When conducting an analysis of nature’s time series, such as meteorological ones, an important matter is a long-range dependence to quantify the global behavior of the series and connect it with other physical characteristics of the region of study. In this paper, we applied the Higuchi fractal dimension and the Hurst exponent (rescaled range) to quantify the relative trend underlying the time series of historical data from 17 of the 34 weather stations located in the Río Bravo-San Juan Basin, Mexico; these data were provided by the National Water Commission CONAGUA) in Mexico. In this way, this work aims to perform a comparative study about the level of persistency obtained by using the Higuchi fractal dimension and Hurst exponent for each station of the basin. The comparison is supported by a climate clustering of the stations, according to the Köppen classification. Results showed a better fitting between the climate of each station and its Higuchi fractal dimension obtained than when using the Hurst exponent. In fact, we found that the more the aridity of the zone the more the persistency of rainfall, according to Higuchi’s values. In turn, we found more relation between the Hurst exponent and the accumulated amount of rainfall. These are relations between the climate and the long-term persistency of rainfall in the basin that could help to better understand and complete the climatological models of the study region. Trends between the fractal exponents used and the accumulated annual rainfall were also analyzed.

Variogram models are a valuable tool used to analyze the variability of a time series; such variability usually entails a spherical or exponential behavior, and so, models based on such functions are commonly used to fit and explain a time series. Variograms have a quasi-periodic structure for rainfall cases, and some extra steps are required to analyze their entire behavior. In this work, we detailed a procedure for a complete analysis of rainfall time series, from the construction of the experimental variogram to curve fitting with well-known spherical and exponential models, and finally proposed a novel model: quadratic–exponential. Our model was developed based on the analysis of 6 out of 30 rainfall stations from our case study: the Río Bravo–San Juan basin, and was constructed from the exponential model while introducing a quadratic behavior near to the origin and taking into account the fact that the maximal variability of the process is known. Considering a sample with diverse Hurst exponents, the stations were selected. The results obtained show robustness in our proposed model, reaching a good fit with and without the nugget effect for different Hurst exponents. This contrasts to previous models, which show good outcomes only without the nugget effect.

Air-quality monitoring and analysis are initial parts of a comprehensive strategy to prevent air pollution in cities. In such a context, statistical tools play an important role in determining the time-series trends, locating areas with high pollutant concentrations, and building predictive models. In this work, we analyzed the spatio-temporal behavior of the pollutant PM₁₀ in the Monterrey Metropolitan Area (MMA), Mexico during the period 2010–2018 by applying statistical analysis to the time series of seven environmental stations. First, we used experimental variograms and scientific visualization to determine the general trends and variability in time. Then, fractal exponents (the Hurst rescaled range and Higuchi algorithm) were used to analyze the long-term dependence of the time series and characterize the study area by correlating that dependence with the geographical parameters of each environmental station. The results suggest a linear decrease in PM₁₀ concentration, which showed an annual cyclicity. The autumn-winter period was the most polluted and the spring-summer period was the least. Furthermore, it was found that the highest average concentrations are located in the western and high-altitude zones of the MMA, and that average concentration is related in a quadratic way to the Hurst and Higuchi exponents, which in turn are related to some geographic parameters. Therefore, in addition to the results for the MMA, the present paper shows three practical statistical methods for analyzing the spatio-temporal behavior of air quality.

Objective. Kinesthetic Motor Imagery (KMI) represents a robust brain paradigm intended for electroencephalography (EEG)-based commands in brain-computer interfaces (BCIs). However, ensuring high accuracy in multi-command execution remains challenging, with data from C3 and C4 electrodes reaching up to 92% accuracy. This paper aims to characterize and classify EEG-based KMI of multilevel muscle contraction without relying on primary motor cortex signals. Approach. A new method based on Hurst exponents is introduced to characterize EEG signals of multilevel KMI of muscle contraction from electrodes placed on the premotor, dorsolateral prefrontal, and inferior parietal cortices. EEG signals were recorded during a hand-grip task at four levels of muscle contraction (0%, 10%, 40%, and 70% of the maximal isometric voluntary contraction). The task was executed under two conditions: first, physically, to train subjects in achieving muscle contraction at each level, followed by mental imagery under the KMI paradigm for each contraction level. EMG signals were recorded in both conditions to correlate muscle contraction execution, whether correct or null accurately. Independent component analysis (ICA) maps EEG signals from the sensor to the source space for preprocessing. For characterization, three algorithms based on Hurst exponents were used: the original (HO), using partitions (HRS), and applying semivariogram (HV). Finally, seven classifiers were used: Bayes network (BN), naive Bayes (NB), support vector machine (SVM), random forest (RF), random tree (RT), multilayer perceptron (MP), and k-nearest neighbors (kNN). Main results. A combination of the three Hurst characterization algorithms produced the highest average accuracy of 96.42% from kNN, followed by MP (92.85%), SVM (92.85%), NB (91.07%), RF (91.07%), BN (91.07%), and RT (80.35%). of 96.42% for kNN. Significance. Results show the feasibility of KMI multilevel muscle contraction detection and, thus, the viability of non-binary EEG-based BCI applications without using signals from the motor cortex.

Kinesthetic Motor Imagery (MKI) has been demonstrated to be a robust paradigm for Brain-Computer Interfaces (BCI). In this paper we present the characterization of KMI paradigm of three tasks of wrist and forearm of the right arm using Hurst exponent and variogram, preceding for ICA to map signals into source space. The results show high persistency an average of 0.76 ± 0.07 for KMI Pronation/Supination (PS), 0.82 ± 0.05 for KMI Flexion-Extension). (FE), and 0.90 ± 0.02 for KMI Abduction-Adduction (AA We found a significant difference between the three KMI tasks, useful for multimodal command in BCI.