Four Methods to Estimate the Fractal Dimension from Self-Affine Signals

Big Data Analysis Framework for Water Quality Indicators with Assimilation of IoT and ML

According to the United Nations, the Sustainable Development Goal ‘6’ seeks to ensure the availability and sustainable management of water for all. Digital technologies, such as big data, Internet of Things (IoT), and machine learning (ML) have a significant role and capability to meet the goal. Water quality analysis in any region is critical to identify and understand the standard of water quality and the quality of water is analyzed based on water quality parameters (WQP). Currently, water pollution and the scarcity of water are two major concerns in the region of Uttarakhand, and the analysis of water before it is supplied for human consumption has gained attention. In this study, a big data analytics framework is proposed to analyze the water quality parameters of 13 districts of Uttarakhand and find the correlation among the parameters with the assimilation of IoT and ML. During the analysis, statistical and fractal methods are implemented to understand the anomalies between the water quality parameters in 13 districts of Uttarakhand. The variation in WQP is analyzed using a random forest (RF) model, and the dataset is segmented location wise and the mean, mode, standard deviation, median, kurtosis, and skewness of time series datasets are examined. The mean of the parameters is adjusted with the coefficient of variation based on the standard values of each parameter. The turbidity in almost all the experimental sites has a normal distribution, with the lowest mean value (0.352 mg/L) and highest (11.9 mg/L) in the Pauri Garhwal and Almora districts, respectively. The pH of the water samples is observed to be in the standard range in all the experimental sites, with average and median values being nearly identical, at 7.189 and 7.20, respectively. However, the pH mode is 0.25. The Cl− concentration varies with mean values from the lowest (0.46 mg/L) to the highest (35.2 mg/L) over the experimental sites, i.e., the Bageshwar and Rudraprayag districts, respectively. Based on the analysis, it was concluded that the water samples were found to be safe to drink and in healthy condition in almost all the districts of the state Uttarakhand, except for the Haridwar district, where some increase in contaminants was observed.

Fractality of tics as a quantitative assessment tool for Tourette syndrome

Tics manifest as brief, purposeless and unintentional movements or noises that, for many individuals, can be suppressed temporarily with effort. Previous work has hypothesized that the chaotic temporal nature of tics could possess an inherent fractality, that is, have neighbour-to-neighbour correlation at all levels of timescale. However, demonstrating this phenomenon has eluded researchers for more than two decades, primarily because of the challenges associated with estimating the scale-invariant, power law exponent—called the fractal dimension D_f—from fractional Brownian noise. Here, we confirm this hypothesis and establish the fractality of tics by examining two tic time series datasets collected 6–12 months apart in children with tics, using random walk models and directional statistics. We find that D_f is correlated with tic severity as measured by the YGTTS total tic score, and that D_f is a sensitive parameter in examining the effect of several tic suppression conditions on the tic time series. Our findings pave the way for using the fractal nature of tics as a robust quantitative tool for estimating tic severity and treatment effectiveness, as well as a possible marker for differentiating typical from functional tics.

The Influence of Silver Nanoparticle Form on the Toxicity in Freshwater Mussels

The contribution of the form of silver nanomaterials (nAg) towards toxicity in aquatic organisms is not well understood. The purpose of this study was to examine the toxicity of various structures (sphere, cube and prism) of nAg in Dreissena bugensis mussels. Mussels were exposed to increasing concentrations of polyvinyl-coated nAg of the same size for 96 h at 15 °C. They were then analyzed for biophysical changes in the cytoplasm (viscosity, protein aggregation and lipids), neuro-activity (fractal kinetics of acetylcholinesterase (AChE)), oxidative stress (labile zinc (Zn) and lipid peroxidation) and inflammation (arachidonate cyclooxygenase). Although some decreasing effects in protein aggregation were observed, viscosity was more strongly decreased in mussels exposed to spheric and prismatic nAg. The activity of AChE was significantly decreased in the following form-dependent manner: prismatic > cubic > spheric nAg. The fractal dimension of AChE reactions was reduced by all geometries of nAg, while dissolved Ag had no effects. For nanoparticles with the same coating and relative size, spheric nAg produced more significant changes towards the fractal dimension of AChE, while prismatic nAg increased both protein aggregation and viscosity, whereas cubic nAg decreased protein aggregation in the cytoplasm. It is concluded that the geometries of nanoparticles could influence toxicity in aquatic organisms.

Non-linear Methods Predominant in Fetal Heart Rate Analysis: A Systematic Review

The analysis of fetal heart rate variability has served as a scientific and diagnostic tool to quantify cardiac activity fluctuations, being good indicators of fetal well-being. Many mathematical analyses were proposed to evaluate fetal heart rate variability. We focused on non-linear analysis based on concepts of chaos, fractality, and complexity: entropies, compression, fractal analysis, and wavelets. These methods have been successfully applied in the signal processing phase and increase knowledge about cardiovascular dynamics in healthy and pathological fetuses. This review summarizes those methods and investigates how non-linear measures are related to each paper's research objectives. Of the 388 articles obtained in the PubMed/Medline database and of the 421 articles in the Web of Science database, 270 articles were included in the review after all exclusion criteria were applied. While approximate entropy is the most used method in classification papers, in signal processing, the most used non-linear method was Daubechies wavelets. The top five primary research objectives covered by the selected papers were detection of signal processing, hypoxia, maturation or gestational age, intrauterine growth restriction, and fetal distress. This review shows that non-linear indices can be used to assess numerous prenatal conditions. However, they are not yet applied in clinical practice due to some critical concerns. Some studies show that the combination of several linear and non-linear indices would be ideal for improving the analysis of the fetus's well-being. Future studies should narrow the research question so a meta-analysis could be performed, probing the indices' performance.

