Multiscale Fluctuation Features of the Dynamic Correlation between Bivariate Time Series

The fluctuation of the dynamic correlation between bivariate time series has some special features on the time-frequency domain. In order to study these fluctuation features, this paper built the dynamic correlation network models using two kinds of time series as sample data. After studying the dynamic correlation networks at different time-scales, we found that the correlation between time series is a dynamic process. The correlation is strong and stable in the long term, but it is weak and unstable in the short and medium term. There are key correlation modes which can effectively indicate the trend of the correlation. The transmission characteristics of correlation modes show that it is easier to judge the trend of the fluctuation of the correlation between time series from the short term to long term.The evolution of media capability of the correlation modes shows that the transmission media in the long term have higher value to predict the trend of correlation. This work does not only propose a new perspective to analyze the correlation between time series but also provide important information for investors and decision makers.


Introduction
A financial time series records the behavior trajectory of a financial market.By analyzing the fluctuation features of time series, we can understand the structures and characteristics of a financial market.There are many studies which focus on the long-term cointegration relationship between two time series [1][2][3][4].However, other scholars have provided evidence that, with the fluctuation of time series, the relationship between any two time series also changes over time [5][6][7].The fluctuation of the correlation between time series can help us to detect the dynamic features of the interaction between them.
Since Zhang and Small first proposed that univariate time series can be transformed to a complex network [8], many literatures have proved that different dynamic characteristics of univariate time series show different topological structures [9][10][11].In the last decade, the complex network has showed its effectiveness in time series analysis in multiple areas, including financial markets [12], engineering [13][14][15], medicine [16][17][18], and geophysics [19].Based on these existing researches, few studies are concerned with another trigger of turning time series to network: the hidden multiscale information in the dynamic relationship between bivariate time series in the frequency domain.Although some scholars have made great progress in how to derive multifrequency complex network to characterize the dynamical behavior of time series [20,21], it is still a challenge to transfer the dynamic relationship between bivariate time series to a complex network involving multiscale information and to explore the underlying fluctuation features with time and frequency change simultaneously.
Many studies indicate that financial time series contain different information in the time domain and frequency domain and have different fluctuation characteristics in different time-scales [22][23][24][25].The fluctuation in the short term, medium term, and long term can provide different reference information for different purposes [26][27][28].To find the multiscale information in the relationship between bivariate time series, the wavelet analysis provides an effective 2 Mathematical Problems in Engineering solution.The wavelet analysis can obtain the seasonal or periodic characteristics by filtering nonlinear data to satisfy the different investors' needs [29][30][31].Thus, it is necessary to decompose the original series using wavelet analysis to obtain the multiscale fluctuation characteristics of time series.
How does one obtain the multiscale dynamic correlation between bivariate time series?As we all know, at each timescale, time series fluctuates over time.If we divide the entire time series into different subperiods, then each subperiod has its own status of the correlation fluctuation.In this study, we defined the correlation modes representing the different correlation statuses [7].The correlation modes change with time and interact with each other successively, forming the correlation modes transmission networks, which will reveal the fluctuation of the dynamic correlation between the bivariate time series.
In this paper, we design an algorithm to combine the complex network with wavelet analysis to investigate the fluctuation of the dynamic correlation between bivariate time series on the time-frequency domain.First, we decompose the original time series into the decomposed sequences at different time-scales.Next, we build a correlation modes transmission network at each time-scale.Finally, we study the multiscale fluctuation characteristics of the dynamic correlation between bivariate time series using the complex network analytical approach, including the recognition of the key correlation modes, the transmission characteristics, and the transmission media.

