Use Of Fractal Analysis Research Articles

The Value of Fractal Analysis in Ultrasound Imaging: Exploring Intricate Patterns

Fractal analysis is a mathematical approach employed to study and describe complex patterns and structures across various disciplines, including mathematics, physics, computer science, biology and finance. Introduced by mathematician Benoit Mandelbrot in the 1970s, fractals are intricate, self-similar patterns that repeat at different scales, exhibiting consistent structures upon magnification or reduction. This analysis involves generating fractals through iterative processes or recursive equations, resulting in highly detailed and complex formations. Fractal analysis enhances medical images by removing noise while preserving details and improving diagnostic quality in magnetic resonance and computed tomography scans. However, there is a lack of comprehensive studies on its application in ultrasound imaging, prompting this narrative review to investigate its use and methodology in this context. Selected papers on the use of fractal analysis in ultrasound imaging were analyzed. Out of 186 records screened, 60 duplicates were removed and 28 were discarded. The text content of 98 potentially eligible papers was checked, with 65 not meeting inclusion criteria. Finally, 33 studies were included in the review. Fractal analysis enhances ultrasound imaging by providing detailed tissue texture characterization, aiding in the diagnosis of conditions like breast and lung cancer, osteoporosis and hypertensive disorders in pregnancy. It quantifies biological structure complexity and improves diagnostic accuracy and reliability. This technique supports clinicians in making informed decisions by offering critical insights into various medical conditions.

Fractal dimensional analysis to reveal traffic flow dynamics as organic metabolism

As urban areas face rapid population and vehicle growth, understanding system dynamics is crucial for managing transportation congestion effectively. Utilizing fractal and metabolic concepts offers fresh perspectives on the behavior and efficiency of complex urban transportation systems. To understand the dynamics of congested traffic flows, we conducted fractal analysis on the traffic pattern in Seoul city with chronic congestion worldwide and investigated its potential for evaluating metabolic efficiency. Our rank and size fractal analysis revealed that Seoul's fractal dimension ranges from 6.7060 to 7.6801. The fractal dimension's curve, when plotted against traffic congestion, shows a peculiar symmetrical relationship. The linear correlation between fractal dimension and metabolic indicators suggests that fractal dimension could potentially serve as a novel metric for evaluating traffic flow efficiency. This study introduces new insights by integrating fractal and metabolic principles to understand traffic system dynamics, marking the first use of fractal analysis in traffic pattern evaluation.

Novel Use of Fractal Analysis for Quantifying Polymethylmethacrylate Distribution Patterns in Osteoporotic and Malignant Vertebral Compression Fractures Following Vertebroplasty.

Purpose: Fractal analysis is a mathematical tool which allows the evaluation of complex microstructural features within materials that cannot be expressed in traditional geometric terms. The purpose of this study is to quantify the differences in polymethylmethacrylate intravertebral cement spatial distribution patterns following vertebroplasty using fractal analysis through the examination of osteoporotic and malignant compression fractures. Methods: Frontal and lateral post-vertebroplasty radiographs were evaluated from 29 patients with osteoporotic and malignant compression fractures who underwent vertebroplasty. The individually treated vertebra were divided into osteoporotic (n = 35) and malignant groups (n = 41). Images underwent segmentation, thresholding, and binarization prior to fractal analysis. Fractal dimension and lacunarity values were derived from the region of interest in treated vertebrae using the "box-counting" and "gliding-box" techniques respectively using ImageJ. The mean values of both parameters were compared between the 2 groups. Results: The mean fractal dimension was significantly higher in the malignant vertebral compression fracture group (1.53 ± 0.08) compared to the osteoporotic group (1.34 ± 0.17; P < .001). Similarly, mean lacunarity values were significantly higher in the malignant fracture group (0.50 ± 0.09) compared to the osteoporotic group (0.37 ± 0.10; P < .001). Conclusions: Fractal dimension and lacunarity values of cement spatial distribution patterns obtained from the post-vertebroplasty radiographs can differentiate between benign osteoporotic and malignant vertebral compression fractures. This novel technique may be useful for evaluating cement spatial distribution patterns in spine augmentation procedures, although further research is warranted in this area.

A Panoramic X-ray as a Supportive Diagnostic Tool for the Screening of Osteoporosis in Patients with Hemophilia A and B.

