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Multifractal Analysis: Identifying the Boundaries Application in the Study of Time Series
Zakharov A. I., Zagaynov A. I.
Military Space Academy named Mozhaiskyi Saint-Petersburg, Russia ana63916157@yandex.ru
Abstract. The work is devoted to revelation of possibilities from multifractal numerical procedures and similar handling techniques of time series developed until now. Application Frac-Lab 2.1 from the mathematic packet MatLab R2008b was used for this purpose, it processes test data (generated, for example, by the Henon map, Lorenz system of equation sand etc.), bases of heart rate variability records from the PhysioNet server and own electrocardiograms (ECG). Three-dimensional graphics of received wavelet transformation, graphs of isolines, multifractal characteristics in the form of Holder exponent and scaling exponent were constructed. All possible data provided in the FracLab (for example, Mhat-wavelet, DoG-wavelet and etc.) were used as a wavelet-forming function in the course of calculating a spectrum. Disadvantages of used realization, received first of all for test data calculating the Holder exponent, were determined. The conception of multifractal methodology widening was proposed for research of time series and creation of own software on its basis.
Keywords: time series, method of wavelet transform modulus maxima.
Introduction
The problem of searching for connection indicators of regulating system - is a perspective direction in the modern fundamental science, whose basic tendency is focused now on existence criterion determination (and detection) of determined chaos in the accentuated give time series of corresponding dynamical system.
As an exampleofsuch time serieswe mentionrecommenda-tions for usage of physiological interpretation of consecutive intervals between QRS-complexes of electrocardiogram (heart rate variability), executed in 1996 by the European Cardiology Society, North American Electrophysiological Community [6] and Committee of Clinicaldiagnostic Devices and Instruments.
Committee of New Medical Technology of the Russian Federation Ministry of Public Health [3] make these time lengths the most perspective in the course of noninvasive diagnostics in the modern practice of scientific investigations. In spite of the recommended (and accepted) range of variability indexes (for example, indexes of variationalpulsometering (Мо, АМо, MxDMn, MxRMn, triangular index) [3, 6], indexes of correlation rhyth-mography (L, w, EllAs, EllSq) [3], indexes of power in various frequency intervals, low and high frequency spectral components, their relations (LF, HF, LF/HF, VLF, ULF, IC) [3, 6] and etc.) the results, received using them have specific disadvantages.
The problem here is in the statement (postulate) about sta-tionarity of impacts of adaptive processes on the cardiac rhythm
Khodakovsky V A.
Petersburg State Transport University Saint-Petersburg, Russian hval 104@mail.ru
regulating systems, such as impact of central and vegetative nervous system, pathologies of circulatory system (resulting in rhythm disturbance), thyroid gland and etc. Certainly, such solution makes a deep impression on possibilities of received results of investigations. At the same time, (including because of the above mentioned reason) the majority of calculated components use traditional (linear) methods of time series analysis (for example, statistic, spectral, correlation and etc.). It, in its turn, simplifies the form evaluation of conformance to considered processes, distorting more their real physiological interpretation.
For example, analyzing the same spectral concentration in the course of routine examination of variability in the various frequency bands, the values of power don't have four evident peaks (to which the frequency components are related). The power coefficients are calculated only thanks to connection with the specific preset interval, and not due to highlighting the specified peculiarity.
This situation is explained by nonlinearity, discontinuity, instability and etc. of the cardiac rhythm reference signal, where it is impossible to exclude both contained phenomena of its internal regulation and procedural error of registering abbreviations itself. To solve the existing difficulty, several researchers (together with our research team) propose usage of non-linear mathematical methods, the majority of which is based on the fractal analysis.
The fractal methods are based, first of all, on examination of scaling invariance (scaling) of the researched process conditions. Conventionally, they may be divided into methods which directly use the idea of fractal (as a geometric entity in the multidimensional phase space) and transferred to it by manipulating initial time series for setting coordinates and methods examining the scale invariance of the initial process direct features. The well known feature of scaling - fractal (correlation) dimensionality may be related to the first ones. But here, there are a number of difficulties connected first of all with convergency of such feature in the finite-dimensional space of enclosure.
Absence of convergence curve saturation of correlation dimensionality are demonstrated in several works appeared in recent years [1, 5, 7]. At the same time, the determined deterministic rhythms - for example, fetal rhythm for periods of gestation 38-40 weeks and ventricular fibrillation, it suggests that the direction of fundamental research vector is correct.