MODWT Wavelet Decomposition.
There are various classifications of the discrete wavelet basis used in existing work, such as MODWT (Maximal Overlap Discrete Wavelet Transform), Haar A-Trous, sym4, and db9 wavelets [32][33][34][35].As one of the discrete wavelet transformations, MODWT is widely used [32].Based on previous studies on the applications of wavelet methods for specific purposes in economics and finance, it appears that MODWT can avoid the adverse effects attributable to the choice of a starting point or an origin for analysis.With help of MODWT, the scale-based analysis of time series can reveal the characteristics of volatility.
Let {  ,  = 0, 1, 2, . . ., −1} be a time series, where  is the length of time series.The MODWT of level  0 for the original time series   yields the column vectors D1 , D2 , . . ., D 0 and S 0 for any positive integer  0 , each of which has  dimensions.D contains the MODWT wavelet coefficients associated with changes in   between scale 2 −1 and scale 2  , and S contains the MODWT scaling coefficients associated with the smoothness of   at the scale 2  0 .
According to the output from the filters at each scale, the time series   can be decomposed and reconstructed into wavelet details and approximation as follows: Wavelet decomposition based on  time-scales ( = 1, 2, . ..) includes  1 ,  2 , . . .,   , representing different deviation subseries from the trend, and   , representing the longterm trend subseries: (2) In this paper, we set the size of a sliding window for 10 days because we want to analyze the short-term fluctuation features of the correlation between bivariate time series.First, we choose day  as start point and get subperiod  which is from day  to day  + 9.Then, we choose day  + 1 as start point and get subperiod +1 which is from day  + 1 to day  + 10.By that analogy, we can obtain a series of subperiods.Then, at each time-scale, we follow four steps to construct the correlation modes transmission networks.

The Multiscale Correlation Modes Transmission
Step 1.We calculate the correlation coefficient between two time series in each subperiod at the same time-scale and then obtain a sequence of the correlation coefficients.The sequence shows the fluctuation of the correlation between bivariate time series at each time-scale [7].In this paper, we use the Pearson Correlation Coefficient to measure the correlation between time series as follows: and  denote the value of the subseries of two time series at each time-scale, where  and  denote the mean of the subseries.  and   denote the value of the subseries at time .Let  denote the values' number of the subseries.The range of the value of Step 2. We symbolize the strength of the correlation between two time series at different time-scales.After calculating the correlation coefficient between two time series at each time-scale, we divide the correlation coefficient into 5 levels.
Then, we get a sequence of symbolized correlation coefficients at each time-scale.The 5 levels are defined as follows [36]: (4) Step 3. We define the correlation modes between two time series by the coarse graining process [39].The coarse graining process is the concept in the phase space.The smallest grain in the phase space is one dot.If we consider roughly a set of dots as a mode, the study of a series of the dots can become the research on modes consisting of a complex system [36,40].Similar to the process for dividing the time series into different subperiods, we use sliding windows to obtain a series of the correlation modes from the symbolized correlation coefficients.We set the size of a sliding window for a transaction period (5 days).Thus, the correlation modes are comprised of 5 symbolized correlation coefficients.Each correlation mode represents the status of the correlation fluctuation of two time series at the corresponding time-scale.
Step 4. We build the correlation modes transmission networks at different time-scales.Like we defined at Step 3, we obtain a series of correlation modes with the moving of the sliding window.One correlation mode converted to another as time goes by: mode 1 → mode 2 → mode 3 → ⋅ ⋅ ⋅ → mode  .Because the conversion between two types of correlation modes would repeat in the transmission process, the trajectory of the conversion among correlation modes forms a network.In the correlation modes transmission network, we take the correlation modes as nodes and the succeeding sequence relations between the correlation modes as edges.The weight of an edge is the frequency of the transformation between two types of correlation modes.The process of building the correlation modes transmission networks is shown in Figure 1.

Data Description.
In this paper, we focus on the relationship between bivariate time series.As the precondition of our work, there should be the theoretical and realistic relationship between two time series.Based on this, we choose the international crude oil spot market and natural gas spot market as the sample markets.The data we choose are the WTI crude oil spot price from NYMEX and natural gas spot price from the Henry Hub.The data cover the period from January 7, 1997, to November 23, 2015.All data are selected from EIA (US Energy Information Administration).  denotes the daily closing price on day .The daily price return is calculated as follows: Node Edge Sliding window