Hemophilia is associated with an increased risk of developing osteoporosis and osteopenia. The aim of the study was to interpret the usefulness of fractal analysis of the trabecular bone of the mandible (FD) and selected radiomorphic indices (the antegonial index (AI) and Klemetti index (KI)) to assess the risk of reduced bone mineral density (BMD) in patients with hemophilia A/B. The study group consisted of 50 patients with type A/B hemophilia. The control group consisted of 25 males without congenital bleeding disorders. The patients had a panoramic radiograph taken with the same X-ray machine (Vistapano S, Durr Dental, Bietigheim-Bissingen, Germany). The AudaXCeph software (Audax d.o.o., Ljubljana, Slovenia) was used to evaluate the AI index, and the ImageJ, software with a specially prepared script, was used to evaluate the FD. The mandibular cortex was assessed distal to the mental foramen using the Klemetti index. There were no statistically significant differences between the AI, KI and FD values between the study group and the control group. This study indicated the lack of usefulness of AI, KI and fractal analysis in identifying patients with hemophilia at risk of reduced bone mass.

Nuclear Fractal Dimension in Diagnosing Oral Cancer- A Systematic Review

Background: Fractal analysis, which derives from fractal geometry, is used to put a number on the degree of shape complexity. Recent research have looked at the use of fractal analysis for tumour characterization.&#x0D; Objective: The systematic review's goal is to determine whether using the fractal dimension as a diagnostic tool can help detect oral squamous cell cancer in its earliest stages.&#x0D; Methodology: By adhering to three main criteria, all research articles published between the years of 2008 and 2020 in three major scientific databases, including PubMed, Scopus, and Web of Science, were included. The selection criteria for the investigation were based on fractal dimension analysis in diagnosing oral cancer, nuclear fractal dimension analysis in identifying the nuclear structure before and after treatment of oral carcinomas.&#x0D; Results: Use of nuclear fractal analysis by dental professionals will help in diagnosing oral cancer at an early stage itself which further has its effect on treatment outcome. All the ten included studies have shown that fractal geometry is a useful objective tool in diagnosing oral cancer. The risk was assessed done by Cochrane risk assessment model based on the several domains used was roughly 53% low risk, 30% ambiguous, and 15% high risk of bias.&#x0D; Conclusion: All three of the objectives of this systematic study that were used as benchmarks improved as a result of the fractal dimension technique. Early detection of the healthy and sick architecture of the nucleus in cases of oral cancer was made possible by the use of fractal dimension principles. It also helps in assessing and comparing nuclear characteristics of cancer cases before and after treatment.&#x0D; Significance: Applying the principles of nuclear fractal dimension improves the diagnosis of oral carcinomas at early stages.

Влияние фрактальной размерности на качество классификации компьютерных атак методами машинного обучения

For building an effective network protection system in computer network against attacks, a promising direction is joint use of fractal analysis and data mining. It is proposed to increase the efficiency of network attacks classification by introducing additional fractal dimension (FD) statistics of attacks along with other attributes. In contrast to the well-known works, it is proposed to further improve the efficiency of classifying network attacks by using not only the average value, but also other statistical characteristics of the DF of attacks and normal traffic as information features. These can be variance, skewness and kurtosis coefficients that characterize the shape and parameters of the distribution of the RF. The effectiveness of the proposed method is evaluated using machine learning algorithms by assessing the quality of the binary classification of network attacks and normal traffic using the UNSW-NB15 database as an example. The following classification algorithms were used to classify the dataset: k-nearest neighbors (k-NN), multiple logistic regression (LR), decision tree (DTC), random forest (RF), ada boost. The following metrics were used to evaluate the effectiveness of the constructed models: accuracy (precision), recall (recall), F-score (F-score), ROC-curves, AUC-ROC. It is shown that the use of mean value, variance, skewness and kurtosis coefficients, which characterize the shape and distribution parameters of the statistical characteristics of the FD distribution as additional information features, makes it possible to increase the efficiency of attack classification by an average of 10%. K-NN and LR classification algorithms. For the DTC and RF algorithms, the greatest effect from the use of additional attributes is in reducing the training and testing time and is about 3.5 times for each of the algorithms.

Magnetic resonance diagnostic criteria of non-compaction cardiomyopathy: new diagnostic criteria still needed?

