Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution.

Probabilistic models with flexible tail behavior have important applications in engineering and earth science. We introduce a nonlinear normalizing transformation and its inverse based on the deformed lognormal and exponential functions proposed by Kaniadakis. The deformed exponential transform can be used to generate skewed data from normal variates. We apply this transform to a censored autoregressive model for the generation of precipitation time series. We also highlight the connection between the heavy-tailed κ-Weibull distribution and weakest-link scaling theory, which makes the κ-Weibull suitable for modeling the mechanical strength distribution of materials. Finally, we introduce the κ-lognormal probability distribution and calculate the generalized (power) mean of κ-lognormal variables. The κ-lognormal distribution is a suitable candidate for the permeability of random porous media. In summary, the κ-deformations allow for the modification of tails of classical distribution models (e.g., Weibull, lognormal), thus enabling new directions of research in the analysis of spatiotemporal data with skewed distributions.

Spatial Modeling of Precipitation Based on Data-Driven Warping of Gaussian Processes.

Modeling and forecasting spatiotemporal patterns of precipitation is crucial for managing water resources and mitigating water-related hazards. Globally valid spatiotemporal models of precipitation are not available. This is due to the intermittent nature, non-Gaussian distribution, and complex geographical dependence of precipitation processes. Herein we propose a data-driven model of precipitation amount which employs a novel, data-driven (non-parametric) implementation of warped Gaussian processes. We investigate the proposed warped Gaussian process regression (wGPR) using (i) a synthetic test function contaminated with non-Gaussian noise and (ii) a reanalysis dataset of monthly precipitation from the Mediterranean island of Crete. Cross-validation analysis is used to establish the advantages of non-parametric warping for the interpolation of incomplete data. We conclude that wGPR equipped with the proposed data-driven warping provides enhanced flexibility and-at least for the cases studied- improved predictive accuracy for non-Gaussian data.

Ising Model for Interpolation of Spatial Data on Regular Grids.

We apply the Ising model with nearest-neighbor correlations (INNC) in the problem of interpolation of spatially correlated data on regular grids. The correlations are captured by short-range interactions between "Ising spins". The INNC algorithm can be used with label data (classification) as well as discrete and continuous real-valued data (regression). In the regression problem, INNC approximates continuous variables by means of a user-specified number of classes. INNC predicts the class identity at unmeasured points by using the Monte Carlo simulation conditioned on the observed data (partial sample). The algorithm locally respects the sample values and globally aims to minimize the deviation between an energy measure of the partial sample and that of the entire grid. INNC is non-parametric and, thus, is suitable for non-Gaussian data. The method is found to be very competitive with respect to interpolation accuracy and computational efficiency compared to some standard methods. Thus, this method provides a useful tool for filling gaps in gridded data such as satellite images.

Geostatistical analysis of precipitation in the island of Crete (Greece) based on a sparse monitoring network.

Based on the predictions of General Circulation Models, significant reduction of precipitation in Mediterranean areas is a possible scenario. Hence, better understanding of the spatial and temporal precipitation patterns is necessary in order to quantify desertification risks and design suitable mitigation measures. This study uses monthly precipitation measurements from a sparse network of 54 monitoring stations on the Mediterranean island of Crete (Greece). The study period extends from 1948 to 2012. The data reveal strong correlations between the western and eastern parts of the island. However, the average annual precipitation in the West is about 450 mm higher than that in the East. We construct a spatial model of average annual precipitation in Crete. The model involves a topographic trend and residuals with anisotropic spatial correlations which are fitted with a recently developed variogram function. We use regression kriging to generate annual precipitation maps and to identify locations of high estimation uncertainty. To our knowledge, this is the most detailed spatial analysis of precipitation in Crete to date. We present the analysis in detail for the year 1971. The trend accounts for ≈ 74% of the total variance. The highest precipitation estimate is 2331 mm in the West and 1781 mm in the East. The highest relative estimation uncertainty (≈ 20%) is observed along the southeastern coastline of the island, where the lowest values of annual precipitation are observed. This region includes one of the major agricultural areas of the island. The same overall patterns are found for other years in the study. Finally, we find no statistical evidence for a decrease in the global (over the entire island) annual precipitation during the study period.

Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle.

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in G. Kaniadakis, Physica A 296, 405 (2001), univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form.

Deep Learning Recurrent Neural Network for Concussion Classification in Adolescents Using Raw Electroencephalography Signals: Toward a Minimal Number of Sensors.

