Degree Of Self-similarity Research Articles

Chaos in Bitcoin Cryptocurrency Metrics: Analysis and Forecasts

AbstractCryptocurrencies, particularly Bitcoin have attracted a lot of attention in the last decades of humanity. Analyzing cryptocurrencies algorithmic differences, chaotic behavior and self-similarity in cryptocurrency metrics might give significant insights for identifying risks and opportunities. Determining the degree of chaos in crypto metrics is critical for understanding complexity, improving prediction capabilities, and supporting decision-making. This study focuses on the analysis of chaos and self-similarity in Bitcoin dynamics for predictability perspective. Return, rate of return and volume quantities in different scales are analyzed with using rescaled range method to reveal the degree of self-similarity. Hurst parameter extracts a comprehensive summary providing information on how current values depend on previous ones to reveal any persistence in Bitcoin metrics. Daily rate of return and return give Hurst degree around 0.64 while they are in between 0.52–0.55 for minutely and hourly based prices. However, an increasing persistence is observed with the increasing time window. Although the largest Lyapunov exponents stay in the positive region for prices and returns of Bitcoin, they are approximately zero for inspected statistics. Periodic characteristics of Bitcoin are also investigated to reveal any dependencies on halving mechanism of Bitcoin. Detailed self-similarity analysis on specific periods shows that bull and bear market seasons don’t make any significant effect on the degree of Hurst parameter. Due to nonlinear and unpredictable characteristics of Bitcoin metrics, distribution fittings are applied to characterize BTC return and rate of return. While Wakeby distribution gives best fitting for daily return, Cauchy distribution gives best for hourly returns.

Effects of Donepezil on Nonlinear Arterial Pressure and Pulse Interval Dynamics in Spontaneously Hypertensive Rats (SHR)

Objectives: This study aimed to evaluate the effects of donepezil, an anticholinesterase agent that potentiates parasympathetic function, in the nonlinear dynamics of blood pressure (BP) and pulse interval (PI) from spontaneously hypertensive rats (SHR). Entropy-based approaches which quantify the irregularity or unpredictability of time series were employed. Additionally, detrended fluctuation analysis (DFA) was applied to estimate the degree of self-similarity in the time series. We hypothesized that donepezil improves both entropy and DFA indices of BP and PI. Methods: Male SHR or Wistar-Kyoto (WKY) rats (5 weeks old) were implanted with subcutaneous osmotic mini-pumps for donepezil (DON) administration (1.4 mg/kg/day) or vehicle. At the end of the treatment (16 weeks), conscious WKY (N=12) and SHR treated with donepezil (N=8), or vehicle (N=9), had their BP directly recorded during 1h. Time series (15 min) from systolic BP and PI were generated and used to calculate sample (SampEn), fuzzy (FuzzyEn), and dispersion (DispEn) entropy, as well as DFA scale exponents in short (α1; 5≤n≤15), mid (α2; 20≤n≤100) and long (α3; 100≤n≤500) windows. The results are shown as mean ± SD. One-Way ANOVA, followed by Tukey’s post-test, evaluated differences between the WKY, SHR, and SHR+DON groups. Results: The data are displayed in the table. Lower PI DispEn was observed in SHR treated with DON compared to WKY. Entropy values of BP series were found smaller in SHR and SHR+DON compared to WKY rats. For DFA of the PI series, SHR+DON showed higher α1 compared to the WKY. Moreover, SHR displayed higher α3 when compared to the WKY. In the series of BP values, the results showed higher α2 and α3 in SHR compared to WKY. [Formula: see text] Conclusion: These findings showed that SHR exhibited lower BP entropy and higher DFA of BP and IP compared with WKY. However, donepezil did not prevent these changes of non-linear indices from BP and PI variability in SHR. Funding Sources: FAPESP, Process 2020/06043-7. CNPq, Process 306994/2021-6 and 423999/2021-4. FAEPA. CAPES. This is the full abstract presented at the American Physiology Summit 2024 meeting and is only available in HTML format. There are no additional versions or additional content available for this abstract. Physiology was not involved in the peer review process.

