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]}:

(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.

(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:

(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:

(4)

(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:

(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.

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

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

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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:

(6)

By admitting logarithm into equation (6):

(7)

Which can be transformed into:

(8)

Or

(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:

(10)

More clearly

(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

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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:

(12)

Or

(13)

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

(14)

(15)

Then,

(16)

(17)

Or

(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)

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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:

(19)

(20)

(21)

(22)

Or

(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:

(24)

Or

(25)

(26)

(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)

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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.

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