Quantifying Soil Complexity Using Fisher Shannon Method on 3D X-ray Computed Tomography Scans.

The conversion of native forest into agricultural land, which is common in many parts of the world, poses important questions regarding soil degradation, demanding further efforts to better understand the effect of land use change on soil functions. With the advent of 3D computed tomography techniques and computing power, new methods are becoming available to address this question. In this direction, in the current work we implement a modification of the Fisher-Shannon method, borrowed from information theory, to quantify the complexity of twelve 3D CT soil samples from a sugarcane plantation and twelve samples from a nearby native Atlantic forest in northeastern Brazil. The distinction found between the samples from the sugar plantation and the Atlantic forest site is quite pronounced. The results at the level of 91.7% accuracy were obtained considering the complexity in the Fisher-Shannon plane. Atlantic forest samples are found to be generally more complex than those from the sugar plantation.

Hydrological changes caused by the construction of dams and reservoirs: The CECP analysis.

We investigated the influence of the construction of cascade dams and reservoirs on the predictability and complexity of the streamflow of the São Francisco River, Brazil, by using complexity entropy causality plane (CECP) in its standard and weighted form. We analyzed daily streamflow time series recorded in three fluviometric stations: São Francisco (upstream of cascade dams), Juazeiro (downstream of Sobradinho dam), and Pão de Açúcar station (downstream of Sobradinho and Xingó dams). By comparing the values of CECP information quantifiers (permutation entropy and statistical complexity) for the periods before and after the construction of Sobradinho (1979) and Xingó (1994) dams, we found that the reservoirs' operations changed the temporal variability of streamflow series toward the less predictable regime as indicated by higher entropy (lower complexity) values. Weighted CECP provides some finer details in the predictability of streamflow due to the inclusion of amplitude information in the probability distribution of ordinal patterns. The time evolution of streamflow predictability was analyzed by applying CECP in 2 year sliding windows that revealed the influence of the Paulo Alfonso complex (located between Sobradinho and Xingó dams), construction of which started in the 1950s and was identified through the increased streamflow entropy in the downstream Pão de Açúcar station. The other streamflow alteration unrelated to the construction of the two largest dams was identified in the upstream unimpacted São Francisco station, as an increase in the entropy around 1960s, indicating that some natural factors could also play a role in the decreased predictability of streamflow dynamics.

Generalized weighted permutation entropy.

A novel heuristic approach is proposed here for time series data analysis, dubbed Generalized weighted permutation entropy, which amalgamates and generalizes beyond their original scope two well established data analysis methods: Permutation entropy and Weighted permutation entropy. The method introduces a scaling parameter to discern the disorder and complexity of ordinal patterns with small and large fluctuations. Using this scaling parameter, the complexity-entropy causality plane is generalized to the complexity-entropy-scale causality box. Simulations conducted on synthetic series generated by stochastic, chaotic, and random processes, as well as real world data, are shown to produce unique signatures in this three dimensional representation.

A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect.

Stock markets can become inefficient due to calendar anomalies known as the day-of-the-week effect. Calendar anomalies are well known in the financial literature, but the phenomena remain to be explored in econophysics. This paper uses multifractal analysis to evaluate if the temporal dynamics of market returns also exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) to the daily returns of market indices worldwide for each day of the week. Our results indicate that distinct multifractal properties characterize individual days of the week. Monday returns tend to exhibit more persistent behavior and richer multifractal structures than other day-resolved returns. Shuffling the series reveals that multifractality arises from a broad probability density function and long-term correlations. The time-dependent multifractal analysis shows that the Monday returns' multifractal spectra are much wider than those of other days. This behavior is especially persistent during financial crises. The presence of day-of-the-week effects in multifractal dynamics of market returns motivates further research on calendar anomalies for distinct market regimes.

Best-fit probability distribution models for monthly rainfall of Northeastern Brazil.