Modified Catalysts and Their Fractal Properties

Obtaining high-area catalysts is in demand in heterogeneous catalysis as it influences the ratio between the number of active surface sites and the number of total surface sites of the catalysts. From this point of view, fractal theory seems to be a suitable instrument to characterize catalysts’ surfaces. Moreover, catalysts with higher fractal dimensions will perform better in catalytic reactions. Modifying catalysts to increase their fractal dimension is a constant concern in heterogeneous catalysis. In this paper, scientific results related to oxide catalysts, such as lanthanum cobaltites and ferrites with perovskite structure, and nanoparticle catalysts (such as Pt, Rh, Pt-Cu, etc.) will be reviewed, emphasizing their fractal properties and the influence of their modification on both fractal and catalytic properties. Some of the methods used to compute the fractal dimension of the catalysts (micrograph fractal analysis and the adsorption isotherm method) and the computed fractal dimensions will be presented and discussed.

CEPS: An Open Access MATLAB Graphical User Interface (GUI) for the Analysis of Complexity and Entropy in Physiological Signals

BACKGROUND

We developed CEPS as an open access MATLAB^® GUI (graphical user interface) for the analysis of Complexity and Entropy in Physiological Signals (CEPS), and demonstrate its use with an example data set that shows the effects of paced breathing (PB) on variability of heart, pulse and respiration rates. CEPS is also sufficiently adaptable to be used for other time series physiological data such as EEG (electroencephalography), postural sway or temperature measurements.

METHODS

Data were collected from a convenience sample of nine healthy adults in a pilot for a larger study investigating the effects on vagal tone of breathing paced at various different rates, part of a development programme for a home training stress reduction system.

RESULTS

The current version of CEPS focuses on those complexity and entropy measures that appear most frequently in the literature, together with some recently introduced entropy measures which may have advantages over those that are more established. Ten methods of estimating data complexity are currently included, and some 28 entropy measures. The GUI also includes a section for data pre-processing and standard ancillary methods to enable parameter estimation of embedding dimension m and time delay τ ('tau') where required. The software is freely available under version 3 of the GNU Lesser General Public License (LGPLv3) for non-commercial users. CEPS can be downloaded from Bitbucket. In our illustration on PB, most complexity and entropy measures decreased significantly in response to breathing at 7 breaths per minute, differentiating more clearly than conventional linear, time- and frequency-domain measures between breathing states. In contrast, Higuchi fractal dimension increased during paced breathing.

CONCLUSIONS

We have developed CEPS software as a physiological data visualiser able to integrate state of the art techniques. The interface is designed for clinical research and has a structure designed for integrating new tools. The aim is to strengthen collaboration between clinicians and the biomedical community, as demonstrated here by using CEPS to analyse various physiological responses to paced breathing.

Crowding Effects of Polystyrene Nanoparticles on Lactate Dehydrogenase Activity in Hydra Attenuata

Plastics pervade our environment and potentially release important quantities of plastic nanoparticles (NPs) from degradation in the environment. The purpose of this study was to examine the crowding effects of polystyrene NPs on lactate dehydrogenase (LDH) in vitro and following exposure to Hydra attenuata. First, LDH activity was measured in vitro in the presence of filamentous (F-)actin and NPs (50 and 100 nm diameter) to determine changes in viscosity and the fractal kinetics of LDH. The fractal dimension (fD) was also determined using the rescaled range analysis procedure. Secondly, these changes were examined in hydra exposed to NPs for 96h to concentrations of NPs. The data revealed that the addition of F-actin increased the rate of LDH at low substrate (pyruvate) concentrations compared to LDH alone with a gradual decrease in the rate with the addition of pyruvate, which is characteristic of the fractal behavior of enzymes in crowded environments. The addition of 50 and 100 nm NPs also produced these changes, which suggest that NPs could change the space properties of the LDH reaction. The fD was reduced to 0.85 and 0.91 with 50 and 100 nm NPs compared to 1.093 with LDH alone. Decrease in the fD was related with increased amplitudes and frequency in viscosity waves in the reaction media. Exposure of hydra to NPs confirmed the increase in LDH activity and the fD was significantly correlated with LDH activity (r = -0.5). Correction of LDH activity (residuals) still revealed an increase in LDH activity in hydra suggesting increased anaerobic metabolism by NPs. In conclusion, the presence of NPs in the intracellular space decreased the fD, which could influence LDH activity in organisms exposed to NPs.

