Exploration of Fractal Dimensions and Measure of Chaos in a Cubic Map

The fractal dimension is defined as a measure of the complexity of the road network. In this study, we calculated five different fractal dimensions for road networks in Jordan. Fractal dimensions of Box-counting, Perimeter area (P-A), Information, Mass, and Ruler were computed by using BENOIT for 12 governorates’ centers in Jordan. The relationships between the Log-Linear functions of calculated fractal dimensions and urban parameters were determined. The results show that the Jordanian road networks have Box-counting dimension (Db), Perimeter area dimension (Dp), Information dimension (Di), Mass dimension (Dm), and Ruler dimension (Dr) between 1 and 2. Box counting dimension (Db) and Information dimension (Di) had strong positive singular linear correlations with the Logs of area, population, and the total number of roads, while Perimeter area dimension (Dp) had moderate negative singular correlations. Mass dimension (Dm) and Ruler dimension (Dr) showed an insignificant correlation with all parameters. This means Box counting dimension (Db), Perimeter area dimension (Dp), and information dimension (Di) yield a good indication of fractal geometry compared with Mass dimension (Dm) and Ruler (Dr). Results confirm that the fractal analysis of the homogeneous network offers a better understanding of the relationship between D and the network complexity as well as the relation between fractal geometry and urban parameters. Also, since no similar works have been carried before in Jordan, the idea and results of this research will help in reviling the fractal characteristics of the study area for the first time.

Aims Taxus chinensis var. mairei is the material from which taxol, an effective low-toxicity cancer-resistance medicine, is distilled. This species reproduces asexually, especially through sprouting from stems, which enables Taxus to expand its spatial occupation and maintain its population. Therefore, study on clonal properties of Taxus populations may provide a scientific basis for protecting and regenerating populations. This study addresses the following questions: 1) what are the fractal properties of population distribution patterns, 2) are clonal architecture and fractal properties correlated and 3) are fractal properties and aggregation correlated? Methods Based on a survey of Taxus, we established four 40 m × 15 m plots in its central distribution area in Yuanbaoshan Nature Reserve. For each plot, we determined the spatial coordinates of each Taxus individual. For Taxus 3 m tall, we measured height, crown size, height to branches and diameter at breast height; for shorter Taxus seedlings and saplings, only height was measured. We also estimated or measured cover and height of shrubs and herbs. Fractal properties of population patterns were analyzed by using the box-counting dimension and the information dimension. Important findings The box-counting dimension ranged from 0.993 1 to 1.353 1, which was far from the theoretical value 2, indicating low degree of spatial occupation. Significant correlation between the clonal architecture index and the box-counting dimension indicates that degree of spatial occupation was closely related to clonal architecture, i.e., populations tending to have phalanx clonal architecture had a stronger degree of spatial occupation than those tending to have guerrilla clonal architecture. The information dimension varied from 1.350 8 to 1.652 1 with community type. The Pearson correlation between information dimension and pattern index is significant, suggesting that differences of information dimension could reflect variation of the aggregation intensity. Information dimension was closely related to number and spatial distribution patterns of ramets.

Optical Coherence Tomography Angiography (OCTA) is an imaging modality that provides threedimensional information of the retinal microvasculature and therefore promises early diagnosis and sufficient monitoring in ophthalmology. However, there is considerable variability between experts analysing this data. Measures for quantitative assessment of the vasculature need to be developed and established, such as fractal dimension. Fractal dimension can be used to assess the complexity of vessels and has been shown to be independently associated with neovascularization, a symptom of diseases such as diabetic retinopathy. This investigation assessed the performance of three fractal dimension algorithms: Box Counting Dimension (BCD), Information Dimension (ID), and Differential Box Counting (DBC). Two of those, BCD and ID, rely on previous vessel segmentation. Assessment of the added value or disturbance regarding the segmentation step is a second aim of this study. The investigation was performed on a data set composed of 9 in vivo human eyes. Since there is no ground truth available, the performance of the methods in differentiating the Superficial Vascular Complex (SVC) and Deep Vascular Complex (DVC) layers apart and the consistency of measurements of the same layer at different time-points were tested. The performance parameters were the ICC and the Mann- Whitney U tests. The three applied methods were suitable to tell the different layers apart and showed consistent values applied in the same slab. Within the consistency test, the non-segmentation-based method, DBC, was found to be less accurate, expressed in a lower ICC value, compared to its segmentation-based counterparts. This result is thought to be due to the DBC’s higher sensitivity when compared to the other methods. This higher sensitivity might help detect changes in the microvasculature, like neovascularization, but is also more likely prone to noise and artefacts.