The analysis of precipitation data is extremely important for strategic planning and decision-making in various natural systems, as well as in planning and preparing for a drought period. The drought is responsible for several impacts on the economy of Northeast Brazil (NEB), mainly in the agricultural and livestock sectors. This study analyzed the fit of 2-parameter distributions gamma (GAM), log-normal (LNORM), Weibull (WEI), generalized Pareto (GP), Gumbel (GUM) and normal (NORM) to monthly precipitation data from 293 rainfall stations across NEB, in the period 1988-2017. The maximum likelihood (ML) method was used to estimate the parameters to fit the models and the selection of the model was based on a modification of the Shapiro-Wilk statistic. The results showed the chosen 2-parameter distributions to be flexible enough to describe the studied monthly precipitation data. The GAM and WEI models showed the overall best fits, but the LNORM and GP models gave the best fits in certain months of the year and regions that differed from the others in terms of their average precipitation.

The Choice of an Appropriate Information Dissimilarity Measure for Hierarchical Clustering of River Streamflow Time Series, Based on Calculated Lyapunov Exponent and Kolmogorov Measures.

The purpose of this paper was to choose an appropriate information dissimilarity measure for hierarchical clustering of daily streamflow discharge data, from twelve gauging stations on the Brazos River in Texas (USA), for the period 1989-2016. For that purpose, we selected and compared the average-linkage clustering hierarchical algorithm based on the compression-based dissimilarity measure (NCD), permutation distribution dissimilarity measure (PDDM), and Kolmogorov distance (KD). The algorithm was also compared with K-means clustering based on Kolmogorov complexity (KC), the highest value of Kolmogorov complexity spectrum (KCM), and the largest Lyapunov exponent (LLE). Using a dissimilarity matrix based on NCD, PDDM, and KD for daily streamflow, the agglomerative average-linkage hierarchical algorithm was applied. The key findings of this study are that: (i) The KD clustering algorithm is the most suitable among others; (ii) ANOVA analysis shows that there exist highly significant differences between mean values of four clusters, confirming that the choice of the number of clusters was suitably done; and (iii) from the clustering we found that the predictability of streamflow data of the Brazos River given by the Lyapunov time (LT), corrected for randomness by Kolmogorov time (KT) in days, lies in the interval from two to five days.

Three-dimensional multifractal analysis of trabecular bone under clinical computed tomography.

PURPOSE: An adequate understanding of bone structural properties is critical for predicting fragility conditions caused by diseases such as osteoporosis, and in gauging the success of fracture prevention treatments. In this work we aim to develop multiresolution image analysis techniques to extrapolate high-resolution images predictive power to images taken in clinical conditions. METHODS: We performed multifractal analysis (MFA) on a set of 17 ex vivo human vertebrae clinical CT scans. The vertebrae failure loads (FFailure) were experimentally measured. We combined bone mineral density (BMD) with different multifractal dimensions, and BMD with multiresolution statistics (e.g., skewness, kurtosis) of MFA curves, to obtain linear models to predict FFailure. Furthermore we obtained short- and long-term precisions from simulated in vivo scans, using a clinical CT scanner. Ground-truth data - high-resolution images - were obtained with a High-Resolution Peripheral Quantitative Computed Tomography (HRpQCT) scanner. RESULTS: At the same level of detail, BMD combined with traditional multifractal descriptors (Lipschitz-Hölder exponents), and BMD with monofractal features showed similar prediction powers in predicting FFailure (87%, adj. R2 ). However, at different levels of details, the prediction power of BMD with multifractal features raises to 92% (adj. R2) of FFailure. Our main finding is that a simpler but slightly less accurate model, combining BMD and the skewness of the resulting multifractal curves, predicts 90% (adj. R2) of FFailure. CONCLUSIONS: Compared to monofractal and standard bone measures, multifractal analysis captured key insights in the conditions leading to FFailure. Instead of raw multifractal descriptors, the statistics of multifractal curves can be used in several other contexts, facilitating further research.