Biophysical effects of polystyrene nanoparticles on Elliptio complanata mussels

The presence of nanoplastics (NPs) in various  products and from the weathering of released plastic materials are of concern for the environment's safety. The purpose of this study was to examine the biophysical effects of polystyrene NPs on freshwater mussels. Mussels were exposed to a range of concentrations of NPs (0.1, 0.5, 1, and 5 mg/L) for 24 h and allowed to depurate for 12 h in clean aquarium water. The digestive gland was isolated and analyzed for NPs, lipids, viscosity, protein aggregation, anisotropic changes (liquid crystals: LCs), and the oscillatory modulation in viscosity during the formation of self-organizing enzyme complex of fumarase, malate dehydrogenase, and citrate synthase. The results revealed that mussels accumulated NPs in the digestive gland and their levels were significantly correlated with lipids levels, LCs, the increase in the malate dehydrogenase/citrate synthase activity ratio, and oscillations in viscosity. Protein aggregation was also found to be correlated with lipid levels. The data suggests that the presence of NPs in the digestive gland involves changes in lipid content and LC formation and perturbs the normal oscillations in viscosity during sequential enzyme reactions of the above enzymes. It is concluded that the uptake of NPs in cells could disrupt the internal organization of cells which can interfere with the normal association of enzymes involved in energy metabolism.

Temporal evolution of sample entropy in thermal lens system

The photothermal phenomenon resulting in thermal lens formation in liquid media involves complex molecular dynamics responsible for temperature and refractive index variation. As a thermodynamic system, the entropy of the medium also changes. In this paper, the time series and phase portrait analysis of the thermal lens signal is carried out to understand the molecular dynamics. The study reveals the increase in complexity, disorder, and antipersistance nature through fractal dimension, sample entropy, and Hurst exponent, respectively. The analysis of the signal on segmentation reveals the evolution of sample entropy and the stochastic nature of the system with time. The phase portrait analysis also is in support of these observations. Thus, the study suggests that the temporal evolution of sample entropy is similar to the temperature-dependent refractive index.

On fractality of functional near-infrared spectroscopy signals: analysis and applications

Significance: The human brain is a highly complex system with nonlinear, dynamic behavior. A majority of brain imaging studies employing functional near-infrared spectroscopy (fNIRS), however, have considered only the spatial domain and have ignored the temporal properties of fNIRS recordings. Methods capable of revealing nonlinearities in fNIRS recordings can provide new insights about how the brain functions. Aim: The temporal characteristics of fNIRS signals are explored by comprehensively investigating their fractal properties. Approach: Fractality of fNIRS signals is analyzed using scaled windowed variance (SWV), as well as using visibility graph (VG), a method which converts a given time series into a graph. Additionally, the fractality of fNIRS signals obtained under resting-state and task-based conditions is compared, and the application of fractality in differentiating brain states is demonstrated for the first time via various classification approaches. Results: Results from SWV analysis show the existence of high fractality in fNIRS recordings. It is shown that differences in the temporal characteristics of fNIRS signals related to task-based and resting-state conditions can be revealed via the VGs constructed for each case. Conclusions: fNIRS recordings, regardless of the experimental conditions, exhibit high fractality. Furthermore, VG-based metrics can be employed to differentiate rest and task-execution brain states.

Nonlinear Methods Most Applied to Heart-Rate Time Series: A Review

The heart-rate dynamics are one of the most analyzed physiological interactions. Many mathematical methods were proposed to evaluate heart-rate variability. These methods have been successfully applied in research to expand knowledge concerning the cardiovascular dynamics in healthy as well as in pathological conditions. Notwithstanding, they are still far from clinical practice. In this paper, we aim to review the nonlinear methods most used to assess heart-rate dynamics. We focused on methods based on concepts of chaos, fractality, and complexity: Poincaré plot, recurrence plot analysis, fractal dimension (and the correlation dimension), detrended fluctuation analysis, Hurst exponent, Lyapunov exponent entropies (Shannon, conditional, approximate, sample entropy, and multiscale entropy), and symbolic dynamics. We present the description of the methods along with their most notable applications.

Compound speckle model detects anti-angiogenic tumor response in preclinical nonlinear contrast-enhanced ultrasonography

PURPOSE

This paper proposes a method for analyzing the first-order speckle statistics of nonlinear contrast-enhanced ultrasound images from tumors.

METHODS

Contrast signal intensity is modeled as a compound distribution of exponential probability density functions with a gamma weighting function. The gamma probability weighting function serves as an approximation for log-normally distributed flow velocities in a vascular network. The model was applied to sub-harmonic bolus-injection images acquired from a mouse breast cancer xenograft model treated with murine version bevacizumab.

RESULTS

The area under curve produced using the compound statistical model could more accurately discriminate anti-VEGF-treated tumors from untreated tumors than conventional contrast-enhanced ultrasound image processing. This result was validated with gold standard histological measures of microvascular density. Fractal vessel geometry was estimated using the gamma weighting function and tested against micro-CT perfusion casting. Treated tumors had a significantly lower vascular fractal dimension than control tumors. Vascular complexity estimated using the ultrasound compound statistical model performed similarly to micro-CT fractal dimension for discriminating treated from control tumors.

CONCLUSION

The proposed technique can quantify tumor perfusion and provide an index of vascular complexity, making it a potentially useful addition for clinical detection of vascular normalization in anti-angiogenic trials.

