Multifractional Brownian Motion Research Articles

On the pointwise regularity of the Multifractional Brownian Motion and some extensions

We study the pointwise regularity of the Multifractional Brownian Motion and, in particular, we obtain the existence of so-called slow points of the process, that is points which exhibit a slow oscillation instead of the a.e. regularity. This result entails that a non self-similar process can also exhibit such a behavior. We also consider various extensions with the aim of imposing weaker regularity assumptions on the Hurst function without altering the regularity of the process.

Computer-aided detection of prostate cancer in early stages using multi-parameter MRI: A promising approach for early diagnosis.

Transrectal ultrasound-guided prostate biopsy is the gold standard diagnostic test for prostate cancer, but it is an invasive examination of non-targeted puncture and has a high false-negative rate. In this study, we aimed to develop a computer-assisted prostate cancer diagnosis method based on multiparametric MRI (mpMRI) images. We retrospectively collected 106 patients who underwent radical prostatectomy after diagnosis with prostate biopsy. mpMRI images, including T2 weighted imaging (T2WI), diffusion weighted imaging (DWI), and dynamic-contrast enhanced (DCE), and were accordingly analyzed. We extracted the region of interest (ROI) about the tumor and benign area on the three sequential MRI axial images at the same level. The ROI data of 433 mpMRI images were obtained, of which 202 were benign and 231 were malignant. Of those, 50 benign and 50 malignant images were used for training, and the 333 images were used for verification. Five main feature groups, including histogram, GLCM, GLGCM, wavelet-based multi-fractional Brownian motion features and Minkowski function features, were extracted from the mpMRI images. The selected characteristic parameters were analyzed by MATLAB software, and three analysis methods with higher accuracy were selected. Through prostate cancer identification based on mpMRI images, we found that the system uses 58 texture features and 3 classification algorithms, including Support Vector Machine (SVM), K-nearest Neighbor (KNN), and Ensemble Learning (EL), performed well. In the T2WI-based classification results, the SVM achieved the optimal accuracy and AUC values of 64.3% and 0.67. In the DCE-based classification results, the SVM achieved the optimal accuracy and AUC values of 72.2% and 0.77. In the DWI-based classification results, the ensemble learning achieved optimal accuracy as well as AUC values of 75.1% and 0.82. In the classification results based on all data combinations, the SVM achieved the optimal accuracy and AUC values of 66.4% and 0.73. The proposed computer-aided diagnosis system provides a good assessment of the diagnosis of the prostate cancer, which may reduce the burden of radiologists and improve the early diagnosis of prostate cancer.

Texture Estimation for Abnormal Tissue Segmentation in Brain MRI.

This chapter discusses multifractal texture estimation and characterization of brain lesions (necrosis, edema, enhanced tumor, nonenhanced tumor, etc.) in magnetic resonance (MR) images. This work formulates the complex texture of tumor in MR images using a stochastic model known as multifractional Brownian motion (mBm). Mathematical derivations of the mBm model and corresponding algorithm to extract the spatially varying multifractal texture feature are discussed. Extracted multifractal texture feature is fused with other effective features to enhance the tissue characteristics. Segmentation of the tissues is performed using a feature-based classification method. The efficacy of the mBm texture feature in segmenting different abnormal tissues is demonstrated using a large-scale publicly available clinical dataset. Experimental results and performance of the methods confirm the efficacy of the proposed technique in an automatic segmentation of abnormal tissues in multimodal (T1, T2, Flair, and T1contrast) brain MRIs.

Properties of Various Entropies of Gaussian Distribution and Comparison of Entropies of Fractional Processes

We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes.

Multifractional Hermite processes: Definition and first properties

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener–Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

Minimal model of diffusion with time changing Hurst exponent

We introduce the stochastic process of incremental multifractional Brownian motion (IMFBM), which locally behaves like fractional Brownian motion with a given local Hurst exponent and diffusivity. When these parameters change as function of time the process responds to the evolution gradually: only new increments are governed by the new parameters, while still retaining a power-law dependence on the past of the process. We obtain the mean squared displacement and correlations of IMFBM which are given by elementary formulas. We also provide a comparison with simulations and introduce estimation methods for IMFBM. This mathematically simple process is useful in the description of anomalous diffusion dynamics in changing environments, e.g. in viscoelastic systems, or when an actively moving particle changes its degree of persistence or its mobility.

General Transfer Formula for Stochastic Integral with Respect to Multifractional Brownian Motion

General Transfer Formula for Stochastic Integral with Respect to Multifractional Brownian Motion

A new hybrid approach for nonlinear stochastic differential equations driven by multifractional Gaussian noise

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems.

Measuring conditional correlation between financial markets' inefficiency

<abstract><p>Assuming that stock prices follow a multi-fractional Brownian motion, we estimated a time-varying Hurst exponent ($ h_t $). The Hurst value can be considered a relative volatility measure and has been recently used to estimate market inefficiency. Therefore, the Hurst exponent offers a level of comparison between theoretical and empirical market efficiency. Starting from this point of view, we adopted a multivariate conditional heteroskedastic approach for modeling inefficiency dynamics in various financial markets during the 2007 financial crisis, the COVID-19 pandemic and the Russo-Ukranian war. To empirically validate the analysis, we compared different stock markets in terms of conditional and unconditional correlations of dynamic inefficiency and investigated the predicted power of inefficiency measures through the Granger causality test.</p></abstract>

Fuzzy clustering of time series with time-varying memory

Little attention has been devoted to the long memory among the different data features considered for clustering time series. Following previous literature, we measure the long memory of a time series through the estimated Hurst exponent. However, we exploit the fact that a constant value for the Hurst exponent h is unrealistic in many practical examples. In this paper, assuming that the time series follows a multifractional Brownian motion, we estimate a time-varying Hurst exponent used as the input for a fuzzy clustering procedure. Motivated by the relevance of long memory in finance, the usefulness of the proposed clustering procedure is shown with an application to stock prices.