Development of the last one may take place both considering various improvements of the mentioned methodology and principally different conception, using only foundations of the previous one. The methods of forming the scaling of the exam-
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ined process direct features on the basis of multifractal formalism conception, replacing the fractal exponent (restored attractor) by the directly built specialized spectrum, analyzed in the article, may be related to the last one.
Materials and methods
The scaling or scaling invariance is a main feature which is searched in the time-series in he analyzed methods of researches. At the same time, there are alternative approaches for its plotting, except the analyzed approach to the n-dimensional mapping. For example, the DFA method (detrended fluctuation analysis) is based on forming the automodeling process using the following sums:
к
bk = Z(a, - a) •
i=1
Afterwards, the formed row is divided in the areas with similar length n, where it is approximated (in the simplest variant) by the simple dependence using the method of the smallest squares. Setting the mean square error, the scaling is built by its comparison with the exponential function of the length value n:
J1 N
-Zb - bk (n)]2,
where bk (n) - is approximation bk at the length n areas.
This approach is not perspective due to the beam analysis of scaling. The more advanced method WTMM (wavelet transform modulus maxima) uses plotting of the whole variety of the local maximum line of wavelet transform:
1 ” r -1
W (t, a) = - { f (rM------)dr,
T a J a
-да
where the initial signal f(x) is divided using the function y(x) generated from the soliton-like one with special features by its scale measurements and shifts [2]. In the simplest variant (Holder’s exponent h) the scaling of one of the lines (for example, the maximum one) is researched:
The more complicated approach to plotting the scaling is based on analyzing all lines by introducing the partial function with the weight degree of all wavelet transform maximums:
Pq (S)=ZW (t,, S)]q ,
i
and plotting the scaling using the scaling function к(q) :
Pq (S ) ~ Sk ( q).
Above approachhas atheoreticallyjustified (and perfect) buildcapacityscaling, ranging from construction ofthe expan-sionof the originaltime series, settingthe scale invariance ofa certain lineand ending withthe introduction ofthe special func-tionlinesfor findinglocal maximaofthe generalizedscaling.This is caused byfrequentfailureresultingin the construction ofscalinga single line.
Results and discussion
To demonstrate usage of multifractal algorithms and determine their description possibilities we used standard time series in the theory of determined chaos - generated by nonlinear mapping (including solutions of nonlinear differential equations). Let’s remind, that the main advantage of such data involves a possibility to describe them using (mono)fractal methods, whose confirmation is convergence of fractal (correlation) dimensionality.
The application FracLab 2.1 of mathematic packet MatLab R2008b [4] was used for these purposes (at the first stage of researches. Results of analyzed methodology usage for Lorenz’s system of equations:
dx
— = —cx + cy,
dt
dy
{ — == rz + rr - y, dt
dz
— = ry - bz.
[ dt
with the values of parameters c = 10, b = 8/3, r = 28 are given in fig. 1, where the graphs for Holder exponent and scaling exponent using Mhat-wavelet are presented.
We would like to note that the solutions of mentioned system are received as a result of numerical integration using the methods Runge-Kutta of the 4th order. Even here we demonstrate unsuitability of the first (simplified) approach in the course of researching the scaling. It may be described by insufficiency of numerical structure of one line of local maximum (the result in the form of nondeterministic influence is possible in case of the described base transition), and inadequacy of the existing numeric base of algorithms. At that, the approach with introducing partial (generalized) function provides valid results - its approach to zero in the negative area of scaling and conation to straight line in the positive one.
Researching the time series (including for the considered system) we used various kinds of wavelet -forming function (Mhat, DoG, Wave, Morlet, Haar and etc.)The given results as well as usage of DoG and Wave wavelet are the most successful for the mentioned mapping.
Usage of discussed methodology for the two-dimensional mapping of Henon:
" xn+1 = 1 - ахП + Уп ,
V УП+1 = brn ,
described with the parameters a=1.4, b=0.3 provide less demonstrative results (fig. 2). Similarly to attractor of Lorenz, the Holder exponent is useless for practical interpretation of results. The scaling exponent behaves a little bit differently (including depending on the type of wavelet-forming basis). Here, there is no obvious conation to the direct correlation in the positive area of scaling. This (together with performed remarks) may be explained by direct forming of the researched attractor.
