We used optical remote sensing data of forest sites to measure biosphere dynamics, specifically the normalized difference vegetation index (NDVI) , which we define as the “target” variable. However, employing NDVI presents drawbacks . Namely, it has a saturating and nonlinear connection to green biomass and responds to greenness rather than the actual plant photosynthesis process. Despite these limitations, this index has been used successfully for various purposes, including, but not limited to, evaluating ecosystem resilience and tracking the decline of vegetation greenness in the Amazon forests . Additionally, NDVI was shown to be a good indicator of vegetation response to extreme climate events .

We used NDVI values obtained from the Moderate Resolution Imaging Spectroradiometer (MODIS) in the FluxnetEO dataset v1.0 . This dataset is a multi-dimensional array of values referred to as a data cube . It comprises a collection of labeled univariate time series at the geographic location of eddy covariance towers. Eddy covariance towers are specialized measurement stations designed to capture and quantify land–atmosphere fluxes and meteorological conditions, providing insights into ecosystem health and functionality . The FluxnetEO datasets are gap-filled using measurements from eddy covariance towers, thus providing a higher-resolution product in time and space compared to the raw MODIS data.

The data cubes in the dataset present a spatial resolution of 500 m and a daily temporal resolution, covering the period of 2000–2020 (inclusive). The cubes span a pixel of size 3km×3km centered on the eddy covariance towers in each location. We further pre-processed the data by averaging the NDVI for each time step as the mean of all the pixels within the cube; see Fig. . Additionally, we smoothed the signal using a Savitzky–Golay filter with a 7 d window and a polynomial order of 3 to eliminate any potential artifacts caused by noise.

We selected study sites based on their forest cover percentage, ensuring over 80 % forest cover in each cube; see Fig. . The forest masks were obtained from the Copernicus Global Land Service (CGLS) product , which has a resolution of 100 m. These sites represent three different forest types: evergreen broadleaved forests (EBF), mixed forests (MF), and deciduous broadleaved forests (DBF). We chose to study forest sites because of their importance to the carbon cycle and because they are the least affected by human influence at a daily timescale. Consequently, we employed the described approach, detailed in Fig. , to minimize further imperfections in the vegetation signal caused by human intervention. As a result of this selection criterion, the number of study sites is reduced from 42 to 20.

In this study, we used climate variables as input variables to predict the target variable. Following machine learning terminology, we will refer to them as “features”. We selected air temperature (mean, minimum, and maximum), mean sea level pressure, mean global solar radiation, and precipitation as features. The climate data were obtained from the E-OBS product v26.0e . Based on in situ observations, this dataset is spatially interpolated to cover most of the European continent. The spatial resolution is 0.1° (11.1km×11.1km), and the temporal resolution is daily. The time length of the feature variables is identical to the target variable and spans from 2000 to 2020 (inclusive). We split the data for training and testing as follows: years from 2000 to 2013 (inclusive) for training and validation. We use the remaining years, from 2014 to 2020, for testing, resulting in a 67 % and 33 % training and testing split ratio.

We aim to learn the NDVI behavior of forests as a proxy for ecosystem response to climate drivers. We use air temperature (mean, minimum, maximum), precipitation, pressure, and global solar radiation. We assume that the knowledge of the target variable is constrained to a certain period, after which only the features are available. Our goal is to build a model that, using those features, can predict the target variable. This is obtained by training the models on the available time interval, which comprises the years 2000–2013 (inclusive) for this study. After the training, the models only use feature variables to predict the NDVI for the remaining period, 2014–2020. Further details for the training setup can be found in Appendix .

