Food security is at stake, with climate change heavily impacting agriculture and food production. In the present context of extreme events and changing conditions, developing advanced crop yield models can learn from all available information and provide interpretable predictions for decision-making is of paramount relevance. This work explores the potential and limitations of developing interpretable crop yield models using long short-term memory (LSTM) neural networks, which typically excel at extracting information from time series. LSTMs were designed and trained with multi-source satellite and meteorological time series over Continental US (CONUS) and corn, soybean, and wheat yield data from the US Department of Agriculture. Two recent attribution methods are used to interpret and extract knowledge from the developed models: integrated gradients (IG), based on back-propagation, and Shapley values (SHAP), based on perturbations. Our results show that (1) LSTM models achieved high accuracy (R ² > 0.56); (2) multi-source combinations outperformed single variable models in all crop models; (3) both attribution methods were equivalent in detecting essential drivers and their contribution; (4) satellite estimates of enhanced vegetation index (EVI) and vegetation optical depth (VOD) together with meteorological estimates of maximum temperature (TMX) were the most relevant input features for crop yield estimations; and finally, (5) we discovered critical periods of the crop growth cycle for the corn, soybean, and wheat models. The suggested strategy constitutes an important step towards modeling and understanding crop production systems and advancing in automatic data-driven and accountable field management.

Current remote sensing image classification problems have to deal with an unprecedented amount of heterogeneous and complex data sources. Upcoming missions will soon provide large data streams that will make land cover/use classification difficult. Machine-learning classifiers can help at this, and many methods are currently available. A popular kernel classifier is the Gaussian process classifier (GPC), since it approaches the classification problem with a solid probabilistic treatment, thus yielding confidence intervals for the predictions as well as very competitive results to the state-of-the-art neural networks and support vector machines. However, its computational cost is prohibitive for large-scale applications, and constitutes the main obstacle precluding wide adoption. This paper tackles this problem by introducing two novel efficient methodologies for GP classification. We first include the standard random Fourier features approximation into GPC, which largely decreases its computational cost and permits large-scale remote sensing image classification. In addition, we propose a model which avoids randomly sampling a number of Fourier frequencies and alternatively learns the optimal ones within a variational Bayes approach. The performance of the proposed methods is illustrated in complex problems of cloud detection from multispectral imagery and infrared sounding data. Excellent empirical results support the proposal in both computational cost and accuracy.

In preparation for new-generation imaging spectrometer missions and the accompanying unprecedented inflow of hyperspectral data, optimized models are needed to generate vegetation traits routinely. Hybrid models, combining radiative transfer models with machine learning algorithms, are preferred, however, dealing with spectral collinearity imposes an additional challenge. In this study, we analyzed two spectral dimensionality reduction methods: principal component analysis (PCA) and band ranking (BR), embedded in a hybrid workflow for the retrieval of specific leaf area (SLA), leaf area index (LAI), canopy water content (CWC), canopy chlorophyll content (CCC), the fraction of absorbed photosynthetic active radiation (FAPAR), and fractional vegetation cover (FVC). The SCOPE model was used to simulate training data sets, which were optimized with active learning. Gaussian process regression (GPR) algorithms were trained over the simulations to obtain trait-specific models. The inclusion of PCA and BR with 20 features led to the so-called GPR-20PCA and GPR-20BR models. The 20PCA models encompassed over 99.95% cumulative variance of the full spectral data, while the GPR-20BR models were based on the 20 most sensitive bands. Validation against in situ data obtained moderate to optimal results with normalized root mean squared error (NRMSE) from 13.9% (CWC) to 22.3% (CCC) for GPR-20PCA models, and NRMSE from 19.6% (CWC) to 29.1% (SLA) for GPR-20BR models. Overall, the GPR-20PCA slightly outperformed the GPR-20BR models for all six variables. To demonstrate mapping capabilities, both models were tested on a PRecursore IperSpettrale della Missione Applicativa (PRISMA) scene, spectrally resampled to Copernicus Hyperspectral Imaging Mission for the Environment (CHIME), over an agricultural test site (Jolanda di Savoia, Italy). The two strategies obtained plausible spatial patterns, and consistency between the two models was highest for FVC and LAI ($R^2=0.91$, $R^2=0.86$) and lowest for SLA mapping ($R^2=0.53$). From these findings, we recommend implementing GPR-20PCA models as the most efficient strategy for the retrieval of multiple crop traits from hyperspectral data streams. Hence, this workflow will support and facilitate the preparations of traits retrieval models from the next-generation operational CHIME.

Kernel methods are powerful machine learning techniques which use generic non-linear functions to solve complex tasks. They have a solid mathematical foundation and exhibit excellent performance in practice. However, kernel machines are still considered black-box models as the kernel feature mapping cannot be accessed directly thus making the kernels difficult to interpret. The aim of this work is to show that it is indeed possible to interpret the functions learned by various kernel methods as they can be intuitive despite their complexity. Specifically, we show that derivatives of these functions have a simple mathematical formulation, are easy to compute, and can be applied to various problems. The model function derivatives in kernel machines is proportional to the kernel function derivative and we provide the explicit analytic form of the first and second derivatives of the most common kernel functions with regard to the inputs as well as generic formulas to compute higher order derivatives. We use them to analyze the most used supervised and unsupervised kernel learning methods: Gaussian Processes for regression, Support Vector Machines for classification, Kernel Entropy Component Analysis for density estimation, and the Hilbert-Schmidt Independence Criterion for estimating the dependency between random variables. For all cases we expressed the derivative of the learned function as a linear combination of the kernel function derivative. Moreover we provide intuitive explanations through illustrative toy examples and show how these same kernel methods can be applied to applications in the context of spatio-temporal Earth system data cubes. This work reflects on the observation that function derivatives may play a crucial role in kernel methods analysis and understanding.