Multifractal Analysis of the Foreign Exchange Markets Application to MENA Countries

The present study focus on the multifractal analysis of the exchange rate for Middle East North Africa (MENA) region from January 1999 to May 2017. The purpose of this paper is to examine the behavior of currency markets and to verify the efficiency hypothesis of FOREX market for these countries. We first estimate the scaling function to detect the multifractal character of each series and then the Hölder exponent, using the Generalized Quadratic Variation (GQV) method, as a function of time H(t). We conclude that there's a multifractal character for all these countries with a difference in the degree of persistence of each market.

Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

<p style='text-indent:20px;'>We prove quantitative statistical stability results for a large class of small <inline-formula><tex-math id="M1">\begin{document}$ C^{0} $\end{document}</tex-math></inline-formula> perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.</p>

Clustering Techniques for Land Use Land Cover Classification of Remotely Sensed Images

Image processing is growing fast and persistently. The idea of remotely sensed image clustering is to categorize the image into meaningful land use land cover classes with respect to a particular application. Image clustering is a technique to group an image into units or categories that are homogeneous with respect to one or more characteristics. There are many algorithms and techniques that have been developed to solve image clustering problems, though, none of the method is a general solution. This chapter will highlight the various clustering techniques that bring together the current development on clustering and explores the potentiality of those techniques in extracting earth surface features information from high spatial resolution remotely sensed imageries. It also will provide an insight about the existing mathematical methods and its application to image clustering. Special emphasis will be given on Hölder exponent (HE) and Variance (VAR). HE and VAR are well-established techniques for texture analysis. This chapter will highlight about the Hölder exponent and variance-based clustering method for classifying land use/land cover in high spatial resolution remotely sensed images.

Individual Tree Detection from UAV Imagery Using Hölder Exponent

This article explores the application of Hölder exponent analysis for the identification and delineation of single tree crowns from very high-resolution (VHR) imagery captured by unmanned aerial vehicles (UAV). Most of the present individual tree crown detection (ITD) methods are based on canopy height models (CHM) and are very effective as far as an accurate digital terrain model (DTM) is available. This prerequisite is hard to accomplish in some environments, such as alpine forests, because of the high tree density and the irregular topography. Indeed, in such conditions, the photogrammetrically derived DTM can be inaccurate. A novel image processing method supports the segmentation of crowns based only on the parameter related to the multifractality description of the image. In particular, the multifractality is related to the deviation from a strict self-similarity and can be treated as the information about the level of inhomogeneity of considered data. The multifractals, even if well established in image processing and recognized by the scientific community, represent a relatively new application in VHR aerial imagery. In this work, the Hölder exponent (one of the parameters related to multifractal description) is applied to the study of a coniferous forest in the Western Alps. The infrared dataset with 10 cm pixels is captured by a UAV-mounted optical sensor. Then, the tree crowns are detected by a basic workflow. This consists of the thresholding of the image on the basis of the Hölder exponent. Then, the single crowns are segmented through a multiresolution segmentation approach. The ITD segmentation was validated through a two-level validation analysis that included a visual evaluation and the computing of quantitative measures based on 200 reference crowns. The results were checked against the ITD performed in the same area but using only spectral, textural, and elevation information. Specifically, the visual assessment included the estimation of the producer’s and user’s accuracies and the F1 score. The quantitative measures considered are the root mean square error (RMSE) (for the area, the perimeter, and the distance between centroids) and the over-segmentation and under-segmentation indices, the Jaccard index, and the completeness index. The F1 score indicates positive results (over 73%) as well as the completeness index that does not exceed 0.23 on a scale of 0 to 1, taking 0 as the best result possible. The RMSE of the extension of crowns is 3 m2, which represents only 14% of the average extension of reference crowns. The performance of the segmentation based on the Hölder exponent outclasses those based on spectral, textural, and elevation information. Despite the good results of the segmentation, the method tends to under-segment rather than over-segment, especially in areas with sloping. This study lays the groundwork for future research into ITD from VHR optical imagery using multifractals.

The self-similarity properties and multifractal analysis of DNA sequences

In this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed by 4 nucleotides (adenine, cytosine, guanine, and tymine). Following Jeffrey’s paper we associated a simple contractive function to each nucleotide, and constructed the Hutchinson’s operator W, which was used to build covers of different sizes of the unitary square Q, thus Wk(Q) is a cover of Q, conformed by 4k squares Qk of size 2−k, as each Qk corresponds to a unique subsequence of nucleotides with length k : b1b2...bk. Besides, it is obtained the optimal cover Ck to the fractal F generated for each DNA sequence was obtained. We made a multifractal decomposition of Ck in terms of the sets Jα conformed by the Qk’s with the same value of the Holder exponent α, and determined f (α), the Hausdorff dimension of Jα, using the curdling theorem.

On Fourier integral operators with Hölder-continuous phase

We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in [Formula: see text] with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling–Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in [Formula: see text] is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from time-frequency analysis.