The task can be formalized as follows. We aim to approximate the target variable v(t)∈Rdv using input data u(t). In the context of this study, we retain a unidimensional approach with dv=1, given that we have a single target variable, the NDVI. We assume that both u(t) and v(t) are components of the same dynamical system. Under this assumption, the behavior v(t) is influenced by u(t), allowing us to leverage the information from u(t) to estimate v(t). Our setup consists of two sets of data, u(t) and v(t), which can be measured over a given time period {1,2,…,T}. After time step t=T only u(t) remains measurable, and the goal of the recurrent architectures is to create a model efficient in modeling v(t) based solely on the available u(t) data . To summarize, the models use meteorological data of a given day to predict the response of the vegetation on that same day. Despite having different training methods, all the models in this study are built to perform this task.

We consider four different recurrent architectures: recurrent neural networks (RNNs), long short-term memory networks (LSTMs), gated recurrent-unit networks (GRUs), and echo state networks (ESNs). Each of these models presents an internal state x(t)∈Rdx, which encodes the temporal dependencies of the input data u(t)∈Rdu, where du=6 in our study, representing the dimension of the input data. The internal size dx is chosen to be dx>du. A defining feature of these architectures is the recursive transmission of internal states, facilitating historical data retention as the model progresses through subsequent steps. This evolution of the generic RNN model is given by 1x(t)=H(x(t-1),u(t);θ), where θ represents the weights and biases of the model, also called parameters, and H represents a generic RNN update function.

Deep learning models like feed-forward neural networks (FFNNs) adjust their weights at each time step t∈{1,2,…,T} during training using a method called backpropagation (BP) . BP relies on a “loss function” L given by L(θ)=∑t=1TLt(θ), where θ stands for the network's parameters. Leveraging the loss function, BP minimizes the difference between the model's predicted and actual output by adjusting the model's weights. To do this, BP calculates the gradient of the loss function with respect to each weight and then updates the network weights in a direction that minimizes the loss. One of the most common approaches to minimizing the loss function is stochastic gradient descent (SGD) . However, one of BP's limitations is that it does not account for time dependencies.

For time-dependent models, such as RNNs, LSTMs, and GRUs, handling sequential data poses additional challenges. Backpropagation through time (BPTT) is a specialized training method designed for these architectures . Central to BPTT is the notion of “unrolling” the network over time, effectively transforming it into a FFNN where backpropagation can be applied. This allows the model to calculate errors and update weights across the entire sequence, making it possible to capture long-term dependencies and patterns in time series data.

However, applying BPTT across the entire sequence can be computationally intense and time-consuming. Moreover, it often reduces the error to a very small amount by the end of the sequence. To avoid this issue, we use a truncated version of BPTT, limiting the backpropagation to a fixed number of steps, denoted by k , which is smaller than the total number of training time steps, T .

This truncated approach ensures efficiency but requires transferring the last hidden state from the truncated section to the initial state in the following sequence. This step maintains some memory and ensures the network's training process continuity.

The final output of all the models considered in this study comes from a feed-forward layer. The parameters of this layer are also trained using BP. The following equation describes the feed-forward layer: 2ṽ(t)=σ(Wvx(t)+bv), where ṽ(t)∈Rdv is the predicted output, x(t)∈Rdx is the hidden state of the model at time step t, Wv∈Rdv×dx is the weight matrix, and bv∈Rdx is a bias vector. Additionally, σ represents the activation function. This procedure is illustrated graphically in Fig. a. Finally, the full details of the models are given in Appendix  for the RNNs, Sect.  for the LSTMs, and Sect.  for the GRUs. The RNNs, LSTMs, and GRUs have been implemented using the PyTorch library accessed through Skorch .

Echo state networks ESNs; , along with liquid state machines , belong to the larger family of reservoir computing (RC) models, based on a shared theoretical background . The fundamental idea of ESNs is to project the training data into a higher-dimensional, nonlinear system named the “reservoir” through an input layer. This process transforms the input data into vectors called “states”. After the data pass through the reservoir, the states are collected. The model is then trained by regressing these states against the target data.

