Fractals in Neuroimaging.

Several natural phenomena can be described by studying their statistical scaling patterns, hence leading to simple geometrical interpretation. In this regard, fractal geometry is a powerful tool to describe the irregular or fragmented shape of natural features, using spatial or time-domain statistical scaling laws (power-law behavior) to characterize real-world physical systems. This chapter presents some works on the usefulness of fractal features, mainly the fractal dimension and the related Hurst exponent, in the characterization and identification of pathologies and radiological features in neuroimaging, mainly, magnetic resonance imaging.

The effect of COVID-19 pandemic on return-volume and return-volatility relationships in cryptocurrency markets.

Understanding the dynamics of cryptocurrency markets during financial crises such as the recent one caused by the COVID-19 pandemic is crucial for policy makers and investors. In this study, the effect of COVID-19 pandemic on the return-volatility and return-volume relationships for the ten most traded cryptocurrencies, namely Tether, Bitcoin, Ethereum, Ripple, Litecoin, Bitcoin Cash, EOS, Chainlink, Cardano, and Monero is examined. Further, the behavior of cryptocurrencies during COVID-19 pandemic is compared with less volatile markets such as Gold, WTI, and BRENT crude oil markets. To study the effect of volatility on cryptocurrency return, an EGARCH-M model is employed while for the return-volume relationships the VAR model and Granger causality tests are utilized. Results show that the return-volatility relationships for Tether, Ethereum, Ripple, Bitcoin Cash, EOS, and Monero are significant during COVID-19 pandemic, while the same relationship is not significant prior to the pandemic for any of the studied cryptocurrencies. Our findings of the return-volume relationship support the availability of causal relations from return to trading volume changes for Chainlink and Monero in the pre-COVID-19 period and for Ethereum, Ripple, Litecoin, EOS, and Cardano during the COVID-19 period. However, considering the absolute values of returns, we found a significant relationship from cryptocurrencies' absolute returns to trading volume changes for both the prior and during COVID-19 periods. From a managerial perspective, gold can be considered a suitable asset for portfolio hedging during the pandemic period and trading volume can help traders and investors identify the effect of momentum and potential trend in cryptocurrencies on their investments.

Nonlinear Statistical Analysis of Normal and Pathological Infant Cry Signals in Cepstrum Domain by Multifractal Wavelet Leaders.

Multifractal behavior in the cepstrum representation of healthy and unhealthy infant cry signals is examined by means of wavelet leaders and compared using the Student t-test. The empirical results show that both expiration and inspiration signals exhibit clear evidence of multifractal properties under healthy and unhealthy conditions. In addition, expiration and inspiration signals exhibit more complexity under healthy conditions than under unhealthy conditions. Furthermore, distributions of multifractal characteristics are different across healthy and unhealthy conditions. Hence, this study improves the understanding of infant crying by providing a complete description of its intrinsic dynamics to better evaluate its health status.

The effect of COVID-19 on long memory in returns and volatility of cryptocurrency and stock markets.

We examine long memory (self-similarity) in digital currencies and international stock exchanges prior and during COVID-19 pandemic. Specifically, ARFIMA and FIGARCH models are respectively employed to evaluate long memory parameter in returns and volatility. The dataset contains 45 cryptocurrency markets and 16 international equity markets. The t-test and F-test are performed to estimated long memory parameters. The empirical findings follow. First, the level of persistence in return series of both markets has increased during the COVID-19 pandemic. Second, during COVID-19 pandemic, variability level in persistence in return series has increased in both digital currencies and stock markets. Third, return series in both markets exhibited comparable level of persistence prior and during the COVID-19 pandemic. Fourth, return series in volatility series of cryptocurrency exhibited high degree of persistence compared to international stock markets during the COVID-19 pandemic. Therefore, it is concluded that COVID-19 pandemic significantly affected long memory in return and volatility of cryptocurrency and international stock markets. In addition, our results suggest that the hybrid long memory model represented by the integration of ARFIMA-FIGARCH is significantly suitable to describe returns and volatility of cryptocurrencies and stocks and to reveal differences before and during COVID-19 pandemic periods.

Randomness, Informational Entropy, and Volatility Interdependencies among the Major World Markets: The Role of the COVID-19 Pandemic.

