Fractal Information Dimension Research Articles

An information fractal dimensional relative entropy

Shannon entropy is used to measure information uncertainty, while the information dimension is used to measure information complexity. Given two probability distributions, the difference can be measured by relative entropy. However, the existing relative entropy does not consider the effect of information dimension. To improve the existing entropy, a new relative entropy is presented in this paper. The information fractal dimension is considered in the proposed relative entropy. The new relative entropy is more generalized than the initial relative entropy. When dimension is not considered, it will degenerate to the initial relative entropy. Another interesting point is that the new relative entropy may have negative values when calculating. The physical meaning is still under exploration. Finally, some application examples are provided to exemplify the utilization of the proposed relative entropy.

Linearity in Deng entropy

Linearity is considered as a fundamental property of significant importance in numerous domains. A linear mathematical expression is clear, concise, and convenient for solving problems, making it a valuable tool for approximating complex issues. In recent years, Deng entropy has been proposed as a generalization of Shannon entropy, applied to measure the uncertainty degree of the mass function in the power set. There are two main contributions in this paper. One is that the linearity can be observed in Deng entropy. We present a set of specific mass functions named power assigned mass function (PAMF), which could generate linear type Deng entropy (LTDE). The other is that we find the slope is nothing else but the information fractal dimension of mass function. Moreover, we discover that for any given slope within the range of 0 to ln3ln2, at least one mass function that yields Deng entropy corresponding to the given slope can be derived through the mass function generator (MFG), in a strict linear way. Some proofs, numerical examples, discussions and an error analysis are provided to validate the effectiveness of our findings.

Information fractal dimension of Random Permutation Set

Random permutation set (RPS) is a recently introduced set based on the Dempster–Shafer evidence theory, which considers all possible permutations of the elements within a given set. The information dimension is a significant fractal dimension which plays a vital role in the information theory. Nevertheless, how to develop the information dimension of a specific permutation mass function in RPS remains an unresolved problem. To solve this problem, we propose a new dimension named information fractal dimension of Random Permutation Set. Moreover, several properties of the proposed dimension are explored and numerical examples are provided to illustrate its effectiveness. The research discovers an interesting property related to the permutation mass function corresponding to the maximum RPS entropy: its information dimension is 2, which is equivalent to the fractal dimension of Brownian motion and Peano curve.

Exponential information fractal dimension weighted risk priority number method for failure mode and effects analysis

Exponential information fractal dimension weighted risk priority number method for failure mode and effects analysis

Modeling Belief Propensity Degree: Measures of Evenness and Diversity of Belief Functions

Based on Klir’s framework of uncertainty, the total uncertainty (also called ambiguity) of belief function is linear addition of discord and nonspecificity. Though uncertainty measures of belief function have been discussed widely, there is no measure that can satisfy the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">monotonicity</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">range consistency</i> properties at the same time. In this article, we discuss uncertainty measure of belief function from the perspective of information fractal dimension. An uncertainty quantity called evenness and its measure <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\rm Eve}$</tex-math> </inline-formula> are proposed, which can represent the belief propensity degree of belief function. We first propose the measures of diversity (normalized nonspecificity) and the element evenness (normalized discord), and then fuse them to calculate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\rm Eve}$</tex-math> </inline-formula> . The proposed method can not only measure the subnormal mass function but also interpret the different views of Klir and Smets on “Uncertainty.” In addition, we extend Klir’s framework of uncertainty based on the proposed information quantities.

INFORMATION FRACTAL DIMENSION OF MASS FUNCTION

Fractals play an important role in nonlinear science. The most important parameter when modeling a fractal is the fractal dimension. Existing information dimension can calculate the dimension of probability distribution. However, calculating the fractal dimension given a mass function, which is the generalization of probability, is still an open problem of immense interest. The main contribution of this work is to propose an information fractal dimension of mass function. Numerical examples are given to show the effectiveness of our proposed dimension. We discover an important property in that the dimension of mass function with the maximum Deng entropy is [Formula: see text], which is the well-known fractal dimension of Sierpiski triangle. The application in complexity analysis of time series illustrates the effectiveness of our method.