Summary Non-compaction cardiomyopathy (NCC) is a rare pathology, but the exact rates of its prevalence are not known due to the lack of a diagnostic gold standard. The purpose of this article is to analyse the available cardiovascular magnetic resonance (CMR) diagnostic criteria of non-compaction described in the literature and to compare their sensitivity and specificity in the diagnosis of NCC. A search of available literature related to the CMR diagnostic criteria of myocardial non-compaction was conducted in the medical database PubMed in February of 2022. The period of publication of scientific articles covered the years from 1996 to 2022. A total of 7 full-text scientific articles were included in the final literature review. The main diagnostic criteria were used: the maximum non-compact (NCM) to compact myocardial layers (CM) ratio (NCM:CM), the percentage of trabeculated left ventricular (LV) myocardial mass, the percentage of trabeculated LV myocardial volume, the non-compact myocardial mass index of the total LV, and the determination of the total LV and the maximal fractal dimension (FD) of the apex with the use of fractal analysis. The lack of accurate diagnostic criteria results in an overdiagnosis of NCC. The highest sensitivity and specificity are associated with the maximum FD &gt; 1.30 of the apex established by applying the fractal analysis method. Fractal analysis requires dedicated software, and this method is difficult to apply in routine clinical practice. Thus, the diagnostic criteria for the NCC using magnetic resonance imaging with higher diagnostic value remain to be sought.

METHOD FOR VIBROACOUSTIC DIAGNOSIS OF BEARING SUPPORTS OF ROTORS OF GAS TURBINE ENGINES ON THE BASIS OF FRACTAL ANALYSIS

Currently, more than 45.000 aircraft are operated in the world, most of them are equipped with gas turbine engines. Aircraft engines that have not exhausted the established resources and service life are removed from operation annually. As a rule, the reasons for early termination of operation are defects and malfunctions of individual engine components, their mechanical damage, as well as human errors that lead to a malfunction of the engines. Based on the analysis of the causes of failures and damages of the engines of tactical aviation aircraft, it can be concluded that in the vast majority of cases, the failures manifested themselves in increased vibration, which indicated the destruction of the bearing supports of the compressor and turbine rotors, the failure of which may lead to shutdown and damage to the gas turbine engine. Such consequences adversely affect flight safety and can lead to both the loss of the aircraft and the death of the crew. The specifics of the location of the bearing supports of the rotors greatly complicates their diagnostics in the framework of non-destructive testing. The article proposes a new method for vibroacoustic diagnostics of bearing supports of rotors of gas turbine engines, based on the use of fractal analysis. A hardware-software complex has been created that allows diagnosing the technical condition of the bearing supports of the rotors of gas turbine engines.

Study on Pore Structure and Fractal Characteristics of Tar-Rich Coal during Pyrolysis by Mercury Intrusion Porosimetry (MIP)

This study aims to show how fractal analysis can be effectively used to characterize the pore structure of porous tar-rich coal. In this study, tar-rich coal pores were obtained by mercury intrusion porosimetry (MIP). The results showed that the sample had a high porosity and large pore diameter after pyrolysis, and the porosity of tar-rich coal was more than 35% at 600°C. The pore-throat ratio at high temperature was large, resulting in the high mercury retention rate. The pore distribution curves of samples at pyrolysis temperatures of 500 and 600°C were unimodal, and those of samples at room temperature and 150, 300, and 400°C were bimodal. The models for calculating fractal dimension based on MIP include Menger sponge model and thermodynamic model. Experiments show that the Menger sponge method is more reasonable when the pore size is less than 50 μm and greater than 350 nm. For the fractal dimension calculation of the whole pore curve section, the thermodynamic method was more reasonable. The use of fractal analysis in conjunction with the results of classical characterization methods leads to a better understanding of pore evolution in the pyrolysis process of tar-rich coal. The average fractal dimension could also be used as a valid parameter to monitor the textural evolution of the coals.

Tool Condition Monitoring Using Machine Tool Spindle Electric Current and Multiscale Analysis while Milling Steel Alloy

In the metal cutting process, the tool condition directly affects the quality of the machined component. To control the quality of the cutting tool and avoid equipment downtime, it is essential to monitor its condition during the machining process. The primary purpose is to send a warning before tool wear reaches a certain level, which could influence product quality. In this paper, tool condition is monitored using fractal analysis of the spindle electric current signal. The current study analyzes the monitoring of the cutting tool when milling AISI 5140 steel with a four-flute solid carbide end mill cutter to develop monitoring techniques for wear classification of metal cutting processes. The spindle electric current signal is acquired using the machine tool internal sensor, which meets industrial constraints in their operating conditions. As a new approach, the fractal theory is referred to analyze the spindle electric current signal and then assess the tool wear condition during the metal cutting process. Fractal parameters were defined to extract significant characteristic features of the signal. This research provides a proof of concept for the use of fractal analysis as a decision-making strategy in monitoring tool condition.