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3D printed bio-inspired self-similar carbon fiber reinforced composite sandwich structures for energy absorption

Fractals widely found in nature possess amazing simple shape cell yet infinitely complex macrostructures. Importantly, all fractals manifest a degree of self-similarity that provides new structural design strategy for composite materials. At present, enhancing energy absorption capacity of lightweight carbon fiber reinforced composite structures is critical for their further engineering applications, and meanwhile biomimetic structures have demonstrated excellent mechanical properties compared to conventional engineering structures. In this work, we constructed bio-inspired fractal structures through three fundamental shape units (curve, circle, and hexagon), and investigated nonlinear mechanical responses of hierarchical self-similar structures inspired by snake-like serpentine, bamboo, and honeycomb, having distinct structure ratios related to cell sizes at different geometry levels. Firstly, such self-similar structures were fabricated into homogenous ones by 3D printing for assessing the energy dissipation contribution of polymer matrix. Moreover, the selected self-similar carbon fiber reinforced composites were additively manufactured and their energy absorption mechanisms of composite sandwich structures were studied via experiment and numerical methods. Additionally, the influences of the self-similar types, structure ratios, and cellular scales on nonlinear compressive behaviors of self-similar composite structures were investigated in detail. The results indicated that the fractal bamboo composite components with the structure ratio of 0.2 exhibit the highest energy absorption rates, with a satisfactory level of energy absorption capacity in potential engineering applications. This study can provide a useful reference in the field of biomimetic fractal design of fiber reinforced composite structures by 3D printing technique.

Metrics for the characteristic length scale in the random bicontinuous microstructure of nanoporous gold

Nanoporous gold (NPG) made by dealloying exemplifies materials with random bicontinuous microstructures that can be approximated by leveled-wave type models. As a distinguishing feature, the characteristic length scale – often quantified by the “ligament size” L – of NPG may be tuned over several orders of magnitude while the microstructural geometry retains a high degree of self-similarity. It is therefore essential to have at hand accurate procedures for determining the size by experiment and to match it to analogous size metrics of model scenarios. Working with a set of NPG samples of widely different size, we compare ligament size distributions determined by analysis of scanning electron micrographs to those of the leveled-wave model. The model is representative of various material types with random bicontinuous microstructures. The size distribution is remarkably uniform over the cross-section of experimental samples. Furthermore, the distribution evolves self-similarly upon coarsening, and the normalized distribution width agrees closely to that of the model. A measure for size determined by the electrochemical capacitance ratio method correlates well with L. This supports a protocol for converting between the two measures. As a dimensionless factor characteristic of the microstructural geometry of random dual phase microstructures, the product of L and the specific surface area is found consistent between experiment and model. The findings suggest conversion factors between the various metrics, and they advertise the combination of NPG and the leveled-wave model as a showcase for characterizing the characteristic length scale of random bicontinuous microstructures.

An approach to detecting anomalies in a self-similar network traffic

Aim. The effects of cyber attacks cause failures of network elements, theft of information and other unlawful actions. Cyber attacks are often accompanied by untypical traffic activity and anomalies. The paper aims to develop an approach to detecting anomalies in network traffic by identifying the degree of self-similarity of the traffic using fractal analysis and statistical methods. Methods. The paper uses methods of mathematical statistics, mathematical analysis, fractal analysis. Results. The paper suggests an approach to identifying anomalies in network traffic by evaluating self-similarity and using statistical methods for improving the accuracy of cyber attack detection. At the first stage, the Hurst exponent is calculated for the reference traffic. At the second stage, actual traffic is divided into optimal time intervals; for each interval, the Hurst exponent is calculated. If the identified value of the Hurst exponent differs from the one obtained for the reference traffic, it is decided that there is an anomaly. At the final stage, statistical analysis is used in order to precisely localise the anomaly. The authors analysed fractal and statistical methods that resulted in the identification of more efficient methods to be used as part of the proposed approach. For fractal analysis, the DFA method was proposed, while for statistical analysis, the ARFIMA method was proposed. Conclusion. The suggested approach allows identifying cyber attacks in real time or near-real time.

