. The baseline random
forest model only considers the green moving window, whereas the full random
forest model includes the red moving windows as well. The pixel corresponds
to a location in North America (lat: 37.5∘, long: -87.5∘).
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. ).
Granger-causal inference
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.
Database creation and variable construction
Global data sets
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.
Data sets used in our experiments. Basic data set characteristics
are provided, including the native spatial and temporal resolutions.
Variable
Product name
Spatial resolution
Temporal resolution
Primary data source
Reference
Temperature
CRU-HR (https://crudata.uea.ac.uk/cru/data/hrg/)
0.5∘
monthly
in situ
UDel (https://www.esrl.noaa.gov/psd/data/gridded/data.UDel_AirT_Precip.html)
0.5∘
monthly
in situ
ISCCP (https://isccp.giss.nasa.gov/pub/data/D2Tars/)
1∘
daily
satellite
ERA-Interim (http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/)
0.75∘
3-hourly
reanalysis
GISS (https://data.giss.nasa.gov/gistemp/)
2∘
monthly
in situ
MLOST (https://www.esrl.noaa.gov/psd/data/gridded/data.mlost.html)
5∘
monthly
in situ
CFSR-Land (http://hydrology.princeton.edu/getdata.php?dataid=9)
0.5∘
daily
satellite
Water availability
CRU-HR (https://crudata.uea.ac.uk/cru/data/hrg/)
0.5∘
monthly
in situ
MSWEP (http://www.gloh2o.org/)
0.25∘
3-hourly
satellite/in situ
UDel (https://www.esrl.noaa.gov/psd/data/gridded/data.UDel_AirT_Precip.html)
0.5∘
monthly
in situ
CMAP (https://www.esrl.noaa.gov/psd/data/gridded/data.cmap.html)
2.5∘
monthly
satellite/in situ
CPC-U (https://climatedataguide.ucar.edu/climate-data/cpc-unified-gauge-based-analysis-global-daily-precipitation)
0.25∘
daily
in situ
GPCC (http://www.dwd.de/EN/ourservices/gpcc/gpcc.html)
0.5∘
monthly
in situ
GPCP (https://www.esrl.noaa.gov/psd/data/gridded/data.gpcp.html)
2.5∘
monthly
satellite/in situ
ERA-Interim (http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/)
0.75∘
3-hourly
reanalysis
GLEAM (http://www.gleam.eu/)
0.25∘
daily
satellite
ESA CCI-PASSIVE (http://www.esa-soilmoisture-cci.org/node/145)
0.25∘
daily
satellite
ESA CCI-COMBINED (http://www.esa-soilmoisture-cci.org/node/145)
0.25∘
daily
satellite
GlobSnow (http://www.globsnow.info/index.php?page=Data)
0.25∘
daily
satellite
Radiation
SRB (https://eosweb.larc.nasa.gov/project/srb/srb_table)
1∘
3-hourly
satellite
ERA-Interim (http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/)
0.75∘
3-hourly
reanalysis
Greenness (NDVI)
GIMMS (https://ecocast.arc.nasa.gov/data/pub/gimms/)
0.25∘
monthly
satellite
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.
Anomaly decomposition
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. .
The three components of the NDVI time series decomposition of a
specific pixel of the Northern Hemisphere (lat: 53.5∘, long:
26.5∘). On top are the linear trend (black continuous line) and the
seasonal cycle (dashed black line) fitted on the raw data (red). On the
bottom are the remaining anomalies; see text for details.
Example of lagged and cumulative variables extracted from a
temperature time series. On top is part of a raw daily time series with its
monthly aggregation. In the middle is the 4-month lag-time monthly time series.
On the bottom is the corresponding 4-month cumulative variable. The pixel
corresponds to a location in Kentucky, USA (lat: 37.5∘, long:
-87.5∘).
Predictor variable construction
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.
Results and discussion
Detecting linear Granger-causal relationships
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).
Extreme indices considered as predictive variables. These indices
are derived from the raw (daily) data and the (daily) anomalies of the data
sets in Table . We also calculate the lagged and
cumulative variables from these extreme
indices.
Name
Description
SD
Standard deviation of daily values per month
DIR
Difference between max and min daily value per month
Xx
Max daily value per month
Xn
Min daily value per month
Max5day
Max over 5 consecutive days per month
Min5day
Min over 5 consecutive days per month
X99p/X95p/X90p
Number of days per month over 99th/95th/90th percentile
X1p/X5p/X10p
Number of days per month under 1st/5th/10th percentile
T25C1
Number of days per month over 25 ∘C
T0C1
Number of days per month below 0 ∘C
R10mm/R20mm2
Number of days per month over 10/20 mm
CHD (Consecutive high-value days)
Number of consecutive days per month over 90th percentile
CLD (Consecutive low-value days)
Number of consecutive days per month below 10th percentile
CDD (Consecutive dry days)2
Number of consecutive days per month when precipitation < 1 mm
CWD (Consecutive wet days)2
Number of consecutive days per month when precipitation ≥ 1 mm
Spatial heterogeneity3
Difference between max and min values within 1∘ box
1 Only for temperature data
sets. 2 Only for precipitation data sets. 3 Only for data sets with
native spatial resolution < 1∘ lat–long.
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.
Linear versus non-linear Granger causality
Linear Granger causality of climate on vegetation.
(a) Explained variance (R2) of NDVI anomalies based on a full
ridge regression model in which all climatic variables are included as
predictors. (b) Improvement in terms of R2 by the full ridge
regression model with respect to the baseline ridge regression model that
uses only past values of NDVI anomalies as predictors; positive values
indicate (linear) Granger causality. (c) A filter approach in which
the variable with the highest squared Pearson correlation against the NDVI
anomalies is selected. (d) Improvement in terms of R2 by the
filter approach with respect to the same baseline ridge regression model that
uses only past values of NDVI anomalies.
Non-linear Granger causality of climate on vegetation.
(a) Explained variance (R2) of NDVI anomalies based on a full
random forest model in which all climatic variables are included as
predictors. (b) Improvement in terms of R2 by the full random
forest model with respect to the baseline random forest model that uses only
past values of NDVI anomalies as predictors; positive values indicate
(non-linear) Granger causality.
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).
Mean R2 and variance per IGBP land cover class for both the
baseline and full random forest model. The green part indicates the
improvement in performance of the full model with respect to the baseline,
i.e. the quantification of Granger causality (as in
Fig. b). The number of pixels per IGBP class is noted in
the parentheses.
Analysis of spatiotemporal aspects of our framework. (a)
Explained variance (R2) of NDVI anomalies based on a full random forest
model in which all climatic variables are included as predictors as in
Fig. a, except for the cumulative variables and the
extreme indices (see Sect. ). (b) Difference in terms
of R2 between the model without cumulative and extreme predictors and the
full random forest model in Fig. a. (c) Explained
variance (R2) of NDVI anomalies based on a full random forest model in
which all climatic variables are included as predictors as in
Fig. a, as well as the predictors from the eight
nearest neighbours. (d) Difference in terms of R2 between this
full random forest model which includes spatial information from neighbouring
pixels and the full random forest model in Fig. a.
Spatial and temporal aspects
Comparison of model performance with R2 as the metric with the raw
NDVI time series as target variable. (a) Full random forest model.
(b) Improvement in terms of R2 of the full random forest model
over the baseline random forest model.
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.
The importance of focusing on vegetation anomalies
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.