a b s t r a c tLet f be a univariate polynomial with real coefficients, f ∈ R[X ].Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm.The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d (L +ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: first, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.

Remote sensing is a useful tool for detecting landscape fragmentation, typically by creating land-use maps from remotely sensed images acquired at different dates. Nonetheless, classification may present a number of drawbacks since it degrades the information content of images leading to the loss of continuous information about fragmentation processes. For exploratory purposes, methods to detect landscape change based on continuous information should not require any a-priori assumptions about landscape characteristics. Accordingly, Fourier transforms may represent the best algorithmic solution. In this paper, we describe a Fourier transform tool developed in a free and open-source environment to detect potential fragmentation over the landscape. We briefly introduce Fourier transforms applied to remotely sensed imagery by further showing their potential application with an empirical example. We argue that Fourier transforms represent a straightforward approach for detecting spatial fragmentation of the landscape, on the strength of their potential to detect trends in increase or decrease of complexity/heterogeneity of the landscape in an objective manner. To our knowledge, this is the first open-source tool for analysing fragmentation of the landscape in multitemporal series based on Fourier transforms, which guarantees a high robustness and reproducibility of the applied algorithms

A spatiotemporal machine learning framework for automated prediction and analysis of long-term Land Use/Land Cover dynamics is presented. The framework includes: (1) harmonization and preprocessing of spatial and spatiotemporal input datasets (GLAD Landsat, NPP/VIIRS) including five million harmonized LUCAS and CORINE Land Cover-derived training samples, (2) model building based on spatial k-fold cross-validation and hyper-parameter optimization, (3) prediction of the most probable class, class probabilities and model variance of predicted probabilities per pixel, (4) LULC change analysis on time-series of produced maps. The spatiotemporal ensemble model consists of a random forest, gradient boosted tree classifier, and an artificial neural network, with a logistic regressor as meta-learner. The results show that the most important variables for mapping LULC in Europe are: seasonal aggregates of Landsat green and near-infrared bands, multiple Landsat-derived spectral indices, long-term surface water probability, and elevation. Spatial cross-validation of the model indicates consistent performance across multiple years with overall accuracy (a weighted F1-score) of 0.49, 0.63, and 0.83 when predicting 43 (level-3), 14 (level-2), and five classes (level-1). Additional experiments show that spatiotemporal models generalize better to unknown years, outperforming single-year models on known-year classification by 2.7% and unknown-year classification by 3.5%. Results of the accuracy assessment using 48,365 independent test samples shows 87% match with the validation points. Results of time-series analysis (time-series of LULC probabilities and NDVI images) suggest forest loss in large parts of Sweden, the Alps, and Scotland. Positive and negative trends in NDVI in general match the land degradation and land restoration classes, with “urbanization” showing the most negative NDVI trend. An advantage of using spatiotemporal ML is that the fitted model can be used to predict LULC in years that were not included in its training dataset, allowing generalization to past and future periods, e.g. to predict LULC for years prior to 2000 and beyond 2020. The generated LULC time-series data stack (ODSE-LULC), including the training points, is publicly available via the ODSE Viewer. Functions used to prepare data and run modeling are available via the eumap library for Python.

A seamless spatiotemporal machine learning framework for automated prediction and analysis of long-term Land Use / Land Cover dynamics is presented. The framework includes: (1) harmonization and preprocessing of high-resolution spatial and spatiotemporal input datasets (GLAD Landsat, NPP/VIIRS) including 5 million harmonized LUCAS and CORINE Land Cover-derived training samples, (2) model building based on spatial k-fold cross-validation and hyper-parameter optimization, (3) prediction of the most probable class, class probabilities and model variance of predicted probabilities per pixel, (4) LULC change analysis on time-series of produced maps. The spatiotemporal ensemble model consists of a random forest, gradient boosted tree classifier, and an artificial neural network, with a logistic regressor as meta-learner. The results show that the most important variables for mapping LULC in Europe are: seasonal aggregates of Landsat green and near-infrared bands, multiple Landsat-derived spectral indices, long-term surface water probability, and elevation. Spatial cross-validation of the model indicates consistent performance across multiple years with overall accuracy (a weighted F1-score) of 0.49, 0.63, and 0.83 when predicting 43 (level-3), 14 (level-2), and 5 classes (level-1). The spatiotemporal model outperforms spatial models on known-year classification by 2.7% and unknown-year classification by 3.5%. Results of the accuracy assessment using 48,365 independent test samples shows 87% match with the validation points. Results of time-series analysis (time-series of LULC probabilities and NDVI images) suggest forest loss in large parts of Sweden, the Alps, and Scotland.Positive and negative trends in NDVI in general match the land degradation and land restoration classes, with “urbanization” showing the most negative NDVI trend. An advantage of using spatiotemporal ML is that the fitted model can be used to predict LULC in years that were not included in its training dataset,allowing generalization to past and future periods, e.g. to predict LULC for years prior to 2000 and beyond 2020. The generated LULC time-series data stack (ODSE-LULC), including the training points, is publicly available via the ODSE Viewer. Functions used to prepare data and run modeling are available via the eumap library for python.

Good estimates of ecosystem complexity are essential for a number of ecological tasks: from biodiversity estimation, to forest structure variable retrieval, to feature extraction by edge detection and generation of multifractal surface as neutral models for e.g. feature change assessment. Hence, measuring ecological complexity over space becomes crucial in macroecology and geography. Many geospatial tools have been advocated in spatial ecology to estimate ecosystem complexity and its changes over space and time. Among these tools, free and open source options especially offer opportunities to guarantee the robustness of algorithms and reproducibility. In this paper we will summarize the most straightforward measures of spatial complexity available in the Free and Open Source Software GRASS GIS, relating them to key ecological patterns and processes