Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Hurst Exponent as a Part of Wavelet Decomposition Coefficients to Measure Long-term Memory Time Series Based on Multiresolution Analysis

Vyacheslav V. Lyashenko, Rami Matarneh*, Valeria Baranova, Zhanna V. Deineko

Department of Computer Science Prince Sattam Bin Abdulaziz University Saudi Arabia, Al-Kharj

Abstract: Processing and analysis of data sequences using wavelet-decomposition and subsequent analysis of the all relevant coefficients of such decomposition is one of strong methods to study various processes and phenomena. The key point of data sequence analysis lies in the concept of Hurst exponent. This is due to the fact that Hurst exponent gives an indication of the complexity and dynamics of the correlation structure of any given time series taking into consideration the importance of Hurst exponent estimation for such analysis. There are various methods and approaches to find the Hurst exponent estimation with varying degrees of accuracy and complexity. Therefore, in this paper we have made an attempt to prove the possibility of considering an estimation of Hurst exponent based on the properties of coefficients of wavelet decomposition of a given time series. The obtained results which mainly based on the properties of detailing coefficients of wavelet decomposition show that estimation is easy to calculate and comparable with classic estimation of Hurst exponent. Also ratios has been obtained, that allow to analyze the self-similarity of a given time series.

Keywords: time series, self-similar, wavelet decomposition, Hurst exponent, wavelet-coefficients, detailing coefficient

Citation: Vyacheslav V. Lyashenko, Rami Matarneh, Valeria Baranova, Zhanna V. Deineko, Hurst Exponent as a Part of Wavelet Decomposition Coefficients to Measure Long-term Memory Time Series Based on Multiresolution Analysis. American Journal of Systems and Software, 4 (2). 51-56. January 2016. doi:10.12691/ajss-4-2-4

1. Introduction

The analysis and processing of a sequence of the data presented in the form of time series is one of the prevalent methodologies in studying various processes and phenomena which are concerned to different fields of activity and researches. This is due to the fact that data about a phenomenon or a process under study can be represented as a time series. Banking sector [1,2], securities market [3,4], health, when studying the dynamics of the analyzed processes [5] and in medicine, when the initial data are represented using time series or when the source data is converted from one form of representation into a time series are all good examples of such domains [6]. One of the important research methods of time series is a Wavelet analysis [7,8,9,10]; because it allows us to highlight the characteristics of time series, where the more important role played in this analysis is by wavelet decomposition, taking into consideration that the tool for decomposition on a set of different wavelet coefficients is the multiresolution analysis [11].

A multiresolution wavelet-analysis transforms time series to hierarchical structure by means of the wavelet transformations which results into a set of wavelet coefficients. On each new level of wavelet- decomposition there is a division of an approximating signal of the previous level (presented by some time series) on its high-frequency component and on more smoothed approximating signal [12]. Thus, the multiresolution analysis splits the time series into two components: (1) approximating coefficients and (2) detailing coefficients.

One of the characteristics of coefficients of wavelet decomposition is the Hurst exponent [13] which connects coefficients of wavelet decomposition at different levels of decomposition, besides that, the Hurst exponent also plays as an indicator of the complexity of dynamics and correlation structure of time series. The pervious discussion makes it possible to talk about the properties of the coefficients of wavelet decomposition and the role of Hurst exponent in determining these properties, while Particular importance in such analysis of time series belongs to detailing coefficients of wavelet decomposition.

2. Materials and Methods

According to discrete wavelet-transformation the time series , () consists of a set of coefficients – detailing and approximating [10,11,13,14]:

[picture] | (1)

Where – Approximating wavelet-coefficients of level

– Detailing wavelet-coefficients of level

– Chosen maximum level of decomposition

– Quantity of detailing coefficients at level of decomposition

– Quantity of approximating coefficients at level

– Mother wavelet-function

– Corresponding scaling-function.

[picture] | (2)

Where – Detailing wavelet-coefficients at level, – Scalar product of investigated sequence of data in the form of time series and a mother wavelet on corresponding level of decomposition .

In this case, the main tool to analyze time series is by processing the detailing coefficients which have been obtained on different levels. As a result the obtained series of the detailing coefficients will have the following properties [10,14,15]:

1. If time series is a self-similar process, then detailing coefficients, at each level of decomposition are all self-similar, which is mean that equal distribution of small series of wavelet coefficients at each level of decomposition with some scale will take the form:

[picture] | (3)

Where .

2. Wavelet coefficients resulting from the decomposition process with fixed increments will be fixed for each level ().

3. If there are moments of order P then the coefficients of small waves that were obtained as a result of the decomposition process must satisfy the following equation:

[picture] | (4)

[picture] | (4a)

Where – estimation value of the process.

4. If time series is self-similar, then the correlation function of wavelet-coefficients of level will decrease according to:

[picture] | (5)

Where – number of zero moments of a mother wavelet.