Advances in heart rate variability signal analysis: joint position statement by the e-Cardiology ESC Working Group and the European Heart Rhythm Association co-endorsed by the Asia Pacific Heart Rhythm Society

Following the publication of the Task Force document on heart rate variability (HRV) in 1996, a number of articles have been published to describe new HRV methodologies and their application in different physiological and clinical studies. This document presents a critical review of the new methods. A particular attention has been paid to methodologies that have not been reported in the 1996 standardization document but have been more recently tested in sufficiently sized populations. The following methods were considered: Long-range correlation and fractal analysis; Short-term complexity; Entropy and regularity; and Nonlinear dynamical systems and chaotic behaviour. For each of these methods, technical aspects, clinical achievements, and suggestions for clinical application were reviewed. While the novel approaches have contributed in the technical understanding of the signal character of HRV, their success in developing new clinical tools, such as those for the identification of high-risk patients, has been rather limited. Available results obtained in selected populations of patients by specialized laboratories are nevertheless of interest but new prospective studies are needed. The investigation of new parameters, descriptive of the complex regulation mechanisms of heart rate, has to be encouraged because not all information in the HRV signal is captured by traditional methods. The new technologies thus could provide after proper validation, additional physiological, and clinical meaning. Multidisciplinary dialogue and specialized courses in the combination of clinical cardiology and complex signal processing methods seem warranted for further advances in studies of cardiac oscillations and in the understanding normal and abnormal cardiac control processes.

Nonlinear complexity analysis of brain FMRI signals in schizophrenia

We investigated the differences in brain fMRI signal complexity in patients with schizophrenia while performing the Cyberball social exclusion task, using measures of Sample entropy and Hurst exponent (H). 13 patients meeting diagnostic and Statistical Manual of Mental Disorders, 4th Edition (DSM IV) criteria for schizophrenia and 16 healthy controls underwent fMRI scanning at 1.5 T. The fMRI data of both groups of participants were pre-processed, the entropy characterized and the Hurst exponent extracted. Whole brain entropy and H maps of the groups were generated and analysed. The results after adjusting for age and sex differences together show that patients with schizophrenia exhibited higher complexity than healthy controls, at mean whole brain and regional levels. Also, both Sample entropy and Hurst exponent agree that patients with schizophrenia have more complex fMRI signals than healthy controls. These results suggest that schizophrenia is associated with more complex signal patterns when compared to healthy controls, supporting the increase in complexity hypothesis, where system complexity increases with age or disease, and also consistent with the notion that schizophrenia is characterised by a dysregulation of the nonlinear dynamics of underlying neuronal systems.

Behavioral and emotional dynamics of two people struggling to reach consensus about a topic on which they disagree

We studied the behavioral and emotional dynamics displayed by two people trying to resolve a conflict. 59 groups of two people were asked to talk for 20 minutes to try to reach a consensus about a topic on which they disagreed. The topics were abortion, affirmative action, death penalty, and euthanasia. Behavior data were determined from audio recordings where each second of the conversation was assessed as proself, neutral, or prosocial. We determined the probability density function of the durations of time spent in each behavioral state. These durations were well fit by a stretched exponential distribution, with an exponent, , of approximately 0.3. This indicates that the switching between behavioral states is not a random Markov process, but one where the probability to switch behavioral states decreases with the time already spent in that behavioral state. The degree of this “memory” was stronger in those groups who did not reach a consensus and where the conflict grew more destructive than in those that did. Emotion data were measured by having each person listen to the audio recording and moving a computer mouse to recall their negative or positive emotional valence at each moment in the conversation. We used the Hurst rescaled range analysis and power spectrum to determine the correlations in the fluctuations of the emotional valence. The emotional valence was well described by a random walk whose increments were uncorrelated. Thus, the behavior data demonstrated a “memory” of the duration already spent in a behavioral state while the emotion data fluctuated as a random walk whose steps did not have a “memory” of previous steps. This work demonstrates that statistical analysis, more commonly used to analyze physical phenomena, can also shed interesting light on the dynamics of processes in social psychology and conflict management.

A comparative analysis of spectral exponent estimation techniques for 1/f(β) processes with applications to the analysis of stride interval time series

BACKGROUND

The time evolution and complex interactions of many nonlinear systems, such as in the human body, result in fractal types of parameter outcomes that exhibit self similarity over long time scales by a power law in the frequency spectrum S(f)=1/f(β). The scaling exponent β is thus often interpreted as a "biomarker" of relative health and decline.

NEW METHOD

This paper presents a thorough comparative numerical analysis of fractal characterization techniques with specific consideration given to experimentally measured gait stride interval time series. The ideal fractal signals generated in the numerical analysis are constrained under varying lengths and biases indicative of a range of physiologically conceivable fractal signals. This analysis is to complement previous investigations of fractal characteristics in healthy and pathological gait stride interval time series, with which this study is compared.

RESULTS

The results of our analysis showed that the averaged wavelet coefficient method consistently yielded the most accurate results.

COMPARISON WITH EXISTING METHODS

Class dependent methods proved to be unsuitable for physiological time series. Detrended fluctuation analysis as most prevailing method in the literature exhibited large estimation variances.