The main purpose of our paper is to evaluate the impact of the COVID-19 pandemic on randomness in volatility series of world major markets and to examine its effect on their interconnections. The data set includes equity (Bitcoin and Standard and Poor's 500), precious metals (Gold and Silver), and energy markets (West Texas Instruments, Brent, and Gas). The generalized autoregressive conditional heteroskedasticity model is applied to the return series. The wavelet packet Shannon entropy is calculated from the estimated volatility series to assess randomness. Hierarchical clustering is employed to examine interconnections between volatilities. We found that (i) randomness in volatility of the S&P500 and in the volatility of precious metals were the most affected by the COVID-19 pandemic, while (ii) randomness in energy markets was less affected by the pandemic than equity and precious metal markets. Additionally, (iii) we showed an apparent emergence of three volatility clusters: precious metals (Gold and Silver), energy (Brent and Gas), and Bitcoin and WTI, and (iv) the S&P500 volatility represents a unique cluster, while (v) the S&P500 market volatility was not connected to the volatility of Bitcoin, energy, and precious metal markets before the pandemic. Moreover, (vi) the S&P500 market volatility became connected to volatility in energy markets and volatility in Bitcoin during the pandemic, and (vii) the volatility in precious metals is less connected to volatility in energy markets and to volatility in Bitcoin market during the pandemic. It is concluded that (i) investors may diversify their portfolios across single constituents of clusters, (ii) investing in energy markets during the pandemic period is appealing because of lower randomness in their respective volatilities, and that (iii) constructing a diversified portfolio would not be challenging as clustering structures are fairly stable across periods.

Renyi entropy and mutual information measurement of market expectations and investor fear during the COVID-19 pandemic.

The COVID-19 pandemic has seriously affected world economies. In this regard, it is expected that information level and sharing between equity, digital currency, and energy markets has been altered due to the pandemic outbreak. Specifically, the resulting twisted risk among markets is presumed to rise during the abnormal state of world economy. The purpose of the current study is twofold. First, by using Renyi entropy, we analyze the multiscale entropy function in the return time series of Bitcoin, S&P500, WTI, Brent, Gas, Gold, Silver, and investor fear index represented by VIX. Second, by estimating mutual information, we analyze the information sharing between these markets. The analyses are conducted before and during the COVID-19 pandemic. The empirical results from Renyi entropy indicate that for all market indices, randomness and disorder are more concentrated in less probable events. The empirical results from mutual information showed that the information sharing network between markets has changed during the COVID-19 pandemic. From a managerial perspective, we conclude that during the pandemic (i) portfolios composed of Bitcoin and Silver, Bitcoin and WTI, Bitcoin and Gold, Bitcoin and Brent, or Bitcoin and S&P500 could be risky, (ii) diversification opportunities exist by investing in portfolios composed of Gas and Silver, Gold and Silver, Gold and Gas, Brent and Silver, Brent and Gold, or Bitcoin and Gas, and that (iii) the VIX exhibited the lowest level of information disorder at all scales before and during the pandemic. Thus, it seems that the pandemic has not influenced the expectations of investors. Our results provide an insight of the response of stocks, cryptocurrencies, energy, precious metal markets, to expectations of investors in the aftermath of the COVID-19 pandemic in terms of information ordering and sharing.

The impact of COVID-19 pandemic upon stability and sequential irregularity of equity and cryptocurrency markets.

We explore the evolution of the informational efficiency in 45 cryptocurrency markets and 16 international stock markets before and during COVID-19 pandemic. The measures of Largest Lyapunov Exponent (LLE) based on the Rosenstein's method and Approximate Entropy (ApEn), which are robust to small samples, are applied to price time series in order to estimate degrees of stability and irregularity in cryptocurrency and international stock markets. The amount of regularity infers on the unpredictability of fluctuations. The t-test and F-test are performed on estimated LLE and ApEn. In total, 36 statistical tests are performed to check for differences between time periods (pre- versus during COVID-19 pandemic samples) on the one hand, as well as check for differences between markets (cryptocurrencies versus stocks), on the other hand. During the COVID-19 pandemic period it was found that (a) the level of stability in cryptocurrency markets has significantly diminished while the irregularity level significantly augmented, (b) the level of stability in international equity markets has not changed but gained more irregularity, (c) cryptocurrencies became more volatile, (d) the variability in stability and irregularity in equities has not been affected, (e) cryptocurrency and stock markets exhibit a similar degree of stability in price dynamics, whilst finally (f) cryptocurrency exhibit a low level of regularity compared to international equity markets. We find that cryptos showed more instability and more irregularity during the COVID-19 pandemic compared to international stock markets. Thus, from an informational efficiency perspective, investing in digital assets during big crises as the COVID-19 pandemic, could be considered riskier as opposed to equities.