МЕТОД ВЫЯВЛЕНИЯ АНОМАЛИЙ В СЕТЕВОМ ТРАФИКЕ

Introduction. Computer networks (CN) are highly developed systems with a multi-level hierarchical structure. The use of information and communication technologies in the CN to collect information allows an attacker to influence networks through cyber-attacks. This is facilitated by the massive use of outdated operating systems, ineffective protection mechanisms and the presence of multiple vulnerabilities in unsecured network protocols. Such vulnerabilities help a potential attacker to change the settings of network devices, listen and redirect traffic, block network interaction and gain unauthorized access to the internal components of the CN. The impact of cyber-attacks leads to the appearance of abnormal traffic activity in the CN. For its constant monitoring and detection in the CN, it is necessary to take into account the presence of a large number of network routes, on which sharp fluctuations in data transmission delays and large packet losses periodically occur, new properties of network traffic appear, which requires ensuring high quality of application service. All this served as an incentive to search for new methods of detecting and predicting cyber-attacks fractal analysis can also be attributed to them. The aim of the work is to develop a conceptual method for detecting anomalies caused by cyber-attacks in network traffic through the use of fractal analysis. Methods used. The main provisions of the fractal theory and the use of self-similarity assessment methods proposed by this theory, such as the extended Dickey-Fuller test, R/S analysis and the DFA method, are applied. When testing fractal methods that allow conducting studies of long-term dependencies in network traffic. The scientific novelty lies in the fact that the proposed method correctly identifies anomalies caused by the impact of cyber-attacks, and also allows you to predict and detect both known and unknown computer attacks at an early stage of their manifestation. Practical significance. The presented methodology can be used as an early detection system for cyber-attacks, based on the detection of anomalies in network traffic and the adoption of effective measures to protect the network.

Usefulness of Fractal Analysis of Kirsch Edge Images for the Tissue Fragment Inner Structure in Breast FNAB

Objective: Recent advances in high-precision mammography and ultrasound screening have led to an increase in the detection of early lesions (ductal carcinoma in situ and small cancers) appearing as microcalcified lesions or microcystic images, and there needs to be an improvement in the accuracy of breast fine-needle aspiration biopsy (FNAB) assessing these lesions. The objective of this study was to investigate whether fractal analysis of Kirsch edge images for the tissue fragment inner structure (FKT) is useful in breast FNAB. FKT measures tissue fragment chromasia of hyperchromatic crowded tissue fragments (HCG), tissue fragment shape unevenness, and tissue fragment inner structure complexity. Study Design Materials: Nineteen epithelial tissue fragments of fibroadenoma (FA) from 7 patients and 52 tissue fragments of invasive breast carcinoma of no special type (IBC-NST) (grade 1–2) from 11 patients were assessed. First, tissue fragments were classified into small (smaller than 60 × 102 μm²), medium, and large (100 × 102 μm² or larger), and the appearance rate of each size was determined. Second, for FKT, the luminance value of tissue fragment chromasia, the unevenness and fractal value, and the tissue fragment inner structure complexity were determined. In statistical analysis, the Steel-Dwass test, nonlinear discriminant analysis, and receiver operating characteristic analysis were performed, setting the significance level at p < 0.05. Results: “Unevenness of the tissue fragment shape,” “fractal value of the tissue fragment shape,” and “fractal value of the tissue fragment inner structure” were significantly higher in small and large tissue fragments in IBC-NST compared with those in FA. The specificity and sensitivity were the highest (100%) in small tissue fragments in multivariate analysis using 4 variables (“luminance value of tissue fragment chromasia,” “unevenness of tissue fragment shape,” “fractal value of the tissue fragment shape,” and “fractal value of the tissue fragment inner structure”). Conclusion: FKT, which evaluates “tissue fragment darkness,” “tissue fragment shape unevenness,” and “tissue fragment inner structure complexity” focusing on small tissue fragments of HCG in breast FNAB, is useful as a system that assists cytopathological assessment of breast FNAB.