CONCLUSIONS

The comparative numerical analysis and experimental applications provide a thorough basis for determining an appropriate and robust method for measuring and comparing a physiologically meaningful biomarker, the spectral index β. In consideration of the constraints of application, we note the significant drawbacks of detrended fluctuation analysis and conclude that the averaged wavelet coefficient method can provide reasonable consistency and accuracy for characterizing these fractal time series.

Detection of architectural distortion in prior mammograms via analysis of oriented patterns

We demonstrate methods for the detection of architectural distortion in prior mammograms of interval-cancer cases based on analysis of the orientation of breast tissue patterns in mammograms. We hypothesize that architectural distortion modifies the normal orientation of breast tissue patterns in mammographic images before the formation of masses or tumors. In the initial steps of our methods, the oriented structures in a given mammogram are analyzed using Gabor filters and phase portraits to detect node-like sites of radiating or intersecting tissue patterns. Each detected site is then characterized using the node value, fractal dimension, and a measure of angular dispersion specifically designed to represent spiculating patterns associated with architectural distortion. Our methods were tested with a database of 106 prior mammograms of 56 interval-cancer cases and 52 mammograms of 13 normal cases using the features developed for the characterization of architectural distortion, pattern classification via quadratic discriminant analysis, and validation with the leave-one-patient out procedure. According to the results of free-response receiver operating characteristic analysis, our methods have demonstrated the capability to detect architectural distortion in prior mammograms, taken 15 months (on the average) before clinical diagnosis of breast cancer, with a sensitivity of 80% at about five false positives per patient.

Analysis of Intracranial Pressure

The monitoring of intracranial pressure (ICP) is an important tool in medicine for its ability to portray the brain’s compliance status. The bedside monitor displays the ICP waveform and intermittent mean values to guide physicians in the management of patients, particularly those having sustained a traumatic brain injury. Researchers in the fields of engineering and physics have investigated various mathematical analysis techniques applicable to the waveform in order to extract additional diagnostic and prognostic information, although they largely remain limited to research applications. The purpose of this review is to present the current techniques used to monitor and interpret ICP and explore the potential of using advanced mathematical techniques to provide information about system perturbations from states of homeostasis. We discuss the limits of each proposed technique and we propose that nonlinear analysis could be a reliable approach to describe ICP signals over time, with the fractal dimension as a potential predictive clinically meaningful biomarker. Our goal is to stimulate translational research that can move modern analysis of ICP using these techniques into widespread practical use, and to investigate to the clinical utility of a tool capable of simplifying multiple variables obtained from various sensors.

Shape Analysis of Breast Masses in Mammograms via the Fractal Dimension

Masses due to benign breast diseases and tumors due to breast cancer present significantly different shapes on mammograms. In general, malignant tumors appear with rough and complex boundaries or contours, whereas benign masses present smooth, round, or oval contours. Fractal analysis may be used to derive shape features to perform pattern classification of breast masses and tumors. Several procedures have been proposed to compute the fractal dimension of various types of objects or regions of interest in biomedical images, among which the box-counting and ruler methods are popular. In this study, we applied the two methods mentioned above to compute the fractal dimension of both the two-dimensional (2D) contours of breast masses and tumors, as well as their one-dimensional (1D) signatures. A comparative analysis was performed to assess the performance of the two methods of computing the fractal dimension and the two methods of representing the boundaries of masses. It was observed that analysis of the 2D contour representation with the ruler method resulted in the highest classification accuracy of up to 0.946, as indicated by the area under the receiver operating characteristics (ROC) curve. The results indicate that the fractal dimension can serve as a good shape feature for the benign-versus-malignant classification of breast masses in mammograms.

ON THE USE OF MULTIFRACTAL ANALYSIS AND GENETIC ALGORITHMS FOR THE SEGMENTATION OF CERVICAL CELL IMAGES

This paper deals with the segmentation problem of cervical cell images. This segmentation consists of separating each cell into its core and its cytoplasm. The mentioned separation is part of a uterus cancer recognition system based on the morphology of both the cell core and the cell cytoplasm. In this paper, we propose to perform the above separation by using a multifractal algorithm. An important feature of the proposed algorithm is its low computation cost. To increase the quality of the segmentation, we propose an optimization step based on genetic algorithms. The proposed processing has been tested on several images. Herein, we present only the results obtained for three different cell images.

Application of fractal analysis to mammography

We report on a morphological study of 192 breast masses as seen in mammograms, with the aim of discrimination between benign masses and malignant tumors. From the contour of each mass, we computed the fractal dimension (FD) and a few shape factors, including compactness, fractional concavity, and spiculation index. We calculated FD using four different methods: the ruler and box-counting methods applied to each 2-dimensional (2D) contour and its 1-dimensional signature. The ANOVA test indicated statistically significant differences in the values of the various shape features between benign masses and malignant tumors. Analysis using receiver operating characteristics indicated the area under the curve, A(z), of up to 0.92 with the individual shape features. The combination of compactness, FD with the 2D ruler method, and the spiculation index resulted in the highest A(z) value of 0.93.

