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Estimated entropies from a limited data set are always biased. Consequently, it is not a trivial task to calculate the entropy in real tasks. In this paper, we used a generalized definition of entropy to evaluate the Hartley, Shannon, and Collision entropies. Moreover, we applied the Miller and Harris estimations of Shannon entropy, which are well known bias approaches based on Taylor series. Finally, these estimates were improved by Bayesian estimation of individual probabilities. These methods were tested and used for recognizing Alzheimer's disease, using the relationship between entropy and the fractal dimension to obtain fractal dimensions of 3D brain scans.

The geometric characteristics of dust clouds provide important information on the physical processes that structure such clouds. One of such characteristics is the 2D fractal dimension D of a cloud projected on to the sky plane. In previous studies, which were mostly based on infrared (IR) data, the fractal dimension of individual clouds was found to be in a range from 1.1 to 1.7 with a preferred value of 1.2-1.4. In this work, we use data from Stripe82 of the Sloan Digital Sky Survey to measure the fractal dimension of the cirrus clouds. This is done here for the first time for optical data with significantly better resolution as compared to IR data. To determine the fractal dimension, the perimeter-area method is employed. We also consider IR (IRAS and Herschel) counterparts of the corresponding optical fields to compare the results between the optical and IR. We find that the averaged fractal dimension across all clouds in the optical is $\langle D \rangle =1.69{+0.05} {-0.05}$ which is significantly larger than the fractal dimension of its IR counterparts $\langle D\rangle =1.38{+0.07} {-0.06}$. We examine several reasons for this discrepancy (choice of masking and minimal contour level, image and angular resolution, etc.) and find that for approximately half of our fields the different angular resolution (point spread function) of the optical and IR data can explain the difference between the corresponding fractal dimensions. For the other half of the fields, the fractal dimensions of the IR and visual data remain inconsistent, which can be associated with physical properties of the clouds, but further physical simulations are required to prove it. © 2021 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society. ; We acknowledge financial support from the Russian Science Foundation (grant no. 20-72-10052). Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah. The SDSS website is www.sdss.org. ; With funding from the Spanish government through the Severo Ochoa Centre of Excellence accreditation SEV-2017-0709. ; Peer reviewed

A hedging strategy is designed to increase the likelihood of desired financial out-comes. Market speculators hedge investment positions if they are worth protecting against potential negative outcomes on the underlying investment. Such negative outcomes cannot be avoided altogether, but effective hedging can reduce impact severity. The investment strategy includes an index held by investors (long position) and uses a fractal dimension indicator to warn when liquidity or sentiment changes are imminent. When the named indicator breaches a certain threshold, a hedging position is taken. This sequence of events triggers the implementation of a hedging strategy by entering a buy put-option position. The daily cumulative returns on using the fractal dimension indicators were 83% more profitable on average when applied to each chosen index respectively.

AbstractIn this article, the risk of epidemic transmission on complex networks is studied from the perspective of effective fractal dimension. First, we introduce the method of calculating the effective fractal dimension of the network by taking a scale‐free network as an example. Second, we propose the construction method of administrative fractal network and calculate the . using the classical susceptible exposed infectious removed (SEIR) infectious disease model, we simulate the virus propagation process on the administrative fractal network. The results show that the larger the is, the higher the risk of virus transmission is. Later, we proposed five parameters P, M, B, F, and D, where P denotes population mobility, M denotes geographical distance, B denotes GDP, F denotes , and D denotes population density. The new epidemic growth index formula was obtained by combining these five parameters, and the validity of I in epidemic transmission risk assessment was demonstrated by parameter sensitivity analysis and reliability analysis. Finally, we also confirmed the reliability of the SEIR dynamic transmission model in simulating early COVID‐19 transmission trends and the ability of timely quarantine measures to effectively control the spread of the epidemic.