Retrospective fractal analyses of one-year follow-up data obtained after single-visit nonsurgical endodontic retreatment on periapical radiographs.

The aim of retrospective study was to evaluate the changes to the periapical bone sites after non-surgical endodontic retreatment by using fractal analysis on periapical radiographs. Thirty patients whose teeth had periapical lesions and underwent primary endodontic treatment were selected for nonsurgical single-visit endodontic retreatment. Two radiographs were taken, the first immediately after retreatment and the second at the 1-year follow-up. An identical region of interest close to the infected root apex was selected from each radiograph. FD was measured using the box-counting method. Periapical status was evaluated using the periapical index (PAI). Treatment outcomes were categorized into healed (PAI < 3), or not healed (PAI ≥ 3) based on radiographic criteria. The mean FD value significantly increased at the 1-year follow-up compared to baseline. No significant sex biases were apparent. According to the PAI, retreatment after 1year resulted in a 63.3% healed. The extent of periapical trabecular bone evident radiographically increased, as revealed by FD data, 1year after endodontic retreatment. Clinical relevance The use of fractal analysis as a quantitative method to evaluate changes in periapical trabecular bone (such as healing and inflammation) after endodontic retreatment can be an important marker in determining the prognosis of endodontic retreatment.

Some aspects of fluctuations dynamics of particles in dusty plasma

Dusty plasmas are formed when micrometre particles of solid matter are introduced into glow discharge plasma. In the present research work, we investigated the fluctuation dynamics of the particles using the dynamic light scattering (DLS) technique and particle tracking with high-speed imaging technique. The DLS technique gives light intensity fluctuation which is analysed by fractal tools namely wavelet scalogram to give the Hurst parameter, “H” being associated with the fractional Brownian motion. The structural phases of particles are analysed using the pair correlation function. Particle trajectory analysis determines the transport characteristics of the charged dust particles. The scaling behaviour of the mean squared displacement (MSD) is investigated for different pressures. Results from both the DLS and MSD analysis are discussed regarding different transport mechanisms based on the fractional Brownian motion as the transport model. This study also highlighted the use of fractal analysis as a powerful tool for investigating fluctuation dynamics in dusty plasmas.

Fractal analysis of human cerebellum based on magnetic resonance imaging data: pixel dilating method

Background. The use of fractal analysis as a morphometric method allows to investigate biological structures that have the properties of fractals, including the human cerebellum. Adaptation of fractal analysis techniques to assess the state of brain structures using magnetic resonance imaging is an important area of modern morphology. Objective: to determine the values of the fractal dimension (FD) of the cerebellar tissue measured by magnetic resonance imaging using the pixel dilation. Methods. The study was conducted on magnetic resonance images of the brain of 120 relatively healthy patients. Fractal analysis of digital magnetic resonance images of the cerebellum was performed using the method of pixel dilation in the author's modification. Results. It was found that the average value of FD of the cerebellar vermis on the midsagittal section on T2 weighed images with a brightness threshold of 100 was 1.691 ± 0.01. The FD values of the cerebellar hemisphere tissue were: in the paravermal zone on the left 1.683 ± 0.01, on the right 1.685 ± 0.01; in the central zone of the hemisphere on the left 1.679 ± 0.01, on the right 1.672 ± 0.01; on the lateral zone of the hemisphere on the left 1.665 ± 0.01, on the right 1.682 ± 0.01. These values do not differ statistically significantly in the symmetrical areas of the right and left hemispheres and do not differ from the FD values of the cerebellar vermis. Conclusion. The method of pixel dilation of fractal analysis can be used for morphometric study of magnetic resonance images of the brain. Fractal analysis allows an objective assessment of the morphofunctional condition of the cerebellum, which can be used to diagnose various diseases of the cerebellum and other CNS structures.

An Innovative Approach to Anomaly Detection in Communication Networks Using Multifractal Analysis

Fractal and multifractal analysis can help to discover the structure of the communication system, and in particular the pattern and characteristics of traffic, in order to understand the threats better and detect anomalies in network operation. The massive increase in the amount of data transmitted by different devices makes these systems the target of various types of attacks by cybercriminals. This article presents the use of fractal analysis in detecting threats and anomalies. The issues related to the construction and functioning of the Security Operations Centre (SOC) are presented. To examine the correctness of SOC, several attacks on virtual systems located in the network were carried out, such as Denial of Service (DoS) attack, brute force, malware infections, exploits. Based on data collected from monitoring and devices, the response to the event was analyzed, and multifractal spectra of network traffic before and during the incident were created. The collected information allows us to verify the theses and confirm the effectiveness of multifractal methods in detecting anomalies in the operation of any Information and Communication Technology (ICT) network. Such solutions will contribute to the development of advanced intrusion detection systems (IDS).

