Differentiating patients with obstructive sleep apnea from healthy controls based on heart rate-blood pressure coupling quantified by entropy-based indices.

We introduce an entropy-based classification method for pairs of sequences (ECPS) for quantifying mutual dependencies in heart rate and beat-to-beat blood pressure recordings. The purpose of the method is to build a classifier for data in which each item consists of two intertwined data series taken for each subject. The method is based on ordinal patterns and uses entropy-like indices. Machine learning is used to select a subset of indices most suitable for our classification problem in order to build an optimal yet simple model for distinguishing between patients suffering from obstructive sleep apnea and a control group.

A review of symbolic dynamics and symbolic reconstruction of dynamical systems.

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.

Permutation group entropy: A new route to complexity for real-valued processes.

This is a review of group entropy and its application to permutation complexity. Specifically, we revisit a new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviors.

Information Geometry, Complexity Measures and Data Analysis.

In the last several years, a new approach to information theory, called information geometry, has emerged [...].

SETApp: A machine learning and image analysis based application to automate the sea urchin embryo test.

Since countless xenobiotic compounds are being found in the environment, ecotoxicology faces an astounding challenge in identifying toxicants. The combination of high-throughput in vivo/in vitro bioassays with high-resolution chemical analysis is an effective way to elucidate the cause-effect relationship. However, these combined strategies imply an enormous workload that can hinder their implementation in routine analysis. The purpose of this study was to develop a new high throughput screening method that could be used as a predictive expert system that automatically quantifies the size increase and malformation of the larvae and, thus, eases the application of the sea urchin embryo test in complex toxicant identification pipelines such as effect-directed analysis. For this task, a training set of 242 images was used to calibrate the size-increase and malformation level of the larvae. Two classification models based on partial least squares discriminant analysis (PLS-DA) were built and compared. Moreover, Hierarchical PLS-DA shows a high proficiency in classifying the larvae, achieving a prediction accuracy of 84 % in validation. The scripts built along the work were compiled in a user-friendly standalone app (SETApp) freely accessible at https://github.com/UPV-EHU-IBeA/SETApp. The SETApp was tested in a real case scenario to fulfill the tedious requirements of a WWTP effect-directed analysis.

Forecasting wind power ramps with prediction coordinates.

To the best of our knowledge, the method of prediction coordinates is the only forecasting method in nonlinear time series analysis that explicitly uses the stochastic characteristics of a system with dynamical noise. Specifically, it generates multiple predictions to jointly infer the current states and dynamical noises. Recent findings based on hypothesis testing show that weather is nonlinear and stochastic and, therefore, so are renewable energy power outputs. This being the case, in this paper, we apply the method of prediction coordinates to forecast wind power ramps, which are rapid transitions in the wind power output that can deteriorate the quality of the electricity supply. First, the method of prediction coordinates is tested using numerical simulations. Then, we present an example of wind power ramp forecasting with empirical data. The results show that the method of prediction coordinates compares favorably with other methods, validating it as a reliable tool for forecasting transitions in nonlinear stochastic dynamics, particularly in the field of renewable energies.

A generalized permutation entropy for noisy dynamics and random processes.

Permutation entropy measures the complexity of a deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or simply permutations. Reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov-Sinai entropy of the underlying dynamics in the limit of ever longer permutations and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as happens when time series are output by dynamical systems with observational or dynamical noise or purely random processes. In this paper, we propose a generalized permutation entropy, belonging to the class of group entropies, that is finite in that situation, which is actually the one found in practice. The theoretical results are illustrated numerically by random processes with short- and long-term dependencies, as well as by noisy deterministic signals.

Attention Mechanisms and Their Applications to Complex Systems.

Deep learning models and graphics processing units have completely transformed the field of machine learning. Recurrent neural networks and long short-term memories have been successfully used to model and predict complex systems. However, these classic models do not perform sequential reasoning, a process that guides a task based on perception and memory. In recent years, attention mechanisms have emerged as a promising solution to these problems. In this review, we describe the key aspects of attention mechanisms and some relevant attention techniques and point out why they are a remarkable advance in machine learning. Then, we illustrate some important applications of these techniques in the modeling of complex systems.

Entropy Monotonicity and Superstable Cycles for the Quadratic Family Revisited.

The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor's Monotonicity Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis. Our proof is based on tools and algorithms previously developed by the authors and collaborators to compute the topological entropy of multimodal maps. Specifically, we use the number of transverse intersections of the map iterations with the so-called critical line. The approach is technically simple and geometrical. The same approach is also used to briefly revisit the superstable cycles of the quadratic maps, since both topics are closely related.

Classification and Quantification of Microplastics (<100 µm) Using a Focal Plane Array-Fourier Transform Infrared Imaging System and Machine Learning.