On the non-commutative multifractional Brownian motion

We define a non-commutative analogue of a real gaussian Volterra-type multifractional Brownian motion (NC-mfBm for short) and show that its trajectories behave locally like non-commutative fractional Brownian motion. We determine the pointwise Hölder exponent as well as a random matrix approximation of this process.

On multifractionality of spherical random fields with cosmological applications

This paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps. doi:10.1017/S1446181122000104

Multi-Fractional Brownian Motion: Estimating the Hurst Exponent via Variational Smoothing with Applications in Finance

Beginning with the basics of the Wiener process, we consider limitations characterizing the “Brownian approach” in analyzing real phenomena. This leads us to first consider the fractional Brownian motion (fBm)—also discussing the Wood–Chan fast algorithm to generate sample paths—to then focus on multi-fBm and methods to generate its trajectories. This is heavily linked to the Hurst exponent study, which we link to real data, firstly considering an absolute moment method, allowing us to obtain raw estimates, to then consider variational calculus approaches allowing to smooth it. The latter smoothing tool was tested in accuracy on synthetic data, comparing it with the exponential moving average method. Previous analyses and results were exploited to develop a forecasting procedure applied to the real data of foreign exchange rates from the Forex market.

Multifractional Brownian motion characterization based on Hurst exponent estimation and statistical learning.

This paper proposes an approach for the estimation of a time-varying Hurst exponent to allow accurate identification of multifractional Brownian motion (MFBM). The contribution provides a prescription for how to deal with the MFBM measurement data to solve regression and classification problems. Theoretical studies are supplemented with computer simulations and real-world examples. Those prove that the procedure proposed in this paper outperforms the best-in-class algorithm.

Molecular cancer classification on microarrays gene expression data using wavelet‐based deep convolutional neural network

AbstractMicroarray data analysis is a most promising and difficult process due to the complex nature of data. It includes higher dimensionality, several unbalanced classes, a smaller sample size, the presence of noise, and a higher variation of feature values. This has resulted in a decrease in classification accuracy as well as an overfitting problem. This work proposed an efficient and hybrid deep learning technique for molecular cancer classification using expression data to solve these limitations. The different steps in the proposed work are preprocessing, clustering, extraction, selection, and classification. The input data is preprocessed using a scalable range adaptive bilateral filter. Then clustering is done with the help of an improved binomial clustering approach. After that, the data is extracted with the help of the multifractal Brownian motion method. Then the important features are selected with the help of an improved cuckoo search optimization algorithm. Finally, the data classification is performed using a wavelet‐based deep convolutional neural network. This work is validated with the help of five publically available datasets using the PYTHON platform. The different performance measures considered here are accuracy, precision, recall, and F‐measure. The classification accuracy obtained is 98.36%, 98.12%, 98.55%, 97.70%, and 95.30% for ovarian, breast, colon, leukemia, and prostate cancer datasets. The overall result showed that the suggested technique is better than the existing methods.

On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional parameter H ( ⋅ ) {\mathcal {H}}(\cdot ) , we show that { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} as well as { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} only differ by a part which is locally more regular than { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} itself. On one hand this result implies that the two non-classical multifractional processes { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} have exactly the same local path behavior as that of the classical MBM { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} . On the other hand it allows to obtain from discrete realizations of { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} strongly consistent statistical estimators for values of their functional parameter.

An optimal control problem for a linear SPDE driven by a multiplicative multifractional Brownian motion

In this paper, we study the existence of the solution of a linear SPDE driven by a multiplicative multifractional Brownian motion. Moreover, we study an optimal control problem with a linear quadratic objective functional involving the solution of the studied SPDE. We prove the existence and uniqueness of the optimal control.

Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters

We conduct cluster analysis of a class of locally asymptotically self-similar stochastic processes with finite covariance structures, which includes Brownian motion, fractional Brownian motion, and multifractional Brownian motion as paradigmatic examples. Given the true number of clusters, a new covariance-based dissimilarity measure is introduced, based on which we obtain approximately asymptotically consistent algorithms for clustering locally asymptotically self-similar stochastic processes. In the simulation study, clustering data sampled from fractional and multifractional Brownian motions with distinct Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets’ equity indexes returns and sovereign CDS spreads provides a successful real world application. Implementations in MATLAB of the proposed algorithms and the simulation study are publicly shared in GitHub.

ON MULTIFRACTIONALITY OF SPHERICAL RANDOM FIELDS WITH COSMOLOGICAL APPLICATIONS

AbstractThis paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps.

A Generalization of Multifractional Brownian Motion

In this article, some properties of multifractional Brownian motion (MFBM) are discussed. It is shown that it has persistence of signs long range dependence (LRD) and persistence of magnitudes LRD properties. A generalization called here nth order multifractional Brownian motion (n-MFBM) that allows to take the functional parameter H(t) values in the range (n−1,n) is discussed. Two representations of the n-MFBM are given and their relationship with each other is obtained.