Local scale exponents of blood pressure and heart rate variability by detrended fluctuation analysis: effects of posture, exercise, and aging

Heart rate self-affinity is often assessed by detrended fluctuations analysis, obtaining two coefficients only: a short-term (alpha(1)) exponent and a long-term (alpha(2)) exponent. Our aim is to show the limits of this approach and alternatively propose the estimation of the whole spectrum of local exponents alpha(n) for heart rate and blood pressure. To illustrate the advantages of this approach, we assess the effects of autonomic activations and age on alpha(n). We measured ECG and arterial pressure in 60 volunteers for 10 min, considering three conditions at increasing sympathetic activation: supine rest, sitting, and sitting during exercise. We computed alpha(n) of R-R intervals and systolic, mean, and diastolic blood pressures, as the slope of the detrended fluctuations function in a log-log plot. Volunteers were divided into age groups and compared. Results indicate that: 1) alpha(1) cannot be defined because short-term coefficients decrease with n, while alpha(2) cannot be defined only for blood pressure during supine rest; 2) heart rate and blood pressure scaling structures differ during supine rest but not during exercise; and 3) age effects appear mainly in supine rest, explaining discrepant results in literature. In conclusion, we recommend estimating the whole alpha(n) spectrum before possibly providing the "two-exponent" description only.

Nonlinear analysis of electroencephalogram in schizophrenia patients with persistent auditory hallucination

OBJECTIVE

The recent nonlinear analyses of electroencephalogram (EEG) data have shown that the correlation dimension (D2) reflects the degree of integration of information processing in the brain. There is now considerable evidence that auditory hallucination (AH) reflects dysfunctional gamma and beta frequency oscillations. Gamma oscillations are thought to reflect internally driven representations of objects, and the occurrence of subsequent beta oscillations can reflect the modification of the neuronal circuitry used to encode the sensory perception. The purpose of this study was to test whether AH in schizophrenia patients is reflected in abnormalities in D2 in their EEG, especially in the gamma and beta frequency bands.

METHODS

Twenty-five schizophrenia patients with a history of treatment-refractory AH over at least the past 2 years, and 23 schizophrenia patients with no AH (N-AH) within the past 2 years were recruited for the study. Artifact-free 30-s EEG epochs during rest were examined for D2.

RESULTS

The AH patients showed significantly increased gamma frequency D2 in Fp2 and decreased beta frequency D2 in the P3 region compared with the N-AH patients. These results imply that gamma frequency D2 in the right prefrontal cortex is more chaotic and that beta frequency D2 in the left parietal cortex is more coherent (less chaotic) in AH patients than in N-AH patients.

CONCLUSION

Our study supports the previous evidence indicating that gamma and beta oscillations are pivotal to AH, and also shows the distinctive dimensional complexity between the right prefrontal and left parietal cortexes as the underlying biological correlates of AH in schizophrenia patients.

Feature extraction from a signature based on the turning angle function for the classification of breast tumors

Malignant breast tumors and benign masses appear in mammograms with different shape characteristics: the former usually have rough, spiculated, or microlobulated contours, whereas the latter commonly have smooth, round, oval, or macrolobulated contours. Features that characterize shape roughness and complexity can assist in distinguishing between malignant tumors and benign masses. Signatures of contours may be used to analyze their shapes. We propose to use a signature based on the turning angle function of contours of breast masses to derive features that capture the characteristics of shape roughness as described above. We propose methods to derive an index of the presence of convex regions (XR ( TA )), an index of the presence of concave regions (VR ( TA )), an index of convexity (CX ( TA )), and two measures of fractal dimension (FD ( TA ) and FDd ( TA )) from the turning angle function. The methods were tested with a set of 111 contours of 65 benign masses and 46 malignant tumors with different parameters. The best classification accuracies in discriminating between benign masses and malignant tumors, obtained for XR ( TA ), VR ( TA ), CX ( TA ), FD ( TA ), and FDd ( TA ) in terms of the area under the receiver operating characteristics curve, were 0.92, 0.92, 0.93, 0.93, and, 0.92, respectively.

Fractal analysis of contours of breast masses in mammograms

Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. Breast masses present shape and gray-scale characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses based on shape. The fractal dimension of the contour of a mass may be computed either directly from the 2-dimensional (2D) contour or from a 1-dimensional (1D) signature derived from the contour. We present a study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box counting method applied to 1D and 2D representations of the contours. The methods were applied to a data set of 111 contours of breast masses. Receiver operating characteristics (ROC) analysis was performed to assess and compare the performance of fractal dimension and four previously developed shape factors in the classification of breast masses as benign or malignant. Fractal dimension was observed to complement the other shape factors, in particular fractional concavity, in the representation of the complexity of the contours. The combination of fractal dimension with fractional concavity yielded the highest area (A ( z )) under the ROC curve of 0.93; the two measures, on their own, resulted in A ( z ) values of 0.89 and 0.88, respectively.