Fault system is a significant evidence of tectonic movement during crust tectonic evolution and may play an more important role in oil-gas accumulation process than other tectonic types in sedimentary basin. Carboniferous surface faults in Junggar Basin developed well and varied in size and distribution. There are about 200 faults in Carboniferous, and 187 of them are thrust faults. Chaos-fractals theories have been widely investigated and great progress has been made in the past three decades. One of the important conception-fractal dimension had become a powerful tool for describing non-linearity dynamical system characteristic. The clustered objects in nature are often fractal and fault system distribution in space is inhomogeneous, always occurs in groups, so we can describe spatial distribution of faults from the point of fractal dimension. Fractal dimension of fault system is a comprehensive factor associated with fault number, size, combination modes and dynamics mechanism, so it can evaluate the complexity of fault system quantitatively. The relationship between fault system and oil-gas accumulation is a focus and difficulty problem in petroleum geology, and fractal dimension is a new tool for describing fault distribution and predicting potential areas of hydrocarbon resources. Geographic Information System (GIS) is a kind of technological system collecting, storing, managing, computing, analyzing, displaying and describing the geospatial information supported by computer software and hardware. In the last 15-20 years, GIS have been increasingly used to address a wide variety of geoscience problems. Weights-of-evidence models use the theory of conditional probability to quantify spatial association between fractal dimension and oil-gas accumulation. The weights of evidence are combined with the prior probability of occurrence of oil-gas accumulation using Bayes'rule in a loglinear form under an assumption of conditional independence of the dimension maps to derive posterior probability of occurrence of oil-gas accumulation. In this paper, we first vectorize the fault system in Carboniferous of Junggar Basin in GIS software and store it as polyline layer in Geodatabase of GIS to manage and analyze, then calculate the fractal dimension of three types which are box dimension, information dimension and cumulative length dimension using spatial functions of GIS, in the last use weights-of-evidence model to calculate the correlation coefficients in GIS environment between oil-gas accumulation and three types of fractal dimension in order to quantity the importance of fault system.

Band the merits of cognitive ultra-wideband (CUWB) and fractal dimensions together, this paper presented a spectrum sensing method employing the difference of fractal box dimension and information dimension between noise and primary signals through taking double threshold strategy. For the purpose of meeting the need of spectrum sensing in CUWB, this method selects information dimension as the recognition characteristic parameter to recognize signal modulation mode. Simulation results show that the spectrum sensing method based on fractal dimensions is not sensitive to the noise and has a good sensing performance with low computational complexity, and it is able to realize modulation recognition effectively, as a whole, the method is fit for CUWB system.

In the chapter limit properties of genetic algorithms and theproblem of their classification are elaborated. Recently one can observe an increasing interest in properties of genetic algorithms modelled by Markov chains (Vose, Rowe). However, the known results are mainly limited to existence theorems. They say that there exists a limit distribution for a Markov chain describing a simple genetic algorithm. In the chapter we perform the next step on this way and present a formula for this limit distribution for a Markov chain. Moreover, we claim that our convergence theorems can be extended to algorithms which admit the change in the mutation rate and others parameters. The formula for a limit distribution requires some knowledge about the distribution of the fitness function on the whole solution space. However, it suggests the methods to control the algorithm parameters to get better convergence rate. The formula can play an important role in deriving new classification tools for genetic algorithms that use methods of the theory of dynamical systems. That tools will exploit real dynamics of the search and be independent of the taxonomic methods of classification that are used nowadays. On the base of the knowledge of the limit distribution we construct an optimal genetic algorithm in the probabilistic sense. Generally this algorithm is impossible to describe. This is an open problem at the moment, however, its existence and its form suggest an improvement of the original algorithm by changing its parameters. Constructed in this way the optimal genetic algorithm is an answer to one of the questions stayed by famous No Free Lunch Theorem. Moreover, it is a complementary result to this theorem. On the base of this theoretical result we perform a classification of algorithms and show empirical (computational) results in getting which the entropy, fractal dimension, or its approximations: the box-counting dimension or information dimension, are used. One of the most difficult, however, of practical importance, problems is the choice of an algorithm to given optimisation problem. The distinguishing between an optimisation problem and the algorithm and its choice creates to the main difficulty. Consequently, the distinguishing is an artificial operation because it abstains from the idea of genetic algorithm (GA), since the fitness function, arises from the cost function (i.e. the function to be optimised) is the main object of the genetic algorithm and it emerges from the formulation of the optimisation problem and it is difficult O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