Factors Affecting Avatar Customization Behavior in Virtual Environments

This research aims to examine the psychology and behavior of users when customizing avatars from the standpoint of user experience and to provide constructive contributions to the Metaverse avatar customization platform. This study analyzed the factors that affect the behavior of user-customized avatars in different virtual environments and compared the differences in public self-consciousness, self-expression, and emotional expression among customized avatars in multiple virtual contexts. Methods: Using a between-subjects experimental design, two random groups of participants were asked to customize avatars for themselves in two contexts, a multiplayer online social game (MOSG) and a virtual meeting (VM). Results: When subjects perceived a more relaxed environment, the customized avatars had less self-similarity, and the subjects exhibited a stronger self-disclosure willingness and enhanced avatar wishful identification; nevertheless, public self-consciousness was not increased. When subjects perceived a more serious environment, the customized avatars exhibited a higher degree of self-similarity, and the subjects exhibited a greater self-presentation willingness, along with enhanced identification of avatar similarity and increased public self-consciousness. Conclusions: Participants in both experiment groups expressed positive emotions. The virtual context affects the self-similarity of user-customized avatars, and avatar self-similarity affects self-presentation and self-disclosure willingness, and these factors will affect the behavior of the user-customized avatar.

Root architecture and visualization model of cotton group with different planting spacing under local irrigation.

Planting spacing plays a key role in the root system architecture of the cotton group under local irrigation. This study used the Cellular Automata (CA) theory to establish a root visualization model for the cotton group at two different planting spacing (30 and 15cm) within a leaching-pond. At a planting spacing of 30cm, the lateral roots grew almost horizontally toward the irrigation point, and a logarithmic relationship was observed between root length density and soil water suction. However, at a planting spacing of 15cm, the lateral roots exhibited overlapping growth and mainly competed for resources, and a power function relationship was observed between root length density and soil water suction. The main parameters of the visualization model for each treatment were essentially consistent with the experimental observations, with respective simulation errors were 6.03 and 15.04%. The findings suggest that the correlation between root length density and soil water suction in the cotton plants is a crucial driving force for the model, leading to a more accurate replication of the root structure development pathway. In conclusion, the root system exhibits a certain degree of self-similarity, which extends into the soil.

Boosting deep image prior by integrating external and internal image priors

Recently, an unsupervised method [denoted as deep image prior (DIP)] that interprets images as the output of a convolutional network with randomly initialized inputs has received much attention, as the DIP can adapt its network parameters to any given noisy image without requiring training pairs, providing a great flexibility for handling those noisy images with complex structures and low degree of self-similarity. However, the DIP method still suffers from a crucial limitation that only the noisy image is used as target image, which implies that only internal prior information is used for noise removal and leads to poor denoising performance, compared with supervised denoising models. In this work, we aim to boost the DIP method by integrating external and internal image priors. Specifically, with a given noisy image, we first exploited a state-of-the-art supervised method to denoise it and obtain its corresponding denoised image called initial denoised image. Then the initial denoised image containing rich external prior information was used as extra target image together with the given noisy image within the standard DIP framework, resulting in a highly effective unsupervised recovery process. Next, to take full advantage of the uncertainty of the DIP network, the above DIP denoising routine with the different random inputs was executed multiple times to generate enough complementary denoised images (samples). Finally, with an unsupervised weight map generative network, the generated samples were fused in a pixel-wise manner. The fused image with better image quality was treated as the final denoised image. We had verified the denoising performance of our method on a large of images from several benchmark databases as well as real-world noisy images. The comparative results show that our method outperforms the original DIP method and other unsupervised methods by a large margin, and surpasses state-of-the-art supervised counterparts with comparable peak signal-to-noise rate. Source code is publicly available at https://github.com/chenxiaojun0101/BDIP.