To this effect it is enough to examine its formation, for example, in the real time mode (realized in one of our projects [1]) and tracking the trajectory to make sure in the disruptive method of forming the next coordinate. In addition to attractors analyzed using multifractal methods we analyzed other attractors received using generators of determined chaos (Rossler equation system, Ikeda mapping and etc.) one.
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Fig. 1. Holder exponent and scaling exponent developed for the Lorenz’s system of equations
Fig. 2. Holderexponent and scaling exponent formed for Henon mapping
Fig. 3. Kind of scaling exponent for processed time series HRV
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Fig. 4. Demonstration of deviation minimization functional in relation to the kind of wavelet-forming function
The second stage of researches consisted of revelation of the scaling function nonlinear dependencies for various groups of cardiac rhythm variability time series. Pursuant to peculiarities mentioned above we interpreted the results received using only the scaling exponent. In addition, here, we determined more detailed potential of used methodology.
We were interested in the problem of possibility to approximate the straight line in various ranges of scaling researching. We used the server bases [8] for these purposes. Let’s demonstrate the most interesting results (from the point of view of the mentioned approach) of their processing (fig. 3).
Let’s highlight the prospectivity of analyzed method in the course of comparing the results with model (test) data - in some cases we observed an expressed peculiarity of possibility to highlight the straight line in the specific areas of scaling analysis, it provides a perspective base for researching in the area of creation of (multifractal) deflection coefficients of scaling exponent from asymptotes, to determine the kind of non-deterministic chaos comparing them with (mono)fractal mapping.
Here, it is necessary to highlight that determination of the asymptotic dependence absence (in the course of analyzed approximation) doesn’t imply (multi)fractality of initial time series. At the same time, indication on possibility of approximation by straight line provides adjustment for revelation of mono (fractal) by well known methods, and for permissibility to receive them by separating time series, accountable only to direct ranges of scaling change.
Fig. 4 proposes an example of optimized scaling exponent together with indication of its deviations from the direct relation
for various types of wavelet (the initial time series is generated by the Lorenz’s equation system).
Conclusions
According to given arguments our research team proposes widening of basic multifraction methodology, connected mainly with optimization of scaling exponent in relation to waveletforming function. For these purposes we create special software, which owns a whole range of functional possibilities. We can declare that it has a special functional for calculation of scaling exponent deviations from the direct relation in relation to the kind of wavelet-forming functions. The mentioned realization is fulfilled under the auspices of RFBR and it will be published in our next article.
Acknowledgment
The reported study was supported by RFBR, research project № 12-08-31108-mol_a.
References
1. Antonov V I., Zagaynov А. I., Vu. van Kuang. From the Basics of Numerical Multifractal Research to Create Automated Software Ensure [Ot osnov chislennych multifraktalnych issle-dovsniy k sozdaniyu avnomatizirovannogo programmnogo obe-specheniya]. Nauchno-technicyeskie vedomosti Sankt-Peterburg-skogo gosudarstvennogo politechnicheskogo universiteta [Sci.
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and Tech. Bull. St. Petersburg state Polytechnical Univ.]. St. Petersburg, Izdatelstvo SPbGPU, 2013, no. 2 (169), pp. 71-77.
2. Аstafeva N. М. Wavelet Analysis: Basic Theory and Examples of Application [Veyvlet-analiz: osnovy teorii i primery primeneniya]. Uspekhifizicheskikh nauk[Adv. Phys. Sci.], 1996, Vol. 166, no. 11, pp. 1145-1170.
3. Baevskiy R. M., Ivanov G. G., Gavrilushkin А. P., Dovga-levskiy P.Ya., Kukushkin Yu. А., МironovaТ. F., Prilutskiy D. А., Semenov А. V, Fedorov V F., Fleyshman А. N., Medvedev М. М., Chireykin L. V. Analysis of Variability of Heart Rhythm using Different Electrocardiographic Systems (part 1) [Analiz Varia-belnosti Serdechnogo Ritma pri Ispolzovanii Razlichnych Elek-trokardiograficheskich Sistem (chast 1)]. Vestnik aritmologii [Bull. Arrhythmol.], 2002, no. 24, pp. 65-86.
