The development of new measures and algorithms to quantify the entropy or related concepts of a data series is a continuous effort that has brought many innovations in this regard in recent years. The ultimate goal is usually to find new methods with a higher discriminating power, more efficient, more robust to noise and artifacts, less dependent on parameters or configurations, or any other possibly desirable feature. Among all these methods, Permutation Entropy (PE) is a complexity estimator for a time series that stands out due to its many strengths, with very few weaknesses. One of these weaknesses is the PE’s disregarding of time series amplitude information. Some PE algorithm modifications have been proposed in order to introduce such information into the calculations. We propose in this paper a new method, Slope Entropy (SlopEn), that also addresses this flaw but in a different way, keeping the symbolic representation of subsequences using a novel encoding method based on the slope generated by two consecutive data samples. By means of a thorough and extensive set of comparative experiments with PE and Sample Entropy (SampEn), we demonstrate that SlopEn is a very promising method with clearly a better time series classification performance than those previous methods.

Sleep is a growing area of research interest in medicine and neuroscience. Actually, one major concern is to find a correlation between several physiologic variables and sleep stages. There is a scientific agreement on the characteristics of the five stages of human sleep, based on EEG analysis. Nevertheless, manual stage classification is still the most widely used approach. This work proposes a new automatic sleep classification method based on unsupervised feature classification algorithms recently developed, and on EEG entropy measures. This scheme extracts entropy metrics from EEG records to obtain a feature vector. Then, these features are optimized in terms of relevance using the Q-α algorithm. Finally, the resulting set of features is entered into a clustering procedure to obtain a final segmentation of the sleep stages. The proposed method reached up to an average of 80% correctly classified stages for each patient separately while keeping the computational cost low.

A wildland fire is an uncontrolled fire that occurs mainly in forest areas, although it can also invade urban or agricultural areas. Among the main causes of wildfires, human factors, either intentional or accidental, are the most usual ones. The number and impact of forest fires are expected to grow as a consequence of the global warming. In order to fight against these disasters, it is necessary to adopt a comprehensive, multifaceted approach that enables a continuous situational awareness and instant responsiveness. This paper describes a hierarchical wireless sensor network aimed atearly fire detection in risky areas, integrated with the fire fighting command centres, geographical information systems, and fire simulators. This configuration has been successfully tested in two fire simulations involving all the key players in fire fighting operations: fire brigades, communication systems, and aerial, coordination, and land means.

Permutation Entropy (PE) is a time series complexity measure commonly used in a variety of contexts, with medicine being the prime example. In its general form, it requires three input parameters for its calculation: time series length N, embedded dimension m, and embedded delay $\tau$ . Inappropriate choices of these parameters may potentially lead to incorrect interpretations. However, there are no specific guidelines for an optimal selection of N, m, or $\tau$ , only general recommendations such as $N>>m!$ , $\tau =1$ , or $m=3,\dots ,7$ . This paper deals specifically with the study of the practical implications of $N>>m!$ , since long time series are often not available, or non-stationary, and other preliminary results suggest that low N values do not necessarily invalidate PE usefulness. Our study analyses the PE variation as a function of the series length N and embedded dimension m in the context of a diverse experimental set, both synthetic (random, spikes, or logistic model time series) and real–world (climatology, seismic, financial, or biomedical time series), and the classification performance achieved with varying N and m. The results seem to indicate that shorter lengths than those suggested by $N>>m!$ are sufficient for a stable PE calculation, and even very short time series can be robustly classified based on PE measurements before the stability point is reached. This may be due to the fact that there are forbidden patterns in chaotic time series, not all the patterns are equally informative, and differences among classes are already apparent at very short lengths.

Two main weaknesses have been identified for permutation entropy (PE): the neglect of subsequence pattern differences in terms of amplitude and the possible ambiguities introduced by equal values in the subsequences. A number of variations or customizations to the original PE method to address these issues have been proposed in the scientific literature recently. Specifically for ties, methods have tried to remove the ambiguity by assigning different weighted or computed orders to equal values. Although these methods are able to circumvent such ambiguity, they can substantially increase the algorithm costs, and a general characterization of their practical effectiveness is still lacking. This paper analyses the performance of PE using several biomedical datasets (electroencephalogram, heartbeat interval, body temperature, and glucose records) in order to quantify the influence of ties on its signal class segmentation capability. This capability is assessed in terms of statistical significance of the PE differences between classes and classification sensitivity and specificity. Being obvious that ties modify the PE results, we hypothesize that equal values are intrinsic to the acquisition process, and therefore, they impact all the classes more or less equally. The experimental results confirm ties are often not the limiting factor for PE, even they can be beneficial as a sort of stochastic resonance, and it can be far more effective to focus on the embedding dimension instead.