The Informational Dynamics of Mean‒Variance Relationships in Fertilizer Markets: An Entropic Investigation.

The risk‒return trade-off is a fundamental relationship that has received a large amount of attention in financial and economic analysis. Indeed, it has important implications for understanding linear dynamics in price returns and active quantitative portfolio optimization. The main contributions of this work include, firstly, examining such a relationship in five major fertilizer markets through different time periods: a period of low variability in returns and a period of high variability such as that during which the recent global financial crisis occurred. Secondly, we explore how entropy in those markets varies during the investigated time periods. This requires us to assess their inherent informational dynamics. The empirical results show that higher volatility is associated with a larger return in diammonium phosphate, potassium chloride, triple super phosphate, and urea market, but not rock phosphate. In addition, the magnitude of this relationship is low during a period of high variability. It is concluded that key statistical patterns of return and the relationship between return and volatility are affected during high variability periods. Our findings indicate that entropy in return and volatility series of each fertilizer market increase significantly during time periods of high variability.

Performance of machine learning methods in diagnosing Parkinson's disease based on dysphonia measures.

Parkinson's disease (PD) is a widespread degenerative syndrome that affects the nervous system. Its early appearing symptoms include tremor, rigidity, and vocal impairment (dysphonia). Consequently, speech indicators are important in the identification of PD based on dysphonic signs. In this regard, computer-aided-diagnosis systems based on machine learning can be useful in assisting clinicians in identifying PD patients. In this work, we evaluate the performance of machine learning based techniques for PD diagnosis based on dysphonia symptoms. Several machine learning techniques were considered and trained with a set of twenty-two voice disorder measurements to classify healthy and PD patients. These machine learning methods included linear discriminant analysis (LDA), k nearest-neighbors (k-NN), naïve Bayes (NB), regression trees (RT), radial basis function neural networks (RBFNN), support vector machine (SVM), and Mahalanobis distance classifier. We evaluated the performance of these methods by means of a tenfold cross validation protocol. Experimental results show that the SVM classifier achieved higher average performance than all other classifiers in terms of overall accuracy, G-mean, and area under the curve of the receiver operating characteristic plot. The SVM classifier achieved higher performance measures than the majority of the other classifiers also in terms of sensitivity, specificity, and F-measure statistics. The LDA, k-NN and RT achieved the highest average precision. The RBFNN method yielded the highest F-measure.; however, it performed poorly in terms of other performance metrics. Finally, t tests were performed to evaluate statistical significance of the results, confirming that the SVM outperformed most of the other classifiers on the majority of performance measures. SVM is a promising method for identifying PD patients based on classification of dysphonia measurements.

High-frequency-based features for low and high retina haemorrhage classification.

Haemorrhages (HAs) presence in fundus images is one of the most important indicators of diabetic retinopathy that causes blindness. In this regard, accurate grading of HAs in fundus images is crucial for appropriate medical treatment. The purpose of this Letter is to assess the relative performance of statistical features obtained with three different multi-resolution analysis (MRA) techniques and fed to support vector machine in grading retinal HAs. Considered MRA techniques are the common discrete wavelet transform (DWT), empirical mode decomposition (EMD), and variational mode decomposition (VMD). The obtained experimental results show that statistical features obtained by EMD, VMD, and DWT, respectively, achieved 88.31% ± 0.0832, 71% ± 0.1782, and 64% ± 0.0949 accuracies. It also outperformed VMD and DWT in terms of sensitivity and specificity. Thus, the EMD-based features are promising for grading retinal HAs.

Denoising techniques in adaptive multi-resolution domains with applications to biomedical images.