MODWT Wavelet Decomposition for the Return Series of Crude Oil and Natural Gas Prices.
To study the multiscale phenomenon in the fluctuation of the correlation between the daily crude oil spot price and natural gas spot price, we decompose the return series of original prices into different subseries.We choose  0 = 6 as the maximum level of MODWT.Many researches show that using 6 levels can achieve an optimal balance between sample size and the length of the filter for financial or economic time series [41,42].In our research, decomposing the return series into 6 levels can divide the time series into three fluctuation periods: short term, medium term, and long term, so the results can provide targeted suggestion for different investors and decision makers.The scale  1 represents the 2-4 days' horizon and the scale  2 represents the 4-8 days' horizon.These two scales represent the short-term variation.The scales  3 (8-16 days) and  4 (16-32 days) represent the medium-term variation.The scales  5 (32-64 days) and  6 (64-128 days) represent the long-term variation. 6 represents the trend of the original series obtained after subtracting  1 ,  2 ,  3 ,  4 ,  5 , and  6 from the series.The results of the wavelet decomposition are shown in Figures 2 and 3. prices, we use sliding windows to calculate the dynamic correlation coefficients between crude oil and natural gas prices at each time-scale (see Figure 4).According to Figure 4, the correlation coefficients change over time.The fluctuation of the correlation has the multiscale feature.For the scales  1 and  2 , in the short term, the correlation between crude oil and natural gas prices strongly fluctuates with small change but high frequency.For the scales  3 and  4 , in the medium term, the change ranges are larger but the frequency is lower.For the long-term  5 and  6 , the correlation has the largest change ranges and lowest frequency.Furthermore, the correlation coefficients tend to −1 and 1 and usually last for a period of time after peaking in the long term.It means that the correlation between crude oil and natural gas prices gradually strengthens and tends to be stable with the increase of the time-scales.

The Statistic
After symbolizing the correlation coefficients, the correlation coefficients are divided into 5 levels which are strongly negatively correlated, weakly negatively correlated, not correlated, weakly positively correlated, and strongly positively correlated.The distribution of the correlation levels at different time-scales over time is shown in Figure 5.The five colors from dark to light represent five levels of the correlation.According to Figure 5, with the increase of the time-scales, the percentage accounted for by the three colors between the darkest and the lightest continues to decrease, but the percentage accounted for by the darkest and the lightest continues to increase.After calculating, the percentage of the strongly correlation for the scales  1 and  2 , respectively, accounts for 3.3% and 8.2%.For the scales  3 and  4 , the percentage, respectively, increases to 26.4% and 47.1%.For the scales  5 and  6 , the percentage is relatively high and it, respectively, accounts for 73% and 82.7%.The same color continues longer with the increase of the timescales.This result also means that the correlation between crude oil and natural gas prices gradually strengthens and tends to be stable with the increase of the time-scales from another perspective.

The Recognition of the Key Correlation Modes.
To recognize the key correlation modes at different time-scales, we choose weighted out-degree to measure the importance of the correlation modes.The weighted out-degree of one node represents that node's weight connecting to adjacent nodes.The weighted out-degree is calculated as follows: denotes the set of the nodes connecting to node . , denotes the weight of the edge from node  to node .The weighted out-degree at all time-scales follows the power-law distribution (Figure 6).The values of weighted out-degree of most correlation modes are small but few modes' values of weighted out-degree are very large.This finding means that there are few modes that have a strong effect on the transmission process of the correlation modes.
Figure 7 shows the distributions of weighted out-degree at all time-scales during the time span.In Figure 7, different colors represent different value of weighted out-degree.The lighter the color is, the larger the value of weighted outdegree is.Every value of weighted out-degree represents a correlation mode, so every color represents a correlation   mode.The lighter the color is, the more important the correlation modes are.According to Figure 7, we can find that the key correlation modes have more and more important influence on the transmission process with the increase of the time-scales.The width of one color bar represents the time a correlation mode continues.Each mode will continue for some time and then convert into another mode.The average time that the correlation modes at each time-scale continue is shown in Table 1.In the short term,  1 and  2 (2-8 days), and in the medium term,  3 and  4 (8-32 days), the color distributions of weighted out-degree are uneven, and the colors all continue for a short time.It means that the transmission speed of the correlation modes is fast.The process that one mode converts to another has stochastic characteristics.In the long term,  5 and  6 (32-128 days), the proportion of the light color increases and the colors continue longer.It means that the transmission speed of the correlation modes slows with the increase of the time-scales.
Ranking the correlation modes at all time-scales by weighted out-degree, we can recognize the key correlation modes for the dynamic process.The key correlation modes for the correlation modes transmission networks at different  Thus, crude oil and natural gas prices are primarily not correlated or are weakly correlated in the short term (2-8 days) and medium term (8-32 days).In the long term (32-128 days), crude oil and natural gas prices are primarily strongly positively correlated.