More specifically, the ESNs have three layers: an input layer Win, a reservoir layer W, and an output layer Wout. A distinctive feature of ESNs is that the weights in the input and reservoir layers are fixed. These weights are generated randomly and do not change during training. This approach contrasts with conventional neural networks, where weights are continuously updated during training to reduce errors. For each training time step t∈{1,2,…,T}, the hidden states, indicated as x(t), are preserved and accumulated in a matrix X∈Rdx×T. Indicated as a “state matrix”, this matrix effectively represents the system's dynamics. The last layer of the ESN is obtained through a linear regression operation that uses the states matrix to generate a feed-forward layer, creating the network's output layer: 3Wout=YtargetX⊤(XX⊤+βI)-1, where I is the identity matrix, and β is a regularization coefficient. The matrix Ytarget∈Rdv×T is generated with the desired output v∈Rvd stacked column-wise. While layers of recurrent models trained through BPTT can be stacked, ESNs are usually computed from a single inner layer (reservoir).

The construction and training of the ESN allow for the faster computational time of all the proposed models while also solving the vanishing and exploding gradients since no derivatives are taken at any step of the process. An illustration of the ESN approach to training is provided in Fig. b. The full details for the ESNs are given in Appendix . For the implementations of the ESNs, we relied on the Julia package ReservoirComputing.jl .

In this study, we adopt the approach outlined by to define anomalies based on the seasonal variability observed in the signal. The process described in this section is depicted in Fig. . The anomalies at a given time l, denoted by A(l), are defined as follows: 4A(l)=s(l)-s‾(l)σ(l). Here, s(l) represents the signal at time l, s‾(l) is the mean of the signal at that time, and σ(l) refers to its standard deviation. In this context, the time variable l denotes the specific day of the year, i.e., the third of March, and is determined based on a multi-year mean. The normalization process ensures that the signal exhibits a zero mean, which facilitates identifying extreme events as data points that fall outside a specified distribution range.

After normalizing the data and delineating the anomalies, we characterize extreme events as data points falling outside a specific quantile, chosen to be 0.90, 0.91, …, or 0.99. Based on the selected quantile, we define two threshold parameters κ+ and κ- to represent positive and negative extremes, respectively. We determine these parameters individually for each site involved in our study.

During the training phase, which spans 2000 to 2014, we determine the threshold values κ+ and κ-. We apply these threshold values to the test dataset comprising the years from 2014 to 2020 to determine the extreme events in this latter dataset. The normalization of the data in the test datasets is done with the mean and SD values obtained from the training dataset.

We present our results in Table , showing that the ESN outperformed all other models for both the FS and GS signals, closely followed by the LSTM. While the GRU exhibited comparable results to the LSTM for the FS signal, the differences became more pronounced in the GS signal. In contrast, the RNN consistently delivered the least favorable results across the board and exhibited the highest standard deviation (SD) among the analyzed models. In Fig. , we include the comparison of these metrics per site, which shows great variation in the models' performance across different sites.

The performance rankings of the models remained consistent when transitioning from the FS signal to the GS signal. Notably, the SMAPE metric indicated increased accuracy for all models, while the NRMSE metric suggested decreased accuracy. Similarly, we observe a reduction in the SD for the SMAPE but an increase in the NRMSE. This increase is very noticeable in the GRU and RNN models.

The models generally exhibited similar performance, with the ESN yielding slightly better results and the RNN demonstrating the least accurate forecasts. The gated methods showed similar performances. While these metrics provide an initial picture of the models' performance in the task, the results do not indicate a clear best performer among the models. These similarities in the results' metrics emphasize the need for further exploration using alternative evaluation tools.

Drawing from information theory, we use entropy–complexity (EC) plots (introduced in Sect. ) to examine the residuals, defined as the differences between the model's predictions and measurements. Residuals convey valuable information about a model's performance. In an ideal scenario, where a model perfectly represents a system's dynamics, these residuals would resemble white noise and be positioned in the lower-right corner of the EC plane .