Variational mode decomposition (VMD) is a new adaptive multi-resolution technique suitable for signal denoising purpose. The main focus of this work has been to study the feasibility of several image denoising techniques in empirical mode decomposition (EMD) and VMD domains. A comparative study is made using 11 techniques widely used in the literature, including Wiener filter, first-order local statistics, fourth partial differential equation, nonlinear complex diffusion process, linear complex diffusion process (LCDP), probabilistic non-local means, non-local Euclidean medians, non-local means, non-local patch regression, discrete wavelet transform and wavelet packet transform. On the basis of comparison of 396 denoising based on peak signal-to-noise ratio, it is found that the best performances are obtained in VMD domain when appropriate denoising techniques are applied. Particularly, it is found that LCDP in combination with VMD performs the best and that VMD is faster than EMD.

Image denoising in bidimensional empirical mode decomposition domain: the role of Student's probability distribution function.

Hybridisation of the bi-dimensional empirical mode decomposition (BEMD) with denoising techniques has been proposed in the literature as an effective approach for image denoising. In this Letter, the Student's probability density function is introduced in the computation of the mean envelope of the data during the BEMD sifting process to make it robust to values that are far from the mean. The resulting BEMD is denoted tBEMD. In order to show the effectiveness of the tBEMD, several image denoising techniques in tBEMD domain are employed; namely, fourth order partial differential equation (PDE), linear complex diffusion process (LCDP), non-linear complex diffusion process (NLCDP), and the discrete wavelet transform (DWT). Two biomedical images and a standard digital image were considered for experiments. The original images were corrupted with additive Gaussian noise with three different levels. Based on peak-signal-to-noise ratio, the experimental results show that PDE, LCDP, NLCDP, and DWT all perform better in the tBEMD than in the classical BEMD domain. It is also found that tBEMD is faster than classical BEMD when the noise level is low. When it is high, the computational cost in terms of processing time is similar. The effectiveness of the presented approach makes it promising for clinical applications.

Computational analyses of arteriovenous malformations in neuroimaging.

Computational models have been investigated for the analysis of the physiopathology and morphology of arteriovenous malformation (AVM) in recent years. Special emphasis has been given to image fusion in multimodal imaging and 3-dimensional rendering of the AVM, with the aim to improve the visualization of the lesion (for diagnostic purposes) and the selection of the nidus (for therapeutic aims, like the selection of the region of interest for the gamma knife radiosurgery plan). Searching for new diagnostic and prognostic neuroimaging biomarkers, fractal-based computational models have been proposed for describing and quantifying the angioarchitecture of the nidus. Computational modeling in the AVM field offers promising tools of analysis and requires a strict collaboration among neurosurgeons, neuroradiologists, clinicians, computer scientists, and engineers. We present here some updated state-of-the-art exemplary cases in the field, focusing on recent neuroimaging computational modeling with clinical relevance, which might offer useful clinical tools for the management of AVMs in the future.

Automated detection of circinate exudates in retina digital images using empirical mode decomposition and the entropy and uniformity of the intrinsic mode functions.

This work presents a new automated system to detect circinate exudates in retina digital images. It operates as follows: the true color image is converted to gray levels, and contrast-limited adaptive histogram equalization (CLAHE) is applied to it before undergoing empirical mode decomposition (EMD) as intrinsic mode functions (IMFs). The entropies and uniformities of the first two IMFs are then computed to form a feature vector that is fed to a support vector machine (SVM) for classification. The experimental results using a set of 45 images (23 normal images and 22 images with circinate exudates taken from the STARE database) and tenfold cross-validation indicate that the proposed approach outperforms previous works found in the literature, with perfect classification. In addition, the image processing time was <4 min, making the presented circinate exudate detection system fit for use in a clinical environment.

New approach for automatic classification of Alzheimer's disease, mild cognitive impairment and healthy brain magnetic resonance images.

