Mathematical Problems in Engineering

Volume 2016, Article ID 4742060, 9 pages

http://dx.doi.org/10.1155/2016/4742060

## Multiscale Fluctuation Features of the Dynamic Correlation between Bivariate Time Series

^{1}School of Humanities and Economic Management, China University of Geosciences, Beijing 100083, China^{2}Key Laboratory of Carrying Capacity Assessment for Resource and Environment, Ministry of Land and Resources, Beijing 100083, China^{3}Open Lab of Talents Evaluation, Ministry of Land and Resources, Beijing 100083, China

Received 1 June 2016; Revised 12 September 2016; Accepted 20 September 2016

Academic Editor: Zhong-Ke Gao

Copyright © 2016 Meihui Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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 [1–4]. However, other scholars have provided evidence that, with the fluctuation of time series, the relationship between any two time series also changes over time [5–7]. 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 [9–11]. In the last decade, the complex network has showed its effectiveness in time series analysis in multiple areas, including financial markets [12], engineering [13–15], medicine [16–18], 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 [22–25]. The fluctuation in the short term, medium term, and long term can provide different reference information for different purposes [26–28]. To find the multiscale information in the relationship between bivariate time series, the wavelet analysis provides an effective solution. The wavelet analysis can obtain the seasonal or periodic characteristics by filtering nonlinear data to satisfy the different investors’ needs [29–31]. 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 time-scale, 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.

#### 2. Algorithm and Data Description

##### 2.1. 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 [32–35]. 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 be a time series, where is the length of time series. The MODWT of level for the original time series yields the column vectors and for any positive integer , each of which has dimensions. contains the MODWT wavelet coefficients associated with changes in between scale and scale , and contains the MODWT scaling coefficients associated with the smoothness of at the scale .

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 () includes , representing different deviation subseries from the trend, and , representing the long-term trend subseries:

##### 2.2. The Multiscale Correlation Modes Transmission Networks Construction

After decomposing time series using MODWT, we obtained the subseries of the time series at different time-scales. At each time-scale, we divide the subseries into different subperiods. In this paper, we use sliding windows. Compared with dividing time series into different individual time periods, the advantage of sliding windows is that they contain the features of memory and transitivity [6, 36–38]. The length of sliding windows depends on the needs of the analysis. If the goal is to study the short-term fluctuation of time series, the length can be set to a smaller value. If the goal is to understand the fluctuation of time series in the long term, the length can be set to a larger value.

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 which is from day to day . Then, we choose day as start point and get which is from day to day . 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.

*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 is .

*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]:

*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: . 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.