INFORMATION DIMENSION OF GALTON BOARD

In this paper, we relax the definition of Rényi information dimension. The power law of the Entropy-Layer in the Galton board is discovered and we calculate its information fractal dimension. When the Galton board is extended to bias or three-dimensional space, we get the same fractal features. In addition, according to the connection between Pascal’s triangle and the Poisson distribution, we find constrained Poisson distribution groups with the same information dimension. This is the first time the information entropy is utilized to explore the fractal features of the Galton board and Pascal’s triangle.

STUDY ACOUSTIC EMISSION SIGNALS AT TENSION STEEL 20

An investigation was made of acoustic emission signals during uniaxial tensile testing of flat specimens of steel 20 used for parts of welded structures with a large volume of welding, as well as pipelines, collectors and other parts operating at temperatures from –40 to 450 °C under pressure. Tensile testing with simultaneous registration of acoustic emission was carried out on a universal testing machine manufactured by Tinius OIlsen Ltd, model H100KU, at a movement speed of the active gripper of 0.05 meters per minute. Registration of AE signals was carried out using wideband GT350 sensors from GlobalTest and an analog-to-digital converter NationalInstruments 6363X with subsequent storage of the registration results in the form of a time series in the computer memory. A comparative analysis of the amplitude distribution of the AE signal for the area of the yield area and the area of destruction was carried out according to the value of information entropy, fractal dimension, and self-organization parameter. It was found that the parameter of self-organization of the amplitude distribution of the signal is the most informative in describing the processes associated with acoustic emission. As additional information, it is advisable to use data on the structure of the self-organization parameter. The results obtained indicate the possibility of using the statistical model of the Dirichlet distribution as a model of processes associated with the appearance of acoustic emission signals from sources of incipient and developing defects during routine tests of products made of structural carbon high-quality steels with a pearlite-ferrite structure. The paper presents a version of the model and modeling algorithms for FE-modeling corrosion cracking processes in structural elements loaded by pressure and exposed to aggressive corrosion media. To assess the effectiveness of the present models and algorithms, the failure process of a thin-walled tubular specimen partly submerged into a chlorine-containing liquid and loaded by axial tension is numerically modeled.

Spatial variation of microtopography and its effect on temporal evolution of soil erosion during different erosive stages

Microtopography significantly influences soil erosion at hillslope scale. Currently, the influence of microtopography on temporal evolution of soil erosion is still unclear. A set of simulated rainfall experiments on three 2 m long, 1 m wide soil boxes under a rainfall intensity of 1.5 mm min−1 on a 20° slope were conducted in the Chinese purple soil region to quantify the spatial variation of microtopography and its effect on temporal variations of runoff and sediment yield during water erosion processes. Three local tillage practices: conventional tillage (CT), artificial digging (AD), and ridge tillage (RT) with different initial microrelief (smooth, rough, and very rough) were designed. The spatial heterogeneity of microtopography was characterized by directional derivatives, semivariogram, and fractal information dimension in terms of morphology and quantity. And we identified the temporal variability characteristics of runoff and sediment yield based on wavelet and rescaled range (R/S) analyses. The results indicated that the spatial variability of different tillage practices had strong or medium autocorrelations during different water erosive stages. The spatial autocorrelation scales of CT, AD, and RT were 6.10, 5.32, and 4.01 m, respectively. The fractal information dimensions of tillage practices at various erosive stages were RT > AD > CT. There was a certain range of Di critical values, which reflected the positive or negative effect on soil erosion. The Hurst indexes of the runoff time series were in the section between 0.632 and 0.718, and that of the sediment yield time series ranged between 0.741 and 0.846. The time series of runoff and sediment yield had long-range persistence and varied periodically every 12–16 min during water erosion processes. Overall, the dominant period and the persistence of runoff tended to decrease and increase respectively with the increase in microrelief, while the temporal variation in sediment yield was opposite to that of runoff. Characterization of the temporal variability characteristics of runoff and sediment yield response to microrelief changes would contribute to a better understanding of the hydro-geomorphological processes at hillslope scale.