By using box dimension, information dimension and correlation dimension, fractal features of Wuyishan Natural and Culture Heritage were studied at a multi-dynamic scale. At the A sampling scale (with 1.474 box dimension, 1.415 information dimension, and 1.048 correlation dimension ), the derived correlation coefficients (R) are 0.994, 0.999 and 0.969. At the B sampling scale, box dimension ranges from B1 to B4, the correlation coefficient ranges from 0.825 ~ 1.200, R values are all above 0.961. The range of information dimension is 0.919 ~ 1.302 with R values above 0.964. Correlation dimension range is 0.822 ~ 1.364 with R values above 0.941. At the C sampling scale, range of box dimension is C1 ~ C16 (excluding plots with spot quantity between 0 ~ 1), the correlation coefficient is 0.175 ~ 0.931 with R values above 0.775. That of information dimension is 0.200 ~ 1.039 with R values above 0.771. Correlation dimension range is 0.506 ~ 1.929 with R values above 0.704. It is thus concluded that: the scenery system of Wuyishan Natural and Culture Heritage has obvious fractal characteristics. Furthermore, the fractal dimensions are sensitive to sampling scale.

The manuscript attempts to explore the periodicity in the distribution of galaxies in the recently reported Saraswati supercluster and the Stripe 82 region containing it as an example. The report of 120 Mpc periodicity in the Abell galaxy clusters by power spectrum analysis is the motivation behind the study. The power spectral analysis across the central part of the Stripe 82 region shows a periodic variation of 3.09° or 71 Mpc in fractal dimension whereas an average angular periodicity of 3.45° or 94 Mpc is observed across the Stripe 82 region. This refers to the periodicity of complexity or cluster density of galaxy distribution. The texture of the distribution pattern understood through lacunarity analysis indicates a near symmetric distribution. Fractal dimensions like box-counting dimension, information dimension and correlation dimension are also found through multifractal analysis. While the information dimension tells about the distribution density of galactic points, the correlation dimension details the distribution of galaxies in the neighbourhood.

In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.

B ENOIT is a fractal analysis software product for Windows 95, Windows 98, or Windows NT used to find order and patterns in seemingly chaotic data, particularly where traditional statistical approaches to data analysis fail. It is widely used in disciplines as diverse as biology, chemistry, physics, economics, medicine, and geology. The U.S. Geological Survey, for example, employs fractal analysis to accurately predict the volume of undiscovered deposits of oil and natural gas, on the basis of data from known deposits. BENOIT measures user-supplied data by standard fractal methods. For a fractal, measures change in value as the scale decreases in size because ever-smaller pieces become included in the analysis. Measures are plotted as a function of ruler size on a log-log plot, and a fractal dimension is calculated from the slope of the resulting line. Users select one of 10 analytical measures with the software. Five of the available measures in the program act upon bitmap images in Windows BMP format. These are described as the “self-similar” or two-dimensional (2D) methods, while the remaining group of routines act upon time-series or 1D data. The latter group requires data to be in a simple but specific data format, such as is available in Excel. The program also features a data generator that produces files with a given fractal dimension. Users may find this useful for testing and control purposes. The self-similar or image methods available in BENOIT measure different characteristics of bitmap objects in ways that should be scale-invariant. A real dataset normally has some fractal limit, and outside the limit, the fractal dimension will return a trivial value (1 for time-series or 2 for image data). Upper and lower fractal limits are controlled by the size of the dataset. Self-similar methods available in BENOIT are well known in fractal analysis: box dimension, perimeter-area dimension, information dimension, and ruler dimension. All methods are explained in standard Help files that contain several pages of information for each topic. The 1D analysis routines use “self-affine” methods of analysis. Self-affine fractals differ from self-similiar fractals in that their parts need to be rescaled by different factors in different coordinates to resemble the original. In the roughness-length method, the root-mean-square variation or roughness of the data is calculated for a variety of horizontal scales. The operation provides an estimate of the Hurst exponent, H , in a log-log plot, which is related to the fractal dimension. Standard self-affine methods available include R/S (Rescaled Range) analysis, power spectrum, roughness-length, variogram, and wavelets. Printing of log-log figures is provided, but without many features that would be found in a spreadsheet. Documentation for the program is available online. BENOIT has a highly visual interface, complete with an animated grid or ruler for self-similar fractal methods, and it gives users control of all calculations that the program performs, unlike other fractal software. Benoit is not without flaws. Some operations, such as name registration with the Windows NT 4.0 taskbar and the Open File requester, do not conform to standard Windows conventions. It would be of help to have an outline or flowchart of the operation of the different routines available in BENOIT for newcomers to fractal analysis. In summary, the variety of fractal analysis methods available in BENOIT, together with generally detailed help files and significant user control of operations, make BENOIT a good resource for learning about and using fractal analysis methods.