Artificial neural networks (ANNs) are showing increasing promise as decision support tools in medicine and particularly in neuroscience and neuroimaging. Recently, there has been increasing work on using neural networks to classify individuals with concussion using electroencephalography (EEG) data. However, to date the need for research grade equipment has limited the applications to clinical environments. We recently developed a deep learning long short-term memory (LSTM) based recurrent neural network to classify concussion using raw, resting state data using 64 EEG channels and achieved high accuracy in classifying concussion. Here, we report on our efforts to develop a clinically practical system using a minimal subset of EEG sensors. EEG data from 23 athletes who had suffered a sport-related concussion and 35 non-concussed, control athletes were used for this study. We tested and ranked each of the original 64 channels based on its contribution toward the concussion classification performed by the original LSTM network. The top scoring channels were used to train and test a network with the same architecture as the previously trained network. We found that with only six of the top scoring channels the classifier identified concussions with an accuracy of 94%. These results show that it is possible to classify concussion using raw, resting state data from a small number of EEG sensors, constituting a first step toward developing portable, easy to use EEG systems that can be used in a clinical setting.

Recurrent neural network-based acute concussion classifier using raw resting state EEG data.

Concussion is a global health concern. Despite its high prevalence, a sound understanding of the mechanisms underlying this type of diffuse brain injury remains elusive. It is, however, well established that concussions cause significant functional deficits; that children and youths are disproportionately affected and have longer recovery time than adults; and that individuals suffering from a concussion are more prone to experience additional concussions, with each successive injury increasing the risk of long term neurological and mental health complications. Currently, the most significant challenge in concussion management is the lack of objective, clinically- accepted, brain-based approaches for determining whether an athlete has suffered a concussion. Here, we report on our efforts to address this challenge. Specifically, we introduce a deep learning long short-term memory (LSTM)-based recurrent neural network that is able to distinguish between non-concussed and acute post-concussed adolescent athletes using only short (i.e. 90 s long) samples of resting state EEG data as input. The athletes were neither required to perform a specific task nor expected to respond to a stimulus during data collection. The acquired EEG data were neither filtered, cleaned of artefacts, nor subjected to explicit feature extraction. The LSTM network was trained and validated using data from 27 male, adolescent athletes with sports related concussion, benchmarked against 35 non-concussed adolescent athletes. During rigorous testing, the classifier consistently identified concussions with an accuracy of > 90% and achieved an ensemble median Area Under the Receiver Operating Characteristic Curve (ROC/AUC) equal to 0.971. This is the first instance of a high-performing classifier that relies only on easy-to-acquire resting state, raw EEG data. Our concussion classifier represents a promising first step towards the development of an easy-to-use, objective, brain-based, automatic classification of concussion at an individual level.

Estimation of the uncertainty of hydrologic predictions in a karstic Mediterranean watershed.

The Koiliaris River basin is a semi-arid Mediterranean karstic watershed where water needs during the summer are exclusively covered by the karstic springs flow. Uncertainty assessment of the hydrologic projections for karstic watersheds may reveal possible water deficits that cannot otherwise be taken into account. The Soil Water Assessment Tool (SWAT) along with a karstic model (Karst-SWAT) is used to assess the composite spring and surface flow. The parameter uncertainty of both the surface and karstic flow models is estimated by combining the SUFI2 interface and the @RISK by PALISADE software. Eleven combinations of five Regional Climate Models (RCMs) and three Representative Concentration Pathways (RCPs) provide input to the hydrologic models. Representative rainfall time series for certain scenarios are stochastically modeled with the LARS weather generator. Monte Carlo simulations are used to investigate the effect of input internal variability on the flow output. The uncertainty of karstic flow due to the parameter uncertainty of the SWAT and Karst-SWAT models is 10.0% (Coefficient of Variation), which is comparable to the estimated uncertainty due to climate change scenarios (10.1%) until 2059. The combined uncertainty for the total flow at the basin exit due to both models' parameter uncertainty is 6.6%, comparable to the uncertainty due to the internal variability (5.6%). The total uncertainty of karstic flow, combining model parameter uncertainty and the internal variability of the climate scenarios is 11.0%. The total uncertainty estimate is used in conjunction with the lowest karstic flow projection to assess the most adverse scenario for the future mean annual karstic flow. This is the first study which estimates the combined uncertainty of surface and karstic flow prediction due to model parameter uncertainty and internal variability. Our study provides a rigorous methodology for uncertainty estimation and analysis which is transferable to other karstic regions of the world.