Figure  shows the EC plots of the models' residuals. Additionally, we show the mean of these metrics per model architecture across all sites. We find that the residuals cluster by model across all locations, with minimal overlap between each model's clusters. The LSTM and GRU models occupy the curve's ascending left side, indicating the presence of more structure in the residuals. In contrast, the ESN and, to a greater extent, the RNN are positioned closer to the white noise region, implying less structure in the residuals.

Comparing the FS signal to the GS signal shows no apparent differences in the results, underscoring the consistency of model performance. Based on the position of RNN residuals in the EC plots, we would expect a substantially improved performance of these models compared to the other model architectures. However, inspecting the predicted time series suggests a different explanation: the RNN model predictions show large variability (Fig. , which obfuscates the underlying dynamics. Because the EC plots capture the resulting residuals' noise, one can misinterpret the positioning of RNNs as favorable results.

In this section, we use the metrics introduced in Sect.  to evaluate the models' ability to capture extreme vegetation dynamics. Fig. a shows a distinct division in the probability of detection (POD): the LSTM and ESN models exhibit low values, indicating an inability to detect extreme events. The RNN and GRU models show higher values, indicating better performance. However, the POD values are generally low for all models. Furthermore, all models display some level of overlap in the confidence bands. Finally, it is possible to observe a worsening of the results with increased quantiles. Figure b shows a lower probability of false detection (POFD) of ESNs, indicating that they perform better at avoiding the prediction of extremes when there are none compared to the other models. Conversely, the RNN, which predicts noisy behavior, as shown in Sect. , exhibits higher POFD values. The probability of false alarms (POFA; Fig. c) follows a similar pattern, with the ESN outperforming the other models. The GRU and LSTM display more comparable behaviors, while the RNN lags behind, showing the least favorable performance among all models. The displayed metric consistently leans towards the higher end of the potential value range. Like in the case of the POD, the values of the POFA get worse with each increase in the quantiles. Finally, Fig. d shows the probability of correctness (PC). The ESN leads the graph despite showing the lowest POD. Following closely are the LSTM and GRU, which exhibit similar performance. The RNN maintains the poorest performance across all metrics, emphasizing its limitations in capturing extreme vegetation dynamics.

To examine the prediction performance of extreme NDVI reduction during the summer season, we focus on June, July, and August using only κ-, showcased in Fig. . In Fig. a, we see the probability of detection (POD) for different models. The RNN performs best, outperforming the LSTM, GRU, and ESN, delivering similar results. Figure b shows the probability of false detection (POFD). Here, the RNN performs the worst, while the ESN stands out with the best values and the lowest standard deviation (SD) among all models. The gating-based models, LSTM and GRU, perform almost identically. We find similar performances among models in Fig. c. The ESN demonstrates a decrease in the probability of false alarms (POFA) values for the 0.92 quantiles. Any small gaps in performance observed for lower quantiles are effectively bridged at higher values. Finally, Fig. d illustrates the ESN as the top-performing model, exhibiting the lowest SD among the models. The gated models, LSTM and GRU, display similar performances, while the RNN ranks as the poorest-performing model in this analysis of NDVI summertime decreases. Figure  shows the variability of the results across all analyzed locations.

Similar to previous studies , this work seeks to understand the ability of recurrent neural network architectures to model vegetation greenness. Based on the analyzed metrics, gated architectures and ESNs appear almost equally effective, with ESNs having a slight advantage. The models' performance varies considerably across sites, as illustrated in Fig. . Such variability can explain the overlap in the standard deviation and is probably due to the different dynamics of the locations investigated that span different climate zones and forest types.