Microplastics are defined as microscopic plastic particles in the range from few micrometers and up to 5 mm. These small particles are classified as primary microplastics when they are manufactured in this size range, whereas secondary microplastics arise from the fragmentation of larger objects. Microplastics are widespread emerging pollutants, and investigations are underway to determine potential harmfulness to biota and human health. However, progress is hindered by the lack of suitable analytical methods for rapid, routine, and unbiased measurements. This work aims to develop an automated analytical method for the characterization of small microplastics (<100 µm) using micro-Fourier transform infrared (µ-FTIR) hyperspectral imaging and machine learning tools. Partial least squares discriminant analysis (PLS-DA) and soft independent modeling of class analogy (SIMCA) models were evaluated, applying different data preprocessing strategies for classification of nine of the most common polymers produced worldwide. The hyperspectral images were also analyzed to quantify particle abundance and size automatically. PLS-DA presented a better analytical performance in comparison with SIMCA models with higher sensitivity, sensibility, and lower misclassification error. PLS-DA was less sensitive to edge effects on spectra and poorly focused regions of particles. The approach was tested on a seabed sediment sample (Roskilde Fjord, Denmark) to demonstrate the method efficiency. The proposed method offers an efficient automated approach for microplastic polymer characterization, abundance numeration, and size distribution with substantial benefits for method standardization.

Fingerprinting of Doppler audio signals from the common carotid artery.

Audio fingerprinting involves extraction of quantitative frequency descriptors that can be used for indexing, search and retrieval of audio signals in sound recognition software. We propose a similar approach with medical ultrasonographic Doppler audio signals. Power Doppler periodograms were generated from 84 ultrasonographic Doppler signals from the common carotid arteries in 22 dogs. Frequency features were extracted from each periodogram and included in a principal component analysis (PCA). From this 10 audio samples were pairwise classified as being either similar or dissimilar. These pairings were compared to a similar classification based on standard quantitative parameters used in medical ultrasound and to classification performed by a panel of listeners. The ranking of sound files according to degree of similarity differed between the frequency and conventional classification methods. The panel of listeners had an 88% agreement with the classification based on quantitative frequency features. These findings were significantly different from the score expected by chance (p < 0.001). The results indicate that the proposed frequency based classification has a perceptual relevance for human listeners and that the method is feasible. Audio fingerprinting of medical Doppler signals is potentially useful for indexing and search for similar and dissimilar audio samples in a dataset.

Data fusion approaches in spectroscopic characterization and classification of PDO wine vinegars.

Spain is one of the major producers of high-quality wine vinegars having three protected designations of origin (a.k.a. PDOs): "Vinagre de Jerez", "Vinagre de Condado de Huelva" and "Vinagre de Montilla-Moriles". Their high prices due to their high quality and their high production costs explain the need for developing an adequate quality control technique and the interest in extensive characterization in order to capture the identity of each denomination. In this framework, methodologies based on non-targeted techniques, such as spectroscopies, are becoming popular in food authentication. Thus, for improving vinegar quality assessment, fusion of data blocks obtained from the same samples but different analytical techniques could be a good strategy, since the quantity and quality of sample knowledge could be enhanced providing new insights into the differentiation of vinegars. Therefore, the aim of this manuscript is the development of a multi-platform methodology and a model able to classify the Spanish wine vinegar PDOs. Sixty-five PDO wine vinegars were analyzed by four spectroscopic techniques: Fourier-transform mid-infrared spectroscopy (MIR), near infrared spectroscopy (NIR), multidimensional fluorescence spectroscopy (EEM) and proton nuclear magnetic resonance (1H-NMR). Two different data fusion strategies were evaluated: Mid-level data fusion with different preprocessing, and Common Component and Specific Weights analysis multiblock method. Exploratory and classification analysis on the data from individual techniques were also performed and compared with data fusion models. The data fusion models improved the classification, providing a more efficient differentiation, than the models based on single methods, and supporting the approach to combine these methods to achieve synergies for an optimized PDO differentiation.

Surrogate Data Preserving All the Properties of Ordinal Patterns up to a Certain Length.

We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the details of the underlying dynamics do not matter beyond the refinements of ordinal patterns finer than a predefined length. The proposed surrogate data help construct a test of determinism that is free from the common linearity assumption for a null-hypothesis.

Detecting directional couplings from multivariate flows by the joint distance distribution.