Multifractal spectrum and lacunarity as measures of complexity of osseointegration.

OBJECTIVES: The goal of this study is to contribute to a better quantitative description of the early stages of osseointegration, by application of fractal, multifractal, and lacunarity analysis. MATERIALS AND METHODS: Fractal, multifractal, and lacunarity analysis are performed on scanning electron microscopy (SEM) images of titanium implants that were first subjected to different treatment combinations of i) sand blasting, ii) acid etching, and iii) exposition to calcium phosphate, and were then submersed in a simulated body fluid (SBF) for 30 days. All the three numerical techniques are applied to the implant SEM images before and after SBF immersion, in order to provide a comprehensive set of common quantitative descriptors. RESULTS: It is found that implants subjected to different physicochemical treatments before submersion in SBF exhibit a rather similar level of complexity, while the great variety of crystal forms after SBF submersion reveals rather different quantitative measures (reflecting complexity), for different treatments. In particular, it is found that acid treatment, in most combinations with the other considered treatments, leads to a higher fractal dimension (more uniform distribution of crystals), lower lacunarity (lesser variation in gap sizes), and narrowing of the multifractal spectrum (smaller fluctuations on different scales). CONCLUSION: The current quantitative description has shown the capacity to capture the main features of complex images of implant surfaces, for several different treatments. Such quantitative description should provide a fundamental tool for future large scale systematic studies, considering the large variety of possible implant treatments and their combinations. CLINICAL RELEVANCE: Quantitative description of early stages of osseointegration on titanium implants with different treatments should help develop a better understanding of this phenomenon, in general, and provide basis for further systematic experimental studies. Clinical practice should benefit from such studies in the long term, by more ready access to implants of higher quality.

Fractal Measure and Microscopic Modeling of Osseointegration.

In this study, the process of osseointegration on titanium implant surfaces with different physicochemical treatments subjected to a simulated corporal fluid submersion was evaluated using the concept of fractal dimension. It was found that different treatments led to rather different calcium phosphate crystal growth patterns, with fractal dimension ranging from 1.68 to 1.93. The observed crystal patterns may be explained by a general deposition, diffusion, and aggregation growth mechanism, where diffusing particle sticking probability plays a fundamental role.

Multifractal properties of a closed contour: a peek beyond the shape analysis.

In recent decades multifractal analysis has been successfully applied to characterize the complex temporal and spatial organization of such diverse natural phenomena as heartbeat dynamics, the dendritic shape of neurons, retinal vessels, rock fractures, and intricately shaped volcanic ash particles. The characterization of multifractal properties of closed contours has remained elusive because applying traditional methods to their quasi-one-dimensional nature yields ambiguous answers. Here we show that multifractal analysis can reveal meaningful and sometimes unexpected information about natural structures with a perimeter well-defined by a closed contour. To this end, we demonstrate how to apply multifractal detrended fluctuation analysis, originally developed for the analysis of time series, to an arbitrary shape of a given study object. In particular, we show the application of the method to fish otoliths, calcareous concretions located in fish's inner ear. Frequently referred to as the fish's "black box", they contain a wealth of information about the fish's life history and thus have recently attracted increasing attention. As an illustrative example, we show that a multifractal approach can uncover unexpected relationships between otolith contours and size and age of fish at maturity.

Multifractal analysis of human retinal vessels.

In this paper, it is shown that vascular structures of the human retina represent geometrical multifractals, characterized by a hierarchy of exponents rather then a single fractal dimension. A number of retinal images from the STARE database are analyzed, corresponding to both normal and pathological states of the retina. In all studied cases, a clearly multifractal behavior is observed, where capacity dimension is always found to be larger then the information dimension, which is in turn always larger then the correlation dimension, all the three being significantly lower then the diffusion limited aggregation (DLA) fractal dimension. We also observe a tendency of images corresponding to the pathological states of the retina to have lower generalized dimensions and a shifted spectrum range, in comparison with the normal cases.