Utility of traditional and alternative EMG-based measures of fatigue during low-moderate level isometric efforts

Traditional electromyographic (EMG) measures (e.g., amplitude, mean and median frequencies of the power spectra) have demonstrated inconsistent abilities in monitoring localized muscle fatigue at relatively low effort levels. In the present study, several alternative EMG-based fatigue indices were evaluated, derived using a logarithmic representation of the power spectrum, the fractal dimension of the raw signal, and a Poisson distribution fit to the power spectrum. These methods, along with traditional approaches, were applied to EMG data obtained from three separate experiments. In the first two experiments, 24 participants performed sustained isometric shoulder abductions and torso extensions at 30% of maximum voluntary strength (MVC). In the third experiment, another group of 12 participants performed similar shoulder exercises at 15% and 30% MVC, with repeatability assessed at 15% MVC. Both traditional and alternative EMG measures were analyzed for their 'utility', in terms of sensitivity to fatigue, variability, repeatability, and predictive ability. Results demonstrated that parameters derived from fractal analysis and the Poisson distribution demonstrated high utility. These alternative approaches appear promising as fatigue indices for low level isometric tasks.

HRV Scaling Exponent Identifies Postinfarction Patients Who Might Benefit From Prophylactic Treatment With Amiodarone

Analysing the Holter recordings collected at baseline during the European Myocardial Infarction Amiodarone Trial (EMIAT), we evaluate the possibility of using alpha, the slope of the power spectrum of heart rate variability signals (HRV) in the vicinity of f = 0, for postinfarction risk stratification. We found no relevant difference in the values of alpha for the placebo population. On the contrary, in the amiodarone arm, the distinction in the survival rates of those with high or low alpha-values was highly significant. Moreover, high risk patients with respect to alpha (higher values) did not seem to benefit from amiodarone. The results suggest that alpha might convey physiologic information that is different than what is expressed by other HRV characteristics, such as the triangular index. When combining high risk patients in term of triangular index (<20) and low risk patients with respect to alpha (<median), the difference in survival on placebo and amiodarone became very substantial (24.2% mortality on placebo, 8.7% on amiodarone, p = 0.017). This might offer a possibility of selecting patients likely to benefit from a prophylactic antiarrhythmic treatment after acute myocardial infarction.

Assessment of the Autonomic Control of Heart Rate Variability in Healthy and Spinal-Cord Injured Subjects: Contribution of Different Complexity-Based Estimators

We investigated how complexity-based estimators of heart rate variability can detect changes in cardiovascular autonomic drive with respect to traditional measures of variability. This was done by analyzing healthy subjects and paraplegic patients with different autonomic impairment due to low (vascular impairment only) or high (cardiac and vascular impairment) spinal cord injury, during progressive autonomic activations. While traditional techniques only quantified the effects of the autonomic activation, not distinguishing the effects of the lesion level, some recently proposed complexity estimators could also reveal the pathologic alterations in the autonomic control of heart rate. These estimators included the detrended fluctuation analysis coefficient (sensitive to both low and high autonomic lesions), sample entropy (sensitive to low-level lesions) and the largest Lyapunov exponent (sensitive to high-level lesions). Thus complexity-based methods provide information on the autonomic function from the heart rate dynamics that cannot be obtained by traditional techniques. This finding supports the combined use of both complexity-based and traditional methods to investigate the autonomic cardiovascular control from a more comprehensive perspective.

Estimation and interpretation of 1/falpha noise in human cognition

Recent analyses of serial correlations in cognitive tasks have provided preliminary evidence of the presence of a particular form of long-range serial dependence known as 1/f noise. It has been argued that long-range dependence has been largely ignored in mainstream cognitive psychology even though it accounts for a substantial proportion of variability in behavior (see, e.g., Gilden, 1997, 2001). In this article, we discuss the defining characteristics of long-range dependence and argue that claims about its presence need to be evaluated by testing against the alternative hypothesis of short-range dependence. For the data from three experiments, we accomplish such tests with autoregressive fractionally integrated moving-average time series modeling. We find that long-range serial dependence in these experiments can be explained by any of several mechanisms, including mixtures of a small number of short-range processes.

Detection of single and clustered microcalcifications in mammograms using fractals models and neural networks

Microcalcifications (microCas) are often early signs of breast cancer. However, detecting them is a difficult visual task and recognizing malignant lesions is a complex diagnostic problem. In recent years, several research groups have been working to develop computer-aided diagnosis (CAD) systems for X-ray mammography. In this paper, we propose a method to detect and classify microcalcifications. In order to discover the presence of microCas clusters, particular attention is paid to the analysis of the spatial arrangement of detected lesions. A fractal model has been used to describe the mammographic image, thus, allowing the use of a matched filtering stage to enhance microcalcifications against the background. A region growing algorithm, coupled with a neural classifier, detects existing lesions. Subsequently, a second fractal model is used to analyze their spatial arrangement so that the presence of microcalcification clusters can be detected and classified. Reported results indicate that fractal models provide an adequate framework for medical image processing; consequently high correct classification rates are achieved.

A methodological note on nonlinear time series analysis: is the open- and closed-loop model of Collins and De Luca (1993) a statistical artifact?