The calculation formula of general dimension least square method is presented by applying the multi-fractal theory and the general dimension calculation sample analysis of sine signals is achieved. The general dimensions of measured time domain signal of rolling bearing are calculated and analyzed, the sequence value of general dimension is obtained, the box dimension, the information dimension and the correlation dimension are obtained. The fault diagnosis and classification of random under-check signals are realized by using general dimension correlation coefficient method, the fault pattern recognized accords with the real state. The theory and experiment analysis show that the rolling bearings vibration signals have similar fractal dimensions in the same working condition, whereas, the fractal dimensions are different when the working conditions are different, the difference is distinct. All the fractal dimensions extracted from general dimensions are valuable in fault state diagnosing and recognizing, fulfilled the demands maximally in engineering application.

The current COVID-19 pandemic mainly affects the upper respiratory tract. People with COVID-19 report a wide range of symptoms, some of which are similar to those of common flu, such as sore throat and rhinorrhea. Additionally, COVID-19 shares many clinical symptoms with severe pneumonia, including fever, fatigue, dry cough, and respiratory distress. Several diagnostic strategies, such as the real-time polymerase chain reaction technique and computed tomography imaging, which are more costly than chest radiography, are employed as diagnostic tools. The purpose of this paper is to describe the role of the d-summable information dimension of X-ray images in differentiating several lesions and lung illnesses better than both fractal and information dimensions. The statistical analysis shows that the d-summable information dimension model better describes the information obtained from the X-ray images. Therefore, it is a more precise measure of complexity than the information and box-counting dimension. The results also show that the X-ray images of COVID-19 pneumonia reveal greater damage than those of tuberculosis, pneumonia, and various lung lesions, where the damage is minor or much focused. Because the d-summable information dimension increases as the image complexity decreases, it could pave the way to formulate a new measure to quantify the lung damage and assist the clinical diagnosis based on the area under the d-summable information model. In addition, the physical meaning of the [Formula: see text] parameter in the d-summable information dimension is given.

Objectives: This study aims to evaluate the fractal dimensions (FDs) (box-count dimension and information dimension (Dinf)) and the number of bifurcations (NOB) in retinography of patients with diabetic retinopathy (DR) of grades 2 and 3, both groups without and with risk of diabetic macular edema. Material and Methods: Forty-six retinographies of patients with DR were selected, and classified according to the number of microaneurysms, hemorrhages, and the presence of neovascularization: Grade 2 and grade 3. The images were skeletonized, and then the FDs and the NOB were calculated. Results: The values of box-counting, Dinf, and the NOB of the retinal vascular networks (whole and in the macular region) did not show statistical differences between the grade 2 groups without and with risk of edema, as well as between the grade 3 groups without and with risk of edema. Conclusion: The FD and the NOB revealed that the architecture of the retinal vascular network of individuals with diabetic retinopathy remains unchanged in the presence or absence of risk of macular edema.

Intelligent fault diagnosis of rolling element bearing based on SVMs and fractal dimension

Spatial pattern analysis reveals intra-specific competition and fractal characteristics of Pinus sylvestris var. mongolica populations in Xiaotaojia Gulley Watershed