Explored is the utility of modelling brain magnetic resonance images as a fractal object for the classification of healthy brain images against those with Alzheimer's disease (AD) or mild cognitive impairment (MCI). More precisely, fractal multi-scale analysis is used to build feature vectors from the derived Hurst's exponents. These are then classified by support vector machines (SVMs). Three experiments were conducted: in the first the SVM was trained to classify AD against healthy images. In the second experiment, the SVM was trained to classify AD against MCI and, in the third experiment, a multiclass SVM was trained to classify all three types of images. The experimental results, using the 10-fold cross-validation technique, indicate that the SVM achieved 97.08% ± 0.05 correct classification rate, 98.09% ± 0.04 sensitivity and 96.07% ± 0.07 specificity for the classification of healthy against MCI images, thus outperforming recent works found in the literature. For the classification of MCI against AD, the SVM achieved 97.5% ± 0.04 correct classification rate, 100% sensitivity and 94.93% ± 0.08 specificity. The third experiment also showed that the multiclass SVM provided highly accurate classification results. The processing time for a given image was 25 s. These findings suggest that this approach is efficient and may be promising for clinical applications.

Comparative study of ECG signal denoising by wavelet thresholding in empirical and variational mode decomposition domains.

Hybrid denoising models based on combining empirical mode decomposition (EMD) and discrete wavelet transform (DWT) were found to be effective in removing additive Gaussian noise from electrocardiogram (ECG) signals. Recently, variational mode decomposition (VMD) has been proposed as a multiresolution technique that overcomes some of the limits of the EMD. Two ECG denoising approaches are compared. The first is based on denoising in the EMD domain by DWT thresholding, whereas the second is based on noise reduction in the VMD domain by DWT thresholding. Using signal-to-noise ratio and mean of squared errors as performance measures, simulation results show that the VMD-DWT approach outperforms the conventional EMD-DWT. In addition, a non-local means approach used as a reference technique provides better results than the VMD-DWT approach.

Automated pathologies detection in retina digital images based on complex continuous wavelet transform phase angles.

An automated diagnosis system that uses complex continuous wavelet transform (CWT) to process retina digital images and support vector machines (SVMs) for classification purposes is presented. In particular, each retina image is transformed into two one-dimensional signals by concatenating image rows and columns separately. The mathematical norm of phase angles found in each one-dimensional signal at each level of CWT decomposition are relied on to characterise the texture of normal images against abnormal images affected by exudates, drusen and microaneurysms. The leave-one-out cross-validation method was adopted to conduct experiments and the results from the SVM show that the proposed approach gives better results than those obtained by other methods based on the correct classification rate, sensitivity and specificity.

Hybrid Discrete Wavelet Transform and Gabor Filter Banks Processing for Features Extraction from Biomedical Images.

A new methodology for automatic feature extraction from biomedical images and subsequent classification is presented. The approach exploits the spatial orientation of high-frequency textural features of the processed image as determined by a two-step process. First, the two-dimensional discrete wavelet transform (DWT) is applied to obtain the HH high-frequency subband image. Then, a Gabor filter bank is applied to the latter at different frequencies and spatial orientations to obtain new Gabor-filtered image whose entropy and uniformity are computed. Finally, the obtained statistics are fed to a support vector machine (SVM) binary classifier. The approach was validated on mammograms, retina, and brain magnetic resonance (MR) images. The obtained classification accuracies show better performance in comparison to common approaches that use only the DWT or Gabor filter banks for feature extraction.

Alzheimer's disease detection in brain magnetic resonance images using multiscale fractal analysis.

We present a new automated system for the detection of brain magnetic resonance images (MRI) affected by Alzheimer's disease (AD). The MRI is analyzed by means of multiscale analysis (MSA) to obtain its fractals at six different scales. The extracted fractals are used as features to differentiate healthy brain MRI from those of AD by a support vector machine (SVM) classifier. The result of classifying 93 brain MRIs consisting of 51 images of healthy brains and 42 of brains affected by AD, using leave-one-out cross-validation method, yielded 99.18% ± 0.01 classification accuracy, 100% sensitivity, and 98.20% ± 0.02 specificity. These results and a processing time of 5.64 seconds indicate that the proposed approach may be an efficient diagnostic aid for radiologists in the screening for AD.

Automatic brain MR images diagnosis based on edge fractal dimension and spectral energy signature.

A new automatic system to detect pathologies in human brain magnetic resonance (MR) images is presented. The goal is to classify normal versus abnormal images affected by Alzheimer, Glioma, Herpes, Metastatic, and Multiple Sclerosis. The extracted features are the fractal dimension of edges in the Hilbert domain, and the skewness and kurtosis of their spectral energy distribution. The proposed system (FDSE) outperforms the popular discrete wavelet transform (DWT) and principal component analysis (PCA).