Research on Degradation State Recognition of Planetary Gear Based on Multiscale Information Dimension of SSD and CNN

Planetary gear is the key part of the transmission system for large complex electromechanical equipment, and in general, a series of degradation states are undergone and evolved into a local fatal fault in its full life cycle. So it is of great significance to recognize the degradation state of planetary gear for the purpose of maintenance repair, predicting development trend, and avoiding sudden fault. This paper proposed a degradation state recognition method of planetary gear based on multiscale information dimension of singular spectrum decomposition (SSD) and convolutional neural network (CNN). SSD can automatically realize the embedding dimension selection and component grouping segmentation, and the original vibration signal being nonlinear and nonstationary can be decomposed into a series of singular spectrum decomposition components (SSDCs), adaptively. Then, the multiscale information dimension which combines multiscale analysis and fractal information dimension is proposed for quantifying and extracting the feature information contained in each SSDC. Finally, CNN is used to achieve the effective recognition of the degradation state of planetary gear. The experimental results show that the proposed method can accurately recognize the degradation state of planetary gear, and the overall recognition rate is up to 97.2%, of which the recognition rate of normal planetary gear reaches 100%.

Mono-fractal analysis of CuXO films: Texture and phase distribution

Several studies on thermally-grown copper oxide films dealing with their crystalline properties and main phases as functions of the growth temperature have been published; notwithstanding, few research works have proposed a description of the surface morphology of the films using mono-fractal analysis. In this work, copper oxide films were grown by direct thermal oxidation in atmospheric air at different temperatures and times to obtain experimental proof of the phases present in the films by XRD, FTIR and Raman. The textural properties of the copper oxide films were analyzed from atomic force microscopy (AFM) images. The box counting and information fractal dimensions were used to perform the mono-fractal analysis to establish a correlation between the textural properties and the phases of the CuxO films.

Fractal Dimension Analysis of Subcortical Gray Matter Structures in Schizophrenia.

A failure of adaptive inference—misinterpreting available sensory information for appropriate perception and action—is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7–54.3 and healthy controls, 24.9–51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07–2.18 vs. median: 2.1730, range: 2.15–2.23, p<0.001; Cohen’s effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05–2.19 vs. median: 2.1760, range: 2.12–2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40–2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM surface detected in schizophrenia.

Spatial Up-Scaling Correction for Leaf Area Index Based on the Fractal Theory

The scaling effect correction of retrieved parameters is an essential and difficult issue in analysis and application of remote sensing information. Based on fractal theory, this paper developed a scaling transfer model to correct the scaling effect of the leaf area index (LAI) estimated from coarse spatial resolution image. As the key parameter of the proposed model, the information fractal dimension (D) of the up-scaling pixel was calculated by establishing the double logarithmic linear relationship between D-2 and the normalized difference vegetation index (NDVI) standard deviation (σNDVI) of the up-scaling pixel. Based on the calculated D and the fractal relationship between the exact LAI and the approximated LAI estimated from the coarse resolution pixel, a LAI scaling transfer model was established. Finally, the model accuracy in correcting the scaling effect was discussed. Results indicated that the D increases with increasing σNDVI, and the D-2 was highly linearly correlated with σNDVI on the double logarithmic coordinate axis. The scaling transfer model corrected the scaling effect of LAI with a maximum value of root-mean-square error (RMSE) of 0.011. The maximum absolute correction error (ACE) and relative correction error (RCE) were only 0.108% and 8.56%, respectively. The spatial heterogeneity was the primary cause resulting in the scaling effect and the key influencing factor of correction effect. The results indicated that the developed method based on fractal theory could effectively correct the scaling effect of LAI estimated from the heterogeneous pixels.