The κ-statistics approach to epidemiology.

A great variety of complex physical, natural and artificial systems are governed by statistical distributions, which often follow a standard exponential function in the bulk, while their tail obeys the Pareto power law. The recently introduced [Formula: see text]-statistics framework predicts distribution functions with this feature. A growing number of applications in different fields of investigation are beginning to prove the relevance and effectiveness of [Formula: see text]-statistics in fitting empirical data. In this paper, we use [Formula: see text]-statistics to formulate a statistical approach for epidemiological analysis. We validate the theoretical results by fitting the derived [Formula: see text]-Weibull distributions with data from the plague pandemic of 1417 in Florence as well as data from the COVID-19 pandemic in China over the entire cycle that concludes in April 16, 2020. As further validation of the proposed approach we present a more systematic analysis of COVID-19 data from countries such as Germany, Italy, Spain and United Kingdom, obtaining very good agreement between theoretical predictions and empirical observations. For these countries we also study the entire first cycle of the pandemic which extends until the end of July 2020. The fact that both the data of the Florence plague and those of the Covid-19 pandemic are successfully described by the same theoretical model, even though the two events are caused by different diseases and they are separated by more than 600 years, is evidence that the [Formula: see text]-Weibull model has universal features.

Classification of missing values in spatial data using spin models.

A problem of current interest is the estimation of spatially distributed processes at locations where measurements are missing. Linear interpolation methods rely on the Gaussian assumption, which is often unrealistic in practice, or normalizing transformations, which are successful only for mild deviations from the Gaussian behavior. We propose to address the problem of missing value estimation on two-dimensional grids by means of spatial classification methods based on spin (Ising, Potts, and clock) models. The "spin" variables provide an interval discretization of the process values, and the spatial correlations are captured in terms of interactions between the spins. The spins at the unmeasured locations are classified by means of the "energy matching" principle: the correlation energy of the entire grid (including prediction sites) is estimated from the sample-based correlations. We investigate the performance of the spin classifiers in terms of computational speed, misclassification rate, class histogram, and spatial correlations reproduction using simulated realizations of spatial random fields, real rainfall data, and a digital test image. We also compare the spin-based methods with standard classifiers such as the k -nearest neighbor, the fuzzy k -nearest neighbor, and the support vector machine. We find that the spin-based classifiers provide competitive choices.

Disrupted Information Flow in Resting-State in Adolescents With Sports Related Concussion.

Children and youths are at a greater risk of concussions than adults, and once injured, take longer to recover. A key feature of concussion is an increase in functional connectivity, yet it remains unclear how changes in functional connectivity relate to the patterns of information flow within resting state networks following concussion and how these relate to brain function. We applied a data-driven measure of directed effective brain connectivity to compare the patterns of information flow in healthy adolescents and adolescents with subacute concussion during the resting state condition. Data from 32 healthy adolescents (mean age =16 years) and 21 concussed adolescents (mean age = 15 years) within 1 week of injury were included in the study. Five minutes of resting state data EEG were collected while participants sat quietly with their eyes closed. We applied the information flow rate to measure the transfer of information between the EEG time series of each individual at different source locations, and therefore between different brain regions. Based on the ensemble means of the magnitude of normalized information flow rate, our analysis shows that the dominant nexus of information flow in healthy adolescents is primarily left lateralized and anterior-centric, characterized by strong bidirectional information exchange between the frontal regions, and between the frontal and the central/temporal regions. In contrast, adolescents with concussion show distinct differences in information flow marked by a more left-right symmetrical, albeit still primarily anterior-centric, pattern of connections, diminished activity along the central-parietal midline axis, and the emergence of inter-hemispheric connections between the left and right frontal and the left and right temporal regions of the brain. We also find that the statistical distribution of the normalized information flow rates in each group (control and concussed) is significantly different. This paper is the first to describe the characteristics of the source space information flow and the effective connectivity patterns between brain regions in healthy adolescents in juxtaposition with the altered spatial pattern of information flow in adolescents with concussion, statistically quantifying the differences in the distribution of the information flow rate between the two populations. We hypothesize that the observed changes in information flow in the concussed group indicate functional reorganization of resting state networks in response to brain injury.

Finite-size effects on return interval distributions for weakest-link-scaling systems.

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.