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

In the article, a comparative analysis is performed regarding the accuracy parameter in determining the degree of self-similarity of fractal processes between the following methods: Variance-Time plot, Rescaled Range (R/S), Wavelet-based, Detrended Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis (MFDFA). To evaluate the methods, fractal processes based of Fractional Gaussian Noise were simulated and the dependence between the length of the simulated process and the degree of self-similarity was investigated by calculating the Hurst exponent (H > 0.5). It was found that the Wavelet-based, DFA and MFDFA methods, with a process length greater than 214 points, have a relative error of the Hurst exponent is less than 1%. A methodology for the Wavelet-based method related to determining the size of the scale and the wavelet algorithm was proposed, and it was investigated in terms of the exact determination of the Hurst exponent of two algorithms: Haar and Daubechies with different number of coefficients and different values of the scale. Based on the analysis, it was determined that the Daubechies algorithm with 10 coefficients and scale (i = 2, j = 10) has a relative error of less than 0.5%. The three most accurate methods are applied to the study of real cardiac signals of two groups of people: healthy and unhealthy (arrhythmia) subjects. The results of the statistical analysis, using the t-test, show that the proposed methods can distinguish the two studied groups and can be used for diagnostic purposes.

Image Restoration using Nonlocal Regularized Variational Model with Spatially Adapted Regularization Parameter

Restoration of blurred images corrupted with impulse noise has attracted increasing attention due to its great potential toward practical applications. To guarantee the noisy image deblurring performance, total variation (TV), one of the most popular regularizers, has been extensively exploited in the literature. This local regularizer is known for its capability of preserving edges and penalizing oscillations but not sharp discontinuities. However, TV regularized variational models commonly suffer from undesirable staircase-like artifacts in homogeneous regions. The reason behind this phenomenon is that TV regularizer favors solutions that are piecewise constant in numerical experiments. To overcome the model-dependent deficiency, we proposed to replace the commonly used local regularizer by its nonlocal extension, i.e., nonlocal total variation (NLTV) regularizer. By taking advantage of the high degree of self-similarity within images, the proposed nonlocal variational model is able to remove both blurring and impulse noise while preserving meaningful image details. To guarantee solution stability and efficiency, the resulting nonsmooth image restoration problem was effectively dealt with using an alternating optimization algorithm. In addition, to further enhance the image quality, a local variance estimator-based automatic method was introduced to calculate the spatially adapted regularization parameters during image restoration. Extensive experiments have demonstrated that the proposed method could achieve superior imaging performance compared with several state-of-the-art image restoration methods.

Error Concealment in the Density Field of a Spatiotemporal Image Sequence.

One of the most difficult challenges of multimedia transmission during the last two decades has been the retrieval of degraded or missing regions of images and videos while maintaining satisfactory perceptual accuracy. The objective is to retrieve lost data by using the similarity between frames. Usually, error concealment (EC) schemes depend on replacing incorrect data with data that are identical to the initial. This is possible because video contains a high degree of self-similarity. This research focuses on applying an EC approach in transform-domain video sequences. To conduct EC on films, they must first be translated to frames and then transformed using one of the available transformations into frequency-domain images. Using successive frames, it is possible to recover lost or incorrect data from images. Intra-coded frames (I-frames) may be used to recreate lost knowledge in predictive (P-frames) and bidirectional predictive frames (B-frames). I-frame knowledge that has been lost may be restored using previous intra-coded frames. The use of wavelet error concealment generated more precise results than the other techniques. In this study, it was discovered that covering faults in the density sector with wavelets produces more reliable results than the other techniques.

Adaptive Hurst-Sensitive Active Queue Management.