4. FracLab: A Fractal Analysis Toolbox for Signal and Image Processing (available at: http://fraclab.saclay.inria.fr).
5. Gudkov G. V. The Role of Deterministic Chaos in the Structure of Variability of the Fetal Heart [Rol determin-irovannogo khaosa v structure variabelnosti ritma serdtsa
ploda]. Sovremennye problemy nauki i obrazovaniya [Mod. Prob. Sci. Educ.], Moscow, Krasnodar, 2008, Suppl. no. 1, pp. 413-423.
6. Malik M., Bigger D. T., Shvarts A. D., Kleyder R. Е., Mali-ani A. M., Moss A. D., Shvarts P. D. Heart Rate Variability. Standards of Measurement, Physiological Interpretation of Clinical use [Variabelnosti serdechnogo ritma. Standarty izmereniya, fiz-iologicheskaya interpretatsyay klinicheskoe ispolzovanie]. Ev-ropeyskiy zhurnal po issledovaniyu serdtsa [Eur. J. Heart Res.], 1996, no. 17, pp. 354-381.
7. Mashin V. A. The Relationship of the Slope of the Regression Line of the Heart Rate Graph with Periodic and Non-Linear Dynamics of Heart in Stationary Short-Time Series [Svyaz tangensa ugla naklona linii regressii grafa serdechnogo ritma s periodicheskoy i nelineynoy dinamikoy serdtsa na korotkich statsionarnych otrezkach]. Biofizika [Biophys.], 2006, Vol. 51, is. 3, pp. 534-538.
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Мультифрактальный анализ: выявление границ применения при исследовании временных рядов
Захаров А. И., Загайнов А. И.
ВКАим. Можайского Санкт-Петербург, Россия zagainov239@gmail.com, ana63916157@yandex.ru
Аннотация. Работа посвящена выявлению возможностей разработанных к настоящему моменту мультифрактальных численных методов и средств подобной обработки временных рядов. Для этих целей использовано приложение FracLab 2.1 математического пакета MatLabR2008b, с помощью которого обработаны тестовые данные (сгенерированные, например, отображением Хенона, системой уравнений Лоренца и др.), базы записей вариабельности сердечного ритма сервера PhysioNet и собственные ЭКГ Построены трехмерные графики полученного вейвлет-преобразования, графики изолиний, мультифрактальные характеристики в виде экспоненты Хёлдера и скейлинго-вой экспоненты. В качестве вейвлет-образующей функции при вычислении спектра были использованы все возможные, предложенные в FracLab (напр. Mhat-вейвлет, DoG-вейвлет и др.). Установлены недостатки используемой реализации, полученные, прежде всего, для тестовых данных при расчетах экспоненты Хёлдера. Предложена концепция расширения мультифрактальной методологии исследования временных рядов и создание на ее основе собственного программного обеспечения.
Ключевые слова временные ряды, метод модулей максимумов вейвлет-преобразования.
Литература
1. Антонов В. И. От основ численных мультифрактальных исследований к созданию автоматизированного программного обеспечения / В. И. Антонов, А. И. Загайнов, Вуван Куанг // Науч.-технич. ведомости Санкт-Петербургского гос. поли-
Ходаковский В. А.
Петербургский государственный университет путей сообщения Императора Александра I Санкт-Петербург, Россия hva1104@mail.ru
технич. ун-та. - СПб.: Изд-во СПбГПУ 2013. - № 2 (169). -C. 71-77.
2. Астафьева Н. М. Вейвлет-анализ: основы теории и примеры применения / Н. М. Астафьева // Успехи физ. наук. -
1996. Т 166, № 11. - С. 1145-1170.
3. Баевский Р. М. Анализ вариабельности сердечного ритма при использовании различных электрокардиографических систем (ч. 1) / Р М. Баевский, Г. Г. Иванов, А. П. Гаврилушкин и др. // Вестн. аритмологии. - 2002. - № 24. - С. 65-86.
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5. Гудков Г. В. Роль детерминированного хаоса в структуре вариабельности ритма сердца плода / Г. В. Гудков // Соврем. проблемы науки и образования. - М.; Краснодар, 2008. - Прил. № 1. - С. 413-423.
6. Малик М. Вариабельности сердечного ритма. Стандарты измерения, физиологическая интерпретации клиническое использования / М. Малик, Д. T. Биггер, А. Д. Шварц и др. // Европейский журнал по исследованию сердца. - 1996. -№ 17. - P 354-381.
7. Машин В.А. Связь тангенса угла наклона линии регрессии графа сердечного ритма с периодической и нелинейной динамикой сердца на коротких стационарных отрезках /
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