The use of Bandt-Pompe probability distributions and descriptors of Information Theory has been presenting satisfactory results with low computational cost in the time series analysis literature [1]-[3]. However, these tools have limitations when applied to data without time dependency. Given this context, we present a newly proposed technique for texture analysis and classification based on the Bandt-Pompe symbolization for SAR data. It consists of (i) linearize a 2-D patch of the image using the Hilbert-Peano curve, (ii) build an Ordinal Pattern Transition Graph that considers the data amplitude encoded into the weight of the edges; (iii) obtain a probability distribution function derived from this graph; (iv) compute Information Theory descriptors (Permutation Entropy and Statistical Complexity) from this distribution and use them as features to feed a classifier. The ordinal pattern graph we propose considers that the edges' weight is related to the absolute difference of observations, which encodes the information about the data amplitude. This modification considers the unfavorable signal-to-noise ratio of SAR images and leads to the characterization of several types of textures. Experiments with data from Munich urban areas, Guatemala forest regions, and Cape Canaveral ocean samples show the effectiveness of our technique in homogeneous areas, achieving satisfactory separability levels. The two descriptors chosen in this work are easy and quick to calculate and are used as input for a k-nearest neighbor classifier. Experiments show that this technique presents results similar to state-of-theart techniques that employ a much larger number of features and, consequently, impose a higher computational cost.
Summary This article serves two purposes. Firstly, it surveys the Bandt and Pompe methodology for the statistical community, stressing topics that are open for research. Secondly, it contributes towards a better understanding of the statistical properties of that approach for time series analysis. The Bandt and Pompe methodology consists of computing information theory descriptors from the histogram of ordinal patterns. Such descriptors lie in a 2D manifold: the entropy–complexity plane. This article provides the first proposal of a test in the entropy–complexity plane for the white noise hypothesis. Our test is based on true white noise sequences obtained from physical devices. The proposed methodology provides consistent results: It assesses sequences of true random samples as random (adequate test size), rejects correlated and contaminated sequences (sound test power) and captures the randomness of generators previously analysed in the literature.
This article proposes a study of the SARS-CoV-2 virus spread and the efficacy of public policies in Brazil. Using both aggregated (from large Internet companies) and fine-grained (from Departments of Motor Vehicles) mobility data sources, our work sheds light on the effect of mobility on the pandemic situation in the Brazilian territory. Our main contribution is to show how mobility data, particularly fine-grained ones, can offer valuable insights into virus propagation. For this, we propose a modification in the SENUR model to add mobility information, evaluating different data availability scenarios (different information granularities), and finally, we carry out simulations to evaluate possible public policies. In particular, we conduct a case study that shows, through simulations of hypothetical scenarios, that the contagion curve in several Brazilian cities could have been milder if the government had imposed mobility restrictions soon after reporting the first case. Our results also show that if the government had not taken any action and the only safety measure taken was the population’s voluntary isolation (out of fear), the time until the contagion peak for the first wave would have been postponed, but its value would more than double.
The ultimate purpose of the statistical analysis of ordinal patterns is to characterize the distribution of the features they induce. In particular, knowing the joint distribution of the pair entropy-statistical complexity for a large class of time series models would allow statistical tests that are unavailable to date. Working in this direction, we characterize the asymptotic distribution of the empirical Shannon’s entropy for any model under which the true normalized entropy is neither zero nor one. We obtain the asymptotic distribution from the central limit theorem (assuming large time series), the multivariate delta method, and a third-order correction of its mean value. We discuss the applicability of other results (exact, first-, and second-order corrections) regarding their accuracy and numerical stability. Within a general framework for building test statistics about Shannon’s entropy, we present a bilateral test that verifies if there is enough evidence to reject the hypothesis that two signals produce ordinal patterns with the same Shannon’s entropy. We applied this bilateral test to the daily maximum temperature time series from three cities (Dublin, Edinburgh, and Miami) and obtained sensible results.
We obtain expressions for the asymptotic distributions of the Rényi and Tsallis of order q entropies and Fisher information when computed on the maximum likelihood estimator of probabilities from multinomial random samples. We verify that these asymptotic models, two of which (Tsallis and Fisher) are normal, describe well a variety of simulated data. In addition, we obtain test statistics for comparing (possibly different types of) entropies from two samples without requiring the same number of categories. Finally, we apply these tests to social survey data and verify that the results are consistent but more general than those obtained with a χ2 test.
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