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[EN] This work investigates the characterization of bright lesions in retinal fundus images using texture analysis techniques. Exudates and drusen are evidences of retinal damage in diabetic retinopathy (DR) and age-related macular degeneration (AMD) respectively. An automatic detection of pathological tissues could make possible an early detection of these diseases. In this work, fractal analysis is explored in order to discriminate between pathological and healthy retinal texture. After a deep preprocessing step, in which spatial and colour normalization are performed, the fractal dimension is extracted locally by computing the Hurst exponent (H) along different directions. The greyscale image is described by the increments of the fractional Brownian motion model and the H parameter is computed by linear regression in the frequency domain. The ability of fractal dimension to detect pathological tissues is demonstrated using a home-made system, based on fractal analysis and Support Vector Machine, able to achieve around a 70% and 83% of accuracy in E-OPHTHA and DIARETDB1 public databases respectively. In a second experiment, the fractal descriptor is combined with texture information, extracted by the Local Binary Patterns, improving the bright lesion detection. Accuracy, sensitivity and specificity values higher than 89%, 80% and 90% respectively suggest that the method presented in this paper is a robust algorithm for describing retina texture and can be useful in the automatic detection of DR and AMD. ; This paper was supported by the European Union's Horizon 2020 research and innovation programme under the Project GALAHAD [H2020-ICT-2016-2017, 732613]. In addition, this work was partially funded by the Ministerio de Economia y Competitividad of Spain, Project SICAP [DPI2016-77869-C2-1-R]. The work of Adrian Colomer has been supported by the Spanish Government under a FPI Grant [BES-2014-067889]. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this ...

One of the most important issues to be settled in the analysis of time series is determining their variability andidentifying the process of shaping their values. In the classical approach, volatility is most often identified with the variance of growth rates.However, risk can be characterisednot only by the variability, but also by the predictability of the changes which can be evaluatedusing thefractal dimension. The aim of this paper is to presentthe applicability of the fractal dimension estimated by the surface division method tothe assessment ofthe properties of time series. The paper presents a method for determining the fractal dimension, its interpretation, significance tables and an example of its application. Fractal dimension has been used here to describe the properties of the time series of the WIG stockexchange index in 2014–2018 and the time series of the growth rates of the largest listed Polish companiesin 2015–2018. The applied methodmakesit possible toclassify a time series into one of three classesof series: persistent, random or antipersistent. Specific cases showthe differences between the use of standard deviation and fractal dimension for riskassessment. Fractal dimension appears here to be a method for assessing the degree of stability of variations.

This paper compares built-up patterns and the urban form of South European cities using fractal dimensions. Fractal dimensions (D) are estimated in two different ways: (a) using binary images with information only on the built-up and non-built-up areas and (b) using grayscale images that represent the different built-up densities. The Urban Atlas and the Imperviousness-Soil Sealing Degree datasets are used to compute fractal dimensions for the 14 cities in Spain, Portugal, Italy, Greece and the Mediterranean France with a population exceeding one million. The results indicate that differences in urban form are reflected in the fractal dimensions. Fractal dimensions are higher in cities characterized by a relatively continuous and homogeneous sprawl than in cities with elongated urban form or discontinuous development in periurban areas. In Spanish cities urban development is fragmented with clustered and contrasted patterns and this leads to lower fractal dimensions. In Italian and Portuguese cities, development follows relatively homogeneous patterns and D values are significantly higher. Other key findings of the research indicate that: (a) grayscale fractal dimensions are lower than the corresponding binary ones, nonetheless the relative ranking of the cities according to D remains about the same regardless of the method used and (b) fractal dimensions are highly correlated to the average built-up density.

<p>This paper presents the finger knuckle based biometric authentication system using the approaches like structure entropy, GSHP (Gaussian smoothed High pass), GSOD (Gaussian Smoothed Oriented Directives) and also the well known method for surface roughness measurement called the fractal profiles represented by Topothesy and fractal dimension which describe not only the roughness but also the affine self similarity. We have also implemented Daisy descriptor for the representation of texture. The results of fractal parameters along with the refined scores are comparable to those of the compcode and impcompcode.</p>