An Active Queue Management (AQM) mechanism, recommended by the Internet Engineering Task Force (IETF), increases the efficiency of network transmission. An example of this type of algorithm can be the Random Early Detection (RED) algorithm. The behavior of the RED algorithm strictly depends on the correct selection of its parameters. This selection may be performed automatically depending on the network conditions. The mechanisms that adjust their parameters to the network conditions are called the adaptive ones. The example can be the Adaptive RED (ARED) mechanism, which adjusts its parameters taking into consideration the traffic intensity. In our paper, we propose to use an additional traffic parameter to adjust the AQM parameters—degree of self-similarity—expressed using the Hurst parameter. In our study, we propose the modifications of the well-known AQM algorithms: ARED and fractional order and the algorithms based on neural networks that are used to automatically adjust the AQM parameters using the traffic intensity and its degree of self-similarity. We use the Fluid Flow approximation and the discrete event simulation to evaluate the behavior of queues controlled by the proposed adaptive AQM mechanisms and compare the results with those obtained with their basic counterparts. In our experiments, we analyzed the average queue occupancies and packet delays in the communication node. The obtained results show that considering the degree of self-similarity of network traffic in the process of AQM parameters determination enabled us to decrease the average queue occupancy and the number of rejected packets, as well as to reduce the transmission latency.

METHODOLOGY FOR THE FORMATION OF INFORMATION SYSTEMS FALSE NETWORK TRAFFIC FOR PROTECTION AGAINST NETWORK RECONNAISSANCE

Simulation of false network traffic in order to protect the structural and functional characteristics of information systems is a difficult task in view of the self-similarity of its statistical properties in IP networks, not only in the current moment, but also retrospectively. A Hurst index based algorithm for assessing the degree of self-similarity of network traffic of information systems has been proposed. The connection between the fractal dimension of the attractor of the model of information system functioning and the Hurst index is shown. A technique has been developed to substantiate the characteristics of false network traffic to simulate the functioning of information systems in the process of reconfiguration of their structural and functional characteristics caused by an intruder conducting network reconnaissance. The methodology allows to solve the problem of improving the protection of information systems from network reconnaissance by providing the maximum likelihood of false network traffic by pseudophase reconstruction of the dynamic system attractor, approximating the time series of information traffic of the protected object. The approaches to the description of the network traffic of the information system are considered, the parameters determining the network interaction between the two nodes of the data transmission network are selected as follows: source IP-address, source port, destination IP-address, destination port, protocol, packet size, duration of connection. The process of functioning of information system in different situations is formalized and the dependences allowing to synthesize parameters of false network traffic, statistically similar to the reference ones are received.

КОМПЛЕКСНАЯ МЕТОДИКА ОБНАРУЖЕНИЯ КИБЕРАТАК НА ОСНОВЕ ИНТЕГРАЦИИ ФРАКТАЛЬНОГО АНАЛИЗА И СТАТИСТИЧЕСКИХ МЕТОДОВ

The article discusses a method for detecting cyber-attacks on computer networks based on detecting anomalies in network traffic by assessing its self-similarity and determining the impact of cyber-attacks using statistical methods. The proposed methodology provides for three stages, within which the analysis of the self-similarity property for reference traffic is performed (using the Dickey-Fuller test, R/S analysis and the DFA method), the analysis of the self-similarity property for real traffic (by the same methods) and additional processing of time series by statistical methods (moving average, Z-Score and CUSUM). The issues of software implementation of the proposed approach and the formation of a data set containing network packets are considered. The results of the experiments demonstrated the presence of self-similarity of network traffic and confirmed the high efficiency of the proposed method, which allows detecting cyber-attacks in real or near real time. Introduction: The use of information and communication technologies for information collection in modern computer networks makes it possible for an attacker to influence the network infrastructure by implementing cyber-attacks. Cyberattacks can achieve their goals due to the massive use of outdated operating systems, ineffective protection mechanisms and the presence of multiple vulnerabilities in unsecured network protocols. Such vulnerabilities give a potential attacker the ability to change the settings of network devices, listen and redirect traffic, block network interaction and gain unauthorized access to internal components of computer networks. The purpose of the work is to develop a methodology for detecting anomalies in network traffic by determining the degree of self-similarity of traffic using fractal analysis and statistical methods. Methods used: software implementation of the proposed methodology and the formation of a data set containing network packets. The results of the experiments demonstrated the presence of self-similarity of network traffic and confirmed the high efficiency of the proposed technique, which allows detecting cyber-attacks in real or near real time. The scientific novelty lies in the fact that the proposed methodology provides for three stages, within which the analysis of the self-similarity property for reference traffic is performed (using the Dickey-Fuller test, R/S analysis and the DFA method), the analysis of the self-similarity property for real traffic (by the same methods) and additional processing of time series by statistical methods (methods moving Average (MA), Z-Score and CUSUM). Result: the presented methodology allows detecting the impact of cyberattacks in real and close to real time, and the use of statistical methods increases the accuracy of determining cyberattacks. Practical significance: the presented methodology is universal and can be applied in the information exchange systems of public administration bodies performing the tasks of ensuring the security of the country.