5. For all different levels of decomposition and for all correlations of detailing coefficients of these levels, and must equal to 0.

6. Detailing coefficients of DWT at each level of decomposition have normal distribution with a zero average .

It is clear that the detailing Hurst exponent () has been used when considering the properties of wavelet decomposition coefficients, which represents a measure of self-similarity. Hurst's exponent lies within the range and represents a key measure for the analysis of long-term dependence duration. In the case when Hurst's exponent lies in range , this mean that the time series is persistent and has a trend-stable behavior, but when Hurst's exponent lies in range , this refers to anti-persistent process (growth in the past means reduction in the future, and the tendency to reduction in the future makes probable increase in the future). At the deviations of the process are really casual and don't depend on the previous values. Therefore, the estimation of Hurst exponent values is an important task. There are various methods to estimate Hurst exponent [16,17,18,19,20], but all of them provide only approximate values, while some of them have a high computational complexity. In this work we propose a new method to estimate the Hurst exponent to expand the existing approaches to analyze the time series, using individual properties of detailing coefficients of wavelet decomposition

3. Data for Analysis

To test the obtained results, we will use the self-similar time series, which are presented in Figure 1, Figure 2 and Figure 3.

Figure 1. A modeling time series - Brownian motion (H=0.68)

Figure 2. A modeling time series - Brownian motion (H=0.75)

Figure 3. A modeling time series - Brownian motion (H=0.92)

4. Results and Discussion

Let us consider the property in point 3 which is presented in equation (4). This property takes into consideration the value of the Hurst exponent and can be represented as follows:

[picture] | (6)

By admitting logarithm into equation (6):

[picture] | (7)

Which can be transformed into:

[picture] | (8)

Or

[picture] | (9)

So equation (9) can be used to calculate the values of the Hurst exponent. But we have to make one comment: When we consider the property in point 3 which is presented in equation (4) for the detailing coefficients of the wavelet decomposition, we, in fact, operate with the module provided by the coefficient values to calculate the mathematical expectation (). At the same time in the traditional case it is the mathematical expectation of the aggregate coefficients, which are presented without a module (see equation. 4a). The assumption that the mathematical expectation can be negative makes equation (9) meaningless. So, it should be borne in mind that:

[picture] | (10)

More clearly

[picture] | (11)

Thus, when calculating the Hurst exponent in accordance with equation (9) an error may occur. Table 1 shows the calculation of the Hurst exponent for time series in accordance with the classical approaches using equation (9), where all calculations were performed in MATLAB. Hereinafter, for the wavelet decomposition of a specified time series, we use wavelet db1, knowing that experiments have shown that the use of other wavelets gives similar results.

Table 1. Values of the Hurst exponent for a specified time series

The data in Table 1 shows that the value of Hurst exponent calculated by equation (9) are comparable with the values of the Hurst exponent which are designed in accordance with conventional approaches. Therefore, equation (9) can be used to estimate the value of the Hurst exponent. Consider the two levels of decomposition () for the same series. Then, using equation (9) we get:

[picture] | (12)

Or

[picture] | (13)

But (on the basis of self-similarity of the time series [14]), therefore:

[picture] | (14)

[picture] | (15)

Then,

[picture] | (16)

[picture] | (17)

Or

[picture] | (18)

The equations (16-18) can be used as an indicators of the self-similar time series. Table 2 shows the calculations for different ratios according to equation (17) for a specific time series.

Table 2. The comparability of values for the equation (17)

Table 2 shows that the values obtained in accordance with equation (17) are comparable within the specified terms of comparison. However, it should be noted that the closer levels of decomposition of time series the more comparable calculated values in accordance with equation (17). By rewriting equation (16) we get:

[picture] | (19)

[picture] | (20)

[picture] | (21)

[picture] | (22)

Or

[picture] | (23)

Based on the above, equation (23) can be used to find the mathematical expectation of detailing coefficients for various levels of wavelet decomposition of the original time series, besides, it is also possible to consider an expression for different values of () with same values on a specific level of decomposition, such that:

[picture] | (24)

Or

[picture] | (25)

[picture] | (26)

[picture] | (27)

The equations (25) – (27) can be used as an indicators of the self-similar of the time series. Table 3 shows calculations for different ratios in accordance with equation (26) for a specified time series.

Table 3. The comparability of values for equation (26)

Table 3 shows that the values obtained in accordance with formula (26) are comparable within the specified terms of comparison. Thus, the data confirm the theoretical calculations, allowing to use equations (25) – (27) in the study and comparison of dynamics of different time series.

5. Conclusions

In this paper, have been highlighted the main points regarding the use of multiresolution wavelet analysis as a method of analysis for data sequence. We also looked at the relationship and the value of the Hurst exponent in accordance with the properties of the decomposition of wavelet coefficients of time series at different levels. Based on the properties of decomposition of detailing wavelet coefficients, we have shown the possibility of settlement to obtain an estimate the Hurst exponent for self-similar time series. For example, we have demonstrated the feasibility to estimate Hurst exponent for time series presented by Brownian motion, and at the same time we have discussed and shown the main causes of errors in the calculated ratios to assess the Hurst exponent values, where the validity of all obtained theoretical results had been confirmed by a number of examples.