A new method to analyse species abundances in space using generalized dimensions

Summary Species–area relationships (SAR) and species‐abundance distributions (SAD) are among the most studied patterns in ecology, due to their application to both theoretical and conservation issues. One problem with these general patterns is that different theories can generate the same predictions, and for this reason, they cannot be used to detect different mechanisms of community assembly. A solution is to search for more sensitive patterns, for example by extending the SAR to the whole SAD. A generalized dimension (Dq) approach has been proposed to study the scaling of SAD, but to date, there has been no evaluation of the ability of this pattern to detect different mechanisms. An equivalent way to express SAD is the rank‐abundance distribution (RAD). Here I introduce a new way to study SAD scaling using a spatial version of RAD: the species‐rank surface (SRS), which can be analysed using Dq. Thus, there is an old Dq based on SAR (), and a new one based on SRS (). I perform spatial simulations to examine the relationship of Dq with SAD, spatial patterns and number of species. Finally, I compare the power of both Dq, SAD, SAR exponent and the fractal information dimension to detect different community patterns using a continuum of hierarchical and neutral spatially explicit models. The SAD, and all had good performance in detecting models with contrasting mechanisms. , however, had a better fit to data and allowed comparisons between hierarchical communities where the other methods failed. The SAR exponent and information dimension had low power and should not be used. SRS and could be interesting methods to study community or macroecological patterns.

Patterns of shrub expansion in Alaskan arctic river corridors suggest phase transition.

Recent increases in deciduous shrub cover are a primary focus of terrestrial Arctic research. This study examined the historic spatial patterns of shrub expansion on the North Slope of Alaska to determine the potential for a phase transition from tundra to shrubland. We examined the historic variability of landscape-scale tall shrub expansion patterns on nine sites within river valleys in the Brooks Range and North Slope uplands (BRNS) between the 1950s and circa 2010 by calculating percent cover (PCTCOV), patch density (PADENS), patch size variability (CVSIZE), mean nearest neighbor distance (MEDIST) and the multi-scale information fractal dimension (dI) to assess spatial homogeneity for shrub cover. We also devised conceptual models for trends in these metrics before, during, and after a phase transition, and compared these to our results. By developing a regression equation between PCTCOV and dI and using universal critical dI values, we derived the PCTCOV required for a phase transition to occur. All nine sites exhibited increases in PCTCOV. Five of the nine sites exhibited an increase in PADENS, seven exhibited an increase in CVSIZE, and five exhibited a decrease in MEDIST. The dI values for each site exceeded the requirements necessary for a phase transition. Although fine-scale heterogeneity is still present, landscape-scale patterns suggest our study areas are either currently in a state of phase transition from tundra to shrubland or are progressing towards spatial homogeneity for shrubland. Our results indicate that the shrub tundra in the river valleys of the north slope of Alaska has reached a tipping point. If climate trends observed in recent decades continue, the shrub tundra will continue towards homogeneity with regard to the cover of tall shrubs.

Particle size distribution of Pt–Cu bimetallic nanoparticles by fractal analysis

The fractal dimensions, obtained from analyzing the transmission electron micrographs (TEM) of the Pt–Cu bimetallic nanoparticles at different weight ratios, stabilized by PVP (polyvinyl pyrrolidone), are computed. The analysis indicates a bimodal fractal behavior characterized by a small fractal dimension at low scales and a higher fractal dimension for higher self-similarity domains. This behavior is probably due to a mixed-fractal structure, rather than a lack in instrument resolution. Results also indicate that both box-counting fractal dimension and information dimension increase as Cu concentration increases. A method to compute the distribution of the number of particles versus the radius from fractal analysis is developed; results are in good agreement with the direct analysis of the images.