Use of fractal analysis to evaluate the surface quality of agricultural machinery parts

The research of the determination of the fractal characteristics of the surface of a material proposes the use of a stationary profilograph and a computer program for calculating the Hurst exponent. The low accuracy of fractal analysis using the well-known computer program Fractan is revealed. A computer program developed in VBA for the fractal analysis of the time series is described. The high accuracy of the algorithms for calculating the Hurst exponent incorporated in this program is shown.

Hybridized Machine Learning based Fractal Analysis Techniques for Breast Cancer Classification

The usefulness of Fractal Analysis (FA) is not limited to a particular area. It is applied in variety of fields and has shown its efficiency towards irregular objects. Fractal dimension is the best measure of the roughness for natural elements and hence, it can be treated as a feature of the natural object. Breast masses are irregular and divers from a malignant tumor to benign; hence breast can be treated as one of the best areas where fractal geometry can be applied. It gives a scope where fractal geometry concept can be used as a feature extraction technique in mammogram. On the other hand, the support vector machine is an emerging technique for classification. The survey shows that few works have done on breast mass classification using support vector machine. In our work two most effective techniques are used in separate operations, FA: Box Count Method (BCM) and Support Vector Machine (SVM) that result well in their fields. Feature extraction is done through Box Count Method. The extracted feature, “fractal dimension”, measures the complexity of the input data set of 42 images. For the next segment, the resulting Fractal Dimensions (FD) are processed under the support vector machine classifier to classify benign and malignant cells. The result analysis shows that the combination of SVM and FD yielded the highest with 98.13% accuracy.

Use of fractal analysis in dental images: a systematic review.

This study reviewed the use of fractal analysis (FA) in dental images. A search was performed using PubMed, MEDLINE, LILACS, Web of Science and SCOPUS databases. The inclusion criteria were human studies in the English language, with no date restriction. 78 articles were found in which FA was applied to panoramic radiographs (34), periapical radiographs (21), bitewing radiographs (4), cephalometric radiograph (1), cone beam CT (15), micro-CT (3), sialography (2), and ultrasound (2). Low bone mineral density (21) and systemic or local diseases (22) around the bone of dental implants were the main subjects of the study of FA. Various sizes and sites of the regions of interest were used to evaluate the bone structure. Different ways were used to treat the image and to calculate FA. FA of 43 articles showed significant differences in the comparison of groups, mainly between healthy and sick patients. FA in Dentistry has been widely applied to the study of images. Panoramic and periapical radiographs were those most frequently used. The Image J software and the box-counting method were extensively adopted in the studies reviewed herein. Further studies are encouraged to improve clarification of the parameters that directly influence FA.

Precursory Analysis of GPS Time Series for Seismic Hazard Assessment

We discuss the use of fractal analysis of geophysical big data as a precursory method for the estimation of current seismic hazard. We begin by analyzing GPS time series of land surface displacements recorded in various regions of the planet exhibiting different degrees of seismic activity. We show how the quantitative measure of chaoticity based on a modified detrended fluctuation analysis (DFA) method can be used to distinguish critical and noncritical regimes of evolution of the Earth’s crust. Having established the reference values for the measure of chaoticity, we then choose California as the case study region and apply modified DFA to over 1200 time series for determining local areas currently exposed to higher seismic hazard. We comprehensively analyze the results obtained and compare them with those produced by conventional risk assessment methods. We demonstrate consistency of our results and yet discover that the suggested approach allows us to quantify two qualitatively different mechanisms involved in the growth of seismic hazards. We propose a hypothesis which relates these mechanisms to the known mechanisms of earthquake generation that evince a growth of collective behavior in the Earth’s crust on the eve of an earthquake. In order to verify the hypothesis, we analyze the same GPS data by an independent statistical method, canonical coherence analysis (CCA), which permits us to quantify the degree of synchronized behavior of constituents of the Earth’s crust. The CCA method confirms our previously obtained estimates, thus supporting fractal analysis of GPS time series as a suitable precursory method for current seismic hazard assessment.