References

[1]   | Bruno, V., & Shin, H. S. 2015. Capital flows and the risk-taking channel of monetary policy. Journal of Monetary Economics, 71, 119-132.
[2]   | Black, L., Correa, R., Huang, X., & Zhou, H. 2016. The systemic risk of European banks during the financial and sovereign debt crises. Journal of Banking & Finance, 63, 107-125.
[3]   | Peia, O., & Roszbach, K. 2015. Finance and growth: time series evidence on causality. Journal of Financial Stability, 19, 105-118.
[4]   | Kenett, D. Y., Huang, X., Vodenska, I., Havlin, S., & Stanley, H. E. 2015. Partial correlation analysis: Applications for financial markets. Quantitative Finance, 15(4), 569-578.
[5]   | Briesacher, B. A., Madden, J. M., Zhang, F., Fouayzi, H., Ross-Degnan, D., Gurwitz, J. H., & Soumerai, S. B. 2015. Did Medicare Part D Affect National Trends in Health Outcomes or Hospitalizations?: A Time-Series Analysis. Annals of internal medicine, 162(12), 825-833.
[6]   | SHEN, J. C., Lei, L. U. O., Li, L. I., JING, Q. L., OU, C. Q., YANG, Z. C., & CHEN, X. G. 2015. The impacts of mosquito density and meteorological factors on dengue fever epidemics in Guangzhou, China, 2006-2014: a time-series analysis.Biomedical and Environmental Sciences, 28(5), 321-329.
[7]   | Pogorelenko, N., Lyashenko, V. and Ahmad, M. 2016 Wavelet Coherence as a Research Tool for Stability of the Banking System (The Example of Ukraine). Modern Economy, 7, 955-965.
[8]   | Hosseinioun, N. 2016 Forecasting Outlier Occurrence in Stock Market Time Series Based on Wavelet Transform and Adaptive ELM Algorithm. Journal of Mathematical Finance, 6, 127-133.
[9]   | Chen, W. Y., Wen, M. J., Lin, Y. H., & Liang, Y. W. 2016. On the relationship between healthcare expenditure and longevity: evidence from the continuous wavelet analyses. Quality & Quantity, 50(3), 1041-1057.
[10]   | Lyashenko, V., Matarneh, R., & Deineko, Z. V. 2016. Using the Properties of Wavelet Coefficients of Time Series for Image Analysis and Processing. Journal of Computer Sciences and Applications, 4(2), 27-34.
[11]   | Abry P. The multiscale nature of network traffic: discovery analysis and modeling / P. Abry, R. Baraniuk, P. Flandrin // IEEE Signal Processing Magazine. 2002. № 4 (2). Р. 5-18.
[12]   | Flandrin P. Wavelet analysis and synthesis of fractional Brownian motion / P. Flandrin // IEEE Transactions on Information Theory. 1992. Vol. 38. P. 910-917.
[13]   | Kirichenko, L., Radivilova, T., & Deineko, Zh. Comparative Analysis for Estimating of the Hurst Exponent for Stationary and Nonstationary Time Series. Information Technologies & Knowledge. – Kiev: ITHEA, 2011. Vol. 5. № 1. P. 371-388.
[14]   | Lyashenko, V., Deineko, Z., & Ahmad, A. 2015. Properties of wavelet coefficients of self-similar time series. International Journal of Scientific and Engineering Research, 6(1), 1492-1499.
[15]   | Abry, P., Flandrin, P., Taqqu, M. S., & Veitch, D. 2003. Self-similarity and long-range dependence through the wavelet lens. Theory and applications of long-range dependence, 527-556.
[16]   | Alvarez-Ramirez, J., Echeverria, J. C., & Rodriguez, E. 2008. Performance of a high-dimensional R/S method for Hurst exponent estimation. Physica A: Statistical Mechanics and its Applications, 387(26), 6452-6462.
[17]   | Cajueiro, D. O., & Tabak, B. M. 2005. The rescaled variance statistic and the determination of the Hurst exponent. Mathematics and Computers in Simulation, 70(3), 172-179.
[18]   | Wang, G., Antar, G., & Devynck, P. 2000. The Hurst exponent and long-time correlation. Physics of Plasmas (1994-present), 7(4), 1181-1183.
[19]   | Delignieres, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M., & Ninot, G. 2006. Fractal analyses for ‘short’time series: a re-assessment of classical methods. Journal of Mathematical Psychology, 50(6), 525-544.
[20]   | Esposti, F., Ferrario, M., & Signorini, M. G. 2008. A blind method for the estimation of the Hurst exponent in time series: theory and application. Chaos: An Interdisciplinary Journal of Nonlinear Science, 1.