Piecewise prediction model for watershed‐scale erosion and sediment yield of individual rainfall events on the Loess Plateau, China

AbstractEstablishing a universal watershed‐scale erosion and sediment yield prediction model represents a frontier field in erosion and soil/water conservation. The research presented here was conducted on the Chabagou watershed, which is located in the first sub‐region of the hill‐gully area of the Loess Plateau, China. A back‐propagation artificial neural model for watershed‐scale erosion and sediment yield was established, with the accuracy of the model, then compared with that of multiple linear regression. The sensitivity degree of various factors to erosion and sediment yield was quantitatively analysed using the default factor test. On the basis of the sensitive factors and the fractal information dimension, the piecewise prediction model for erosion and sediment yield of individual rainfall events was established and further verified. The results revealed the back‐propagation artificial neural network model to perform better than the multiple linear regression model in terms of predicting the erosion modulus, with the former able to effectively characterize dynamic changes in sediment yield under comprehensive factor conditions. The sensitivity of runoff erosion power and runoff depth to the erosion and sediment yield associated with individual rainfall events was found to be related to the complexity of surface topography. The characteristics of such a hydrological response are thus closely related to topography. When the fractal information dimension is greater than the topographic threshold, the accuracy of prediction using runoff erosion power is higher than that of using runoff depth. In contrast, when the fractal information dimension is smaller than the topographic threshold, the accuracy of prediction using runoff depth is higher than that of using runoff erosion power. The developed piecewise prediction model for watershed‐scale erosion and sediment yield of individual rainfall events, which introduces runoff erosion power and runoff depth using the fractal information dimension as a boundary, can be considered feasible and reliable and has a high prediction accuracy. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Analysis of fetal cortical complexity from MR images using 3D entropy based information fractal dimension

The fetal cortical complexity is a significant quantification for assessing the development of fetal brain. This study attempts to quantify the development of fetal cortical complexity using the concept of fractal dimension (FD) analysis. Thirty-two fetal MR images were selected from Taipei Veterans General Hospital at 27–37 weeks of gestational age (GA). To investigate the FD of fetal cortical complexity, the entropy based information fractal dimension method (FDEBI), which is modified from Box-Counting method, was adopted and extended from 2D to 3D. The FD results from overall whole fetal brains show that the increase of cortical complexity is highly correlated with the gestational age of the fetus. Moreover, the FD values of right hemispheric brain are larger than those of left hemispheric brain, show that the development of right hemispheric fetal cortical complexity earlier than the left. These results are in good agreement with normal fetal brain development and suggest that the FD is an effective means for the quantification of fetal cortical complexity.

Can the fractal dimension be applied for the early diagnosis of non-proliferative diabetic retinopathy?

The fractal dimension has been employed as a useful parameter in the diagnosis of retinal disease. Avakian et al. (Curr Eye Res 2002; 24: 274-280), comparing the vascular pattern of normal patients with mild to moderate non-proliferative diabetic retinopathy (NPDR), found a significant difference between them only in the macular region. This significant difference in the box-counting fractal dimension of the macular region between normal and mild NPDR patients has been proposed as a method of precocious diagnosis of NPDR. The aim of the present study was to determine if fractal dimensions can really be used as a parameter for the early diagnosis of NPDR. Box-counting and information fractal dimensions were used to parameterize the vascular pattern of the human retina. The two methods were applied to the whole retina and to nine anatomical regions of the retina in 5 individuals with mild NPDR and in 28 diabetic but opthalmically normal individuals (controls), with age between 31 and 86 years. All images of retina were obtained from the Digital Retinal Images for Vessel Extraction (DRIVE) database. The results showed that the fractal dimension parameter was not sensitive enough to be of use for an early diagnosis of NPDR.

AN ITERATING MODEL OF INDUSTRIES DISTRIBUTION IN REAL SPACE

An industry distribution model provides an important reference point for enterprises to extend business scope, and for governments to make industrial area planning. Current industry development models usually ignore the spatial distribution of enterprises, or build the models in virtual spaces. This paper proposes an iterating Model of Industry development in Real Space (MIRS) based on national division standards. Utilizing a multi-level iterating structure and a random concentration distribution approach, MIRS presents the fractal characteristics of log-normal distribution, self-similarity, and increasing trend of fractal information dimension. The coincidence between MIRS and actual industries is verified by the data collected from three industries in China, this allows MIRS to be an effective approach for predicting industry behaviors and a quantitative reference for decision-makers of companies.