The authors reexamined, theoretically and empirically, the method proposed by J. J. Collins and C. D. De Luca (1993) for the analysis of center-of-pressure trajectories. The main argument in this article is that Collins and De Luca's approach is not adapted to the analysis of bounded time series and leads to statistical artifacts such as underestimation of the diffusion process for long-term intervals. The open- and closed-loop model developed by Collins and De Luca is a direct consequence of those statistical problems. Applying more classical methods, such as rescaled range analysis or detrended fluctuation analysis, the authors show that center-of-pressure trajectories can be modeled as continuous, antipersistent fractional Brownian motion. More specifically, those trajectories behave like 1/f noise, a ubiquitous feature in adaptive biological systems.

Fractal characterization of complexity in temporal physiological signals

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Fractal features of dark, maintained, and driven neural discharges in the cat visual system

We employ a number of statistical measures to characterize neural discharge activity in cat retinal ganglion cells (RGCs) and in their target lateral geniculate nucleus (LGN) neurons under various stimulus conditions, and we develop a new measure to examine correlations in fractal activity between spike-train pairs. In the absence of stimulation (i.e., in the dark), RGC and LGN discharges exhibit similar properties. The presentation of a constant, uniform luminance to the eye reduces the fractal fluctuations in the RGC maintained discharge but enhances them in the target LGN discharge, so that neural activities in the pair cease to be mirror images of each other. A drifting-grating stimulus yields RGC and LGN driven spike trains similar in character to those observed in the maintained discharge, with two notable distinctions: action potentials are reorganized along the time axis so that they occur only during certain phases of the stimulus waveform, and fractal activity is suppressed. Under both uniform-luminance and drifting-grating stimulus conditions (but not in the dark), the discharges of pairs of LGN cells are highly correlated over long time scales; in contrast discharges of RGCs are nearly uncorrelated with each other. This indicates that action-potential activity at the LGN is subject to a common fractal modulation to which the RGCs are not subjected.

Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images

Fractal analysis is a method of characterizing complex shapes such as the trabecular structure of bone. Numerous algorithms for estimating fractal dimension have been described, but the Fourier power spectrum method is particularly applicable to self-affine fractals, and facilitates corrections for the effects of noise and blurring in an image. We found that it provided accurate estimates of fractal dimension for synthesized fractal images. For natural texture images fractality is limited to a range of scales, and the fractal dimension as a function of spatial frequency presents as a fractal signature. We found that the fractal signature was more successful at discriminating between these textures than either the global fractal dimension or other metrics such as the mean width and root-mean-square width of the spectral density plots. Different natural textures were also readily distinguishable using lacunarity plots, which explicitly characterize the average size and spatial organization of structural sub-units within an image. The fractal signatures of small regions of interest (32x32 pixels), computed in the frequency domain after corrections for imaging system noise and MTF, were able to characterize the texture of vertebral trabecular bone in CT images. Even small differences in texture due to acquisition slice thickness resulted in measurably different fractal signatures. These differences were also readily apparent in lacunarity plots, which indicated that a slice thickness of 1 mm or less is necessary if essential architectural information is not to be lost. Since lacunarity measures gap size and is not predicated on fractality, it may be particularly useful for characterizing the texture of trabecular bone.

Variability in isochronous tapping: higher order dependencies as a function of intertap interval

Isochronous serial interval production (ISIP) data, as from unpaced finger tapping, exhibit higher order dependencies (drift). This fact has largely been ignored by the timing literature, one reason probably being that influential timing models assume random variability. Men and women, 22-36 years old, performed a synchronization-continuation task with intertap intervals (ITI) from 0.4 s to 2.2 s. ISIP variability was partitioned into components attributable to drift and 1st-order serial correlation, and the results indicate that (a) drift contributes substantially to the dispersion for longer ITIs, (b) drift and 1st-order correlation are different functions of the ITI, and (c) drift exhibits break close to 1.0 s and 1.4 s ITI. These breaks correspond to qualitative changes in performance for other temporal tasks, which suggests common timing processes across modalities and tasks.

Origins of timing errors in human sensorimotor coordination

The authors analyzed fluctuations in timing errors when 8 human participants attempted to coordinate movement with external rhythmic signals. The temporal dynamics of the errors is usually described in terms of simple, self-correcting models. Here the authors demonstrate that timing errors are characterized by a 1/f(alpha) type of long memory process. The value of the exponent alpha differentiates different types of coordination states: synchronization and syncopation. More interesting, evidence was found that alpha can be changed when participants use different coordination strategies. Together with the authors' understanding of the generation mechanism for long memory processes, these results suggest that 1/f(alpha) type of long-range correlated timing errors are of higher cortical origin and are likely the outcome of distributed neural processes acting on multiple time scales.

Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes

Stochastic fractal signals can be characterized by the Hurst coefficient H, which is related to the exponents of various power-law statistics characteristic of these processes. Two techniques widely used to estimate H are spectral analysis and detrended fluctuation analysis (DFA). This paper examines the analytical link between these two measures and shows that they are related through an integral transform. Numerical simulations confirm this relationship for ideal synthesized fractal signals. Their performance as estimators of H is compared based on a mean square error criterion and found to be similar. DFA measures are derived for physiological signals of heartbeat R-R intervals through the integral transform of a spectral density estimate. These agree with directly calculated DFA estimates, indicating that the relationship holds for signals with nonideal fractal properties. It is concluded that DFA and spectral measures provide equivalent characterizations of stochastic signals with long-term correlation.