On the meaning of Hurst entropy applied to EEG data series

The Hurst exponent estimates the degree of self-similarity and predictability of a time series, which, under this nonlinear statistical model, can adopt two opposing tendencies with respect to the way these data series are mobilized over time. Persistent and anti-persistent series are thus described, depending on whether the Hurst value describing them is greater or less than 0.5 in a range of 0 to 1. Those series that show "Hurst effect" (persistent series with H>0.5) are potentially predictable in the short or medium term, which makes them especially interesting for any predictive interest in science or economics. In the brain, when the oscillatory electrical activity of an EEG is analyzed, the time series that make up a complete recording include a variable number of sources, electrodes or channels, which detect the electrical activity of millions of pyramidal neurons at the level of the cerebral cortex. When this signal is decomposed into its frequency components, its spectrum ranges from the slowest waves, or Delta (0.1-4Hz), to the fastest, Gamma (>30 Hz). In the intermediate range, the Theta (4-8Hz); Alpha (8-12Hz); and Beta (13-30Hz) oscillations complete the range of the EEG spectrum. When estimating the Hurst exponent by filtered intervals of the total frequency range, only for the Delta band, the H values exceed H=0.5. The Hurst values for Delta were close to H = 0.85 and all the rest of the spectrum falls to values less than 0.5, in a range from 1.5<H<3.5.In the present work, we extend the interpretation of the Hurst exponent applied to electroencephalographic time series, and we propose to pay special attention to the so-called anti-persistent series (H<0.5) which, overshadowed by the predictive promises of the Hurst effect of persistent series, we believe may hide the very long-term persistence of complex homeostatic processes, stabilized in the human brain over millions of years of evolution.

Анализ поведения трафика при переходе с одного уровня иерархии сети на другой

The hierarchical network model is the most preferable in the design of computer networks, as it allows you to create a more stable structure of network, rationally allocate available resources, and also provide a higher degree of data protection. In this work, the study of the behavior of the traffic during the transition from one level of the network hierarchy to another, based on the study of the values of the traffic self-similarity degree during this transition. For the study, a simulation model of a computer network with a hierarchical topology was developed using the NS-3 simulator. Also, a window application was developed in the Visual C# programming language. With the help of this application the degree of self-similarity of the traffic was investigated using the files obtained as a result of processing the trace file. Thus, as a result of the study, it can be stated that any changes in the degree of self-similarity of the network traffic when this traffic moves from one level of the hierarchy to another level depends on such a condition as the direction of traffic movement. The initial degree of selfsimilarity of network traffic also effects on the network traffic self-similarity degree.