The Transmission Characteristics of the Correlation Modes.
The transmission characteristics of the correlation modes include two parts: the transmission speed and the transmission direction.We choose the average path length to measure the transmission speed.The transmission direction is described by the average weighted clustering coefficient.The average path length equals the average number of the edges that one correlation mode will pass to reach another.If one correlation mode will pass few correlation modes to another correlation mode, it means that the transmission speed is fast.The average weighted clustering coefficient represents the tightness of the network.The larger the average weighted clustering coefficient is, the more closely the correlation modes connect each other.In the loose network, the paths formed by the transmission of the correlation modes are mostly catenarian.In the tight network, the transmission of the correlation modes is more random.The principle of the weighted clustering coefficient is shown in Figure 8.The average path length is calculated as follows: denotes the number of the nodes in the network and  , denotes the shortest path from node  to node .
The weighted clustering coefficient of node  is calculated as follows [43]: is the weighted degree of node .  is the degree of node .  is the weight of the edge connecting node  to node .  is the adjacent matrix of node  and node .The mean of the weighted clustering coefficients of all nodes is the average weighted clustering coefficient of the network.According to Figure 9, the general trend is that the average path length increases and the average weighted clustering coefficient decreases with the increase of the time-scales.This result means that transmission speed becomes slower with the increase of the time-scales.In the short term and medium term, the transmission of the correlation modes has the stochastic features.In the long term, the transmission of the correlation modes has the characteristics that have a certain direction.

The Recognition of the Transmission Media.
In the transmission of the correlation modes, some correlation modes connect two or more correlation modes which do not have a direct connection, and these modes are called the transmission media.Recognizing the important transmission media can help us to judge the tendency of the correlation between crude oil and natural gas prices.In this paper, we choose the betweenness centrality to measure the media capability of the correlation modes in the correlation modes transmission networks at different time-scales.The mean of the betweenness centrality of all correlation modes is the average betweenness centrality of the network, which represents the average level of the media capability that the modes have in the networks.The standard betweenness centrality is calculated as follows [5]: () denotes the number of the shortest paths that pass node  between node  and node .  denotes the number of the shortest paths between node  and node .
Figure 10 shows that the average betweenness centrality is stable in the short term and medium term, and the average betweenness centrality in the long term is clearly higher.Therefore, the modes have stronger media capability in the long term, and it explains that the transmission of the correlation modes has the characteristics that have a certain direction from another perspective.There are some key transmission media in the correlation modes transmission networks at different time-scales, shown in Table 3.When these modes appear in the transmission, they usually predict that the correlation between crude oil and natural gas prices is in the midst of transformation.

Conclusion
For studying the multiscale fluctuation features of the dynamic correlation between bivariate time series, we proposed an algorithm to map the fluctuation of the correlation into networks which can show us the nonlinear and nonstationary moving trajectory of the correlation modes.After analyzing the structure characteristics of the correlation modes transmission networks at different time-scales, we got some findings that can help us understand the fluctuation characteristics of time series on both the time domain and frequency domain.
The correlation between bivariate time series is in the fluctuation, which means that the correlation is dynamic.The fluctuation of the correlation at different time-scales has different features.In the short term, the correlation between crude oil and natural gas prices strongly fluctuates with small change ranges but high frequency.In the long term, the change ranges are large but the frequency is low.There are few key correlation modes which show the main status of the correlation at different time-scales.In the short term, medium term, and long term, the key correlation modes are, strongly negative correlated)  2 (−0.8 <   ≤ −0.3, weakly negative correlated)  3 (−0.3<   < 0.3, no correlation)  4 (0.3 ≤   < 0.8, weakly positive correlated)  5 (0.8 ≤   ≤ 1, strongly positive correlated) .
The fluctuation of the correlation coefficients

Figure 1 :
Figure 1: Schematic illustration of building the correlation modes transmission networks.Step 1: calculate the correlation coefficient in each subperiod.Step 2: symbolize the strength of the correlation coefficients.Step 3: abstract the correlation modes from symbolized correlation coefficients.Step 4: build the correlation modes transmission networks.

Figure 2 :Figure 3 :
Figure 2: The wavelet decomposition for the return series of crude oil price.

S 6 Figure 4 :
Figure 4: The fluctuation of the correlation between crude oil and natural gas prices at different time-scales.

6 Figure 5 :
Figure 5: The distribution of the symbolized correlation coefficients over time at different time-scales.

Figure 6 :
Figure 6: The distribution of weighted out-degree.

6 Figure 7 :
Figure 7: The distribution of weighted out-degree on the time span.

Figure 9 :Figure 10 :
Figure 9: The evolution of the transmission characteristics of the networks.

Table 1 :
The average time that the correlation modes at all timescales continue.