Additionally, we also provide a comparison of entropy–complexity (EC) plots in Fig. . Contrary to the previous metrics, RNNs seem to outperform all other models in the EC plots. However, the presence of disproportionate high noise in the EC plots skewed the results towards the white noise range. Generally, this would indicate the absence of structure in the residuals and point to a good behavior of the prediction. Further investigation into the actual dynamics of the prediction showed that this was due to the high levels of noise. Furthermore, the EC plots clear the distinction between similarly performing models, i.e., gated architectures and ESNs. While these models show comparable performances of NRMSE and SMAPE, they are clearly distinct in the EC plots, highlighting the better performance of the ESNs. Thus, our results illustrate that it is crucial not to rely solely on a single set of metrics when evaluating results in similar studies.

Our results for the analysis of vegetation extreme responses to climate drivers using daily data showed that none of the models notably outperformed the others across all the metrics considered, as seen in Fig. . The ESNs performed well in avoiding false alarms but failed to obtain a high accuracy when predicting extremes, performing similarly to LSTMs. ESNs also showed the lowest standard deviation among all the models considered. RNNs performed worst among all metrics analyzed. Overall, all the models showed poor results for this task. When considering negative extremes in the summer season, indicating a decrease in the productivity of vegetation, the results showed again that the recurrent architectures could not correctly capture extremes in vegetation responses to climate forcing (Fig. ). The ESNs performed best in avoiding false extremes and the overall accuracy of the task's execution, while the RNNs showed the worst prediction performance.

Motivated by recent ESN applications to data from laboratory experiments , we expected ESNs to outperform the other models in the extreme prediction task. Instead, the results showed that, while the ESN provided good results, they were not outstanding compared to the BPTT gated models. Additionally, none of the models showed good results, contrary to other applications in hydrology showcasing the successful application of recurrent models to extremes .

Often, remote sensing studies are subject to a series of data limitations. However, in this case, the data went through rigorous quality checks and corrections . When sensed from space in discrete wavelengths, such optical data remain a coarse abstraction of the biological states of the forest under scrutiny. More critical limitation arises from the availability of daily data solely spanning the last 21 years. Although this is a long time span from a remote sensing perspective, it can be regarded as a short period for training machine learning models, which require abundant data points for optimal performance.

The metrics we used are known to be sensitive to noise. Using various metrics, we identified limitations in methods such as the EC plots (Fig. ). For instance, the noise in the RNN's prediction was incorrectly attributed to residual noise, distorting the outcomes. Moreover, standard metrics like NRMSE and SMAPE provide a narrowed view. This is evident when comparing ESNs and LSTMs: even though they had similar scores with these metrics, their distinctions were clear in the EC plots.

Most existing studies center on creating a single global model, which is key to understanding performances beyond the training sample . However, we opted to train individual models for each site to understand how, under optimal conditions, each architecture could capture the specific dynamics of an ecosystem without the need for extrapolation. In this way, we could focus on the models' inherent capability to understand each ecosystem's dynamics, and we are sure that model differences do not arise due to different data demands for generalizations. Increasing the locations and features and expanding the models are necessary future steps for the continued investigation of modeling extremes in vegetation with ML. It has been shown that including additional features and multiple locations improves the ML model performance in hydrological applications . While similar studies do exist in the context of biosphere dynamics , more investigation is needed. For example, including more locations, as highlighted in represents the easiest way to improve performance. Investigating the effects of more location on the performance of ML for vegetation extremes would be an important contribution for the continued adoption of ML models in predicting biosphere dynamics.

ESN and LSTM demonstrated comparable performance in modeling vegetation greenness. However, these models exhibited limited predictive capability for extreme events. Spatial information has proven helpful in similar studies , and it could be beneficial to explore the extent to which it contributes to extreme conditions. Furthermore, the models used could be tailored more to the task. In , the authors point out the limitations in using the mean-squared-error loss function, a practice still widely diffused in the field and adopted in this paper. Furthermore, show the superior performance of loss functions based on weighting and relative entropy compared to other loss functions in learning extreme events. Finally, further investigations could benefit from deeper explorations of ESNs. With their user-friendly nature, straightforward hyperparameter tuning, and fast training, they stand out as a robust modeling method, able to compete with top-tier deep learning architectures such as LSTMs.