The identification of directional couplings (or drive-response relationships) in the analysis of interacting nonlinear systems is an important piece of information to understand their dynamics. This task is especially challenging when the analyst's knowledge of the systems reduces virtually to time series of observations. Spurred by the success of Granger causality in econometrics, the study of cause-effect relationships (not to be confounded with statistical correlations) was extended to other fields, thus favoring the introduction of further tools such as transfer entropy. Currently, the research on old and new causality tools along with their pitfalls and applications in ever more general situations is going through a time of much activity. In this paper, we re-examine the method of the joint distance distribution to detect directional couplings between two multivariate flows. This method is based on the forced Takens theorem, and, more specifically, it exploits the existence of a continuous mapping from the reconstructed attractor of the response system to the reconstructed attractor of the driving system, an approach that is increasingly drawing the attention of the data analysts. The numerical results with Lorenz and Rössler oscillators in three different interaction networks (including hidden common drivers) are quite satisfactory, except when phase synchronization sets in. They also show that the method of the joint distance distribution outperforms the lowest dimensional transfer entropy in the cases considered. The robustness of the results to the sampling interval, time series length, observational noise, and metric is analyzed too.

Sampling methods for the study of volatile profile of PDO wine vinegars. A comparison using multivariate data analysis.

High-quality wine vinegars have been registered in Spain under protected designation of origin (PDO): "Vinagre de Jerez", "Vinagre de Condado de Huelva" and "Vinagre de Montilla-Moriles". The raw material, production and aging processes determine their quality and their aromatic composition. Vinegar volatile profile is usually analyzed by gas chromatography-mass spectrometry (GC-MS), being necessary a previous extraction step. Thus, three different sampling methods (Headspace solid phase microextraction "HS-SPME", Headspace stir bar sorptive extraction "HSSE" and Dynamic headspace extraction "DHS") were studied for the analysis of the volatile composition of Spanish PDO wine vinegars. Multivariate curve resolution (MCR) was used to solve chromatographic problems, improving the results obtained. Principal component analysis (PCA) showed that not all the sampling methods were equally suitable for the characterization and differentiation between PDOs and categories, being HSSE the technique that made able the best vinegar characterization.

A Brief Review of Generalized Entropies.

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon-Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

On the limits of probabilistic forecasting in nonlinear time series analysis II: Differential entropy.

In a previous paper, the authors studied the limits of probabilistic prediction in nonlinear time series analysis in a perfect model scenario, i.e., in the ideal case that the uncertainty of an otherwise deterministic model is due to only the finite precision of the observations. The model consisted of the symbolic dynamics of a measure-preserving transformation with respect to a finite partition of the state space, and the quality of the predictions was measured by the so-called ignorance score, which is a conditional entropy. In practice, though, partitions are dispensed with by considering numerical and experimental data to be continuous, which prompts us to trade off in this paper the Shannon entropy for the differential entropy. Despite technical differences, we show that the core of the previous results also hold in this extended scenario for sufficiently high precision. The corresponding imperfect model scenario will be revisited too because it is relevant for the applications. The theoretical part and its application to probabilistic forecasting are illustrated with numerical simulations and a new prediction algorithm.

Mathematical methods in medicine: neuroscience, cardiology and pathology.

The application of mathematics, natural sciences and engineering to medicine is gaining momentum as the mutual benefits of this collaboration become increasingly obvious. This theme issue is intended to highlight the trend in the case of mathematics. Specifically, the scope of this theme issue is to give a general view of the current research in the application of mathematical methods to medicine, as well as to show how mathematics can help in such important aspects as understanding, prediction, treatment and data processing. To this end, three representative specialties have been selected: neuroscience, cardiology and pathology. Concerning the topics, the 12 research papers and one review included in this issue cover biofluids, cardiac and virus dynamics, computational neuroscience, functional magnetic resonance imaging data processing, neural networks, optimization of treatment strategies, time-series analysis and tumour growth. In conclusion, this theme issue contains a collection of fine contributions at the intersection of mathematics and medicine, not as an exercise in applied mathematics but as a multidisciplinary research effort that interests both communities and our society in general.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Characterization and authentication of Spanish PDO wine vinegars using multidimensional fluorescence and chemometrics.

This work assesses the potential of multidimensional fluorescence spectroscopy combined with chemometrics for characterization and authentication of Spanish Protected Designation of Origin (PDO) wine vinegars. Seventy-nine vinegars of different categories (aged and sweet) belonging to the Spanish PDOs "Vinagre de Jerez", "Vinagre de Montilla-Moriles" and "Vinagre de Condado de Huelva", were analyzed by excitation-emission fluorescence spectroscopy. A visual assessment of fluorescence landscapes pointed out different trends with vinegar categories. PARAllel FACtor analysis (PARAFAC) extracted the potential fluorophores and their values in the PDO vinegars. This information, coupled with different classification methods (Partial Least Square Discrimination Analysis "PLS-DA" and Support Vectors Machines "SVM"), was able to discriminate the wine vinegar category within each PDO, for which SVM models obtained better results (>92% of classification). In each category, SVM also allows the differentiation between PDOs. The proposed methodology could be used as an analysis method for the authentication of Spanish PDO wine vinegars.