Expertise-Related Differences in Cyclic Motion Patterns in Drummers: A Kinematic Analysis

BackgroundAt present only little information is available concerning the acquisition of skilled movements in musicians. Although optimally a longitudinal study of changing movement patterns during the process of increasing expertise is required, long-term follow up over several years is difficult to manage. Therefore, in the present cross-sectional study a comparative kinematic analysis of skilled movements in drummers with different levels of expertise was carried out.AimsThe aim of the investigation was (1) to analyze the kinematic differences between beginners, students and expert drummers, and (2) to deduce from the results general rules related to the acquisition of drumming expertise and (3) to discuss the implications for drum teaching.MethodTwo highly skilled experts, eight professional drumming students and five beginners participated in the experiment. Fast repetitive drumming movements were assessed using an active infrared measurement setup (SELSPOT-System). Recording was obtained from LEDs positioned over the shoulder-, elbow-, wrist- and MCP-joints and close to the tip of the stick at a sampling rate of 300 Hz. Kinematic analysis included calculation of angles, velocities and accelerations and assessment of the relation between velocity and acceleration as phase diagrams.ResultsTemporal accuracy of the drumming movements was related to expertise. In contrast to beginners, experts and students revealed a high degree of self-similarity of movements and a predominant use of low-mass distal joints, resulting in a whiplash-like movement when hitting the pad.ConclusionIntense training in students and experts results in economic utilization of forces. Percussion teachers can take advantage of the kinematic analysis and improve their instructions according to the student’s observed motor pattern.

A New Measure to Characterize the Degree of Self-Similarity of a Shape and Its Applicability.

We propose a new measure (Γ) to quantify the degree of self-similarity of a shape using branch length similarity (BLS) entropy which is defined on a simple network consisting of a single node and its branches. To investigate the properties of this measure, we computed the Γ values for 70 object groups (20 shapes in each group) in the MPEG-7 shape database and performed grouping on the values. With relatively high Γ values, identical groups had visually similar shapes. On the other hand, the identical groups with low Γ values had visually different shapes. However, the aspect of topological similarity of the shapes also warrants consideration. The shapes of statistically different groups exhibited significant visual difference from each other. Also, in order to show that the Γ can have a wide variety of applicability when properly used with other variables, we showed that the finger gestures in the (Γ, Z) space are successfully classified. Here, the Z means a correlation coefficient value between entropy profiles for gesture shapes. As shown in the applications, Γ has a strong advantage over conventional geometric measures in that it captures the geometrical and topological properties of a shape together. If we could define the BLS entropy for color, Γ could be used to characterize images expressed in RGB. We briefly discussed the problems to be solved before the applicability of Γ can be expanded to various fields.

Regional features of topographic relief over the Loess Plateau, China: evidence from ensemble empirical mode decomposition

Landforms with similar surface matter compositions, endogenic and exogenic forces, and development histories tend to exhibit significant degrees of self-similarity in morphology and spatial variation. In loess hill-gully areas, ridges and hills have similar topographic relief characteristics and present nearly periodic variations of similar repeating structures at certain spatial scales, which is termed the topographic relief period (TRP). This is a relatively new concept, which is different from the degree of relief, and describes the fluctuations of the terrain from both horizontal and vertical (cross-section) perspectives, which can be used for in-depth analysis of 2-D topographic relief features. This technique provides a new perspective for understanding the macro characteristics and differentiation patterns of loess landforms. We investigate TRP variation features of different landforms on the Loess Plateau, China, by extracting catchment boundary profiles (CBPs) from 5 m resolution digital elevation model (DEM) data. These profiles were subjected to temporal-frequency analysis using the ensemble empirical mode decomposition (EEMD) method. The results showed that loess landforms are characterized by significant regional topographic relief; the CBP of 14 sample areas exhibited an overall pattern of decreasing TRPs and increasing topographic relief spatial frequencies from south to north. According to the TRPs and topographic relief characteristics, the topographic relief of the Loess Plateau was divided into four types that have obvious regional differences. The findings of this study enrich the theories and methods for digital terrain data analysis of the Loess Plateau. Future study should undertake a more in-depth investigation regarding the complexity of the region and to address the limitations of the EEMD method.