Fluctuations of the nonlinear time series are driven by the traverses of multiscale conformations from one state to another. Aiming to characterize the evolution of multiscale conformations with changes in time and frequency domains, we present an algorithm that combines the wavelet transform and the complex network. Based on defining the multiscale conformation using a set of fluctuation states from different frequency components at each time point rather than the single observable value, we construct the conformational evolution complex network. To illustrate, using data of Shanghai’s composition index with daily frequency from 1991 to 2014 as an example, we find that a few major conformations are the main contributors of evolution progress, the whole conformational evolution network has a clustering effect, and there is a turning point when the size of the chain of multiscale conformations is 14. This work presents a refined perspective into underlying fluctuation features of financial markets.
Detecting the dynamical features of a timedependent complex system mainly depends on time series analysis. This problem is complicated by the nonlinear characteristic of the original time series [
In this paper, we focus on financial time series. As we know, financial markets consist of a number of stakeholders with objects in various time horizons, which results in financial time series comprising a combination of different frequency components [
In regard to the multiscale conformation problem, wavelets offer an effective solution: representing the original time series as a function with two variables, namely, time and frequency [
Aiming to encode the underlying multiscale conformation evolution features of financial time series, we propose a new algorithm incorporating wavelet transform and the complex network. First, we use the wavelet transform to decompose an original time series into timefrequency domain. We then define the multiscale conformation for one time point with a set of frequency components; a process which offers us a detailed description for current time points rather than for a singular number. Multiscale conformations varying as time changes together form an evolutionary process. We identify the multiscale conformations as nodes, the transmissions over time as edges, and the edges’ weight as the frequency of transmission. Hence, we construct the multiscale conformation evolution process as a multiscale evolution complex network. A structural features analysis could help us to explore the underlying dynamical features of financial time series.
First, we use the continuous wavelet transform to obtain the wavelet power spectrum of an original financial time series [
According to the Heisenberg uncertainty principle, there is always a tradeoff between the localization of time and scale. For the purpose of extracting features, the Morlet wavelet with
The continuous wavelet transform could be obtained by projecting the original time series onto the specific wavelet
From the continuous wavelet transform, we can obtain further information about the time series: namely, amplitude. The square of the amplitude
Based on wavelet power results, we define the multiscale fluctuation conformation at each time point. The frequency band of the wavelet power matrix ranges from 2 to 512 days, and we discretize the successive frequency bands as sets of 9 separate frequency bands including 2 days, 4 days, 8 days, 16 days, 32 days, 64 days, 128 days, 256 days, and 512 days to represent the multiscale components. The discretized wavelet power matrix is defined as
According to the actual value of
Each time point has one corresponding multiscale conformation that consists of nine fluctuation states from nine frequency components. For example, the multiscale conformation for the first time point is
We consider the multiscale conformation for each time point to be a node, transmissions denoted with the corresponding time of multiscale conformations to be edges, and the frequency of the same transmission between conformations to be weight.
We choose the Shanghai (security) composite index (SHCI) from January of 1991 to December of 2014 in daily frequency to serve as a data source. The SHCI represents the fluctuation in the Shanghai stock market comprehensively (Figure
Original time series of Shanghai (security) composite index from January 1999 to December 2014.
The wavelet power spectrum of the SHCI can help us to understand the fluctuation of the SHCI as it varies with time and frequency (Figure
The wavelet power spectrum of the SHCI.
Based on the wavelet power spectrum, we obtain 9 frequency bands through discretion (Figure
The discretion results of the wavelet power spectrum of SHCI.
According to the above, there are 5870 multiscale conformations for 5870 data points. These multiscale fluctuation conformations transform into each other with changes over time, forming the multiscale fluctuation evolution process. In fact, there are fewer than 5870 types of multiscale conformations due to the repetitive attendance of some conformations. We can thus obtain a directed and weighted multiscale fluctuation evolution network (Figure
The multiscale fluctuation evolution network.
Based on the multiscale fluctuation evolution network, we first use an index of the weighted outdegree to identify the major multiscale conformations. Then, the index of betweenness centralities is used to describe the transmission capability of multiscale conformations. The index of the modularity class could be implemented to explore the clustering effect of multiscale conformations. Finally, we divide the multiscale evolution network into multiscale conformation chains with different lengths and characterize the fluctuation features of SHCI by analyzing the evolution process of these chains.
For the time series containing 5870 time points, there are supposed to be 5870 multiscale conformations and 5869 edges. In fact, due to the repetitiveness of some multiscale conformations, there are 417 nodes and 1165 edges in reality. To identify major multiscale conformations, we introduce the index of the weighted outdegree, which not only can depict the number of neighbor conformations of one multiscale conformation but can also demonstrate the weight between multiscale conformations and their neighbors. We define the weight outdegree of one multiscale conformation as follows:
Distributions of the weighted outdegree of node. (a) Cumulative distribution of the weighted outdegree of the node (sorted by the value of the weighted outdegree of the nodes in descending order,
The result shows that 111 types of multiscale conformations occur during the evolution process with high frequency and are the main constructors of the SHCI fluctuation status. (Here we list the top 10 conformations in Table
The top ten multiscale conformations ranked by the weighted outdegree.
Number  Conformations  Weighted outdegree  Percentage (%) accounts for total weighted outdegree 

1 

649  10.87 
2 

291  4.87 
3 

228  3.82 
4 

152  2.55 
5 

151  2.53 
6 

149  2.50 
7 

115  1.93 
8 

101  1.69 
9 

99  1.66 
10 

99  1.66 
The transmission capability is another crucial characteristic of multiscale conformations during the evolution process and can be described by the index of betweenness centralities. In the complex network of multiscale conformations, the average length of characteristic paths is 11.805, which means that the average shortest transmission path between any two random multiscale conformations must pass another 11 or 12 conformations. In a smallworld network, the shortest transmission path is less than 6 on average and the transmission between any two nodes is easy. Hence, in the evolution complex network of multiscale conformations, transmissions between any two multiscale conformations are not convenient and have to be connected by other multiscale conformations that are playing a medium role in the evolution process. Multiscale conformations with high attendance on the shortest transmission path have good transmission capabilities, meaning that these conformations could control more information in the multiscale evolution process. The transmission capability of multiscale conformations can be described by betweenness centralities. The normalized definition of betweenness centralities
The evolution of the betweenness centrality of multiscale conformations over time is depicted in Figure
The weighted outdegree of the top ten conformations ranked by betweenness centrality.
Number  Conformations  Betweenness centrality  Weighted outdegree 

1 

0.6445  649 
2 

0.2808  291 
3 

0.2386  62 
4 

0.2382  13 
5 

0.2035  39 
6 

0.1813  93 
7 

0.1787  30 
8 

0.1762  19 
9 

0.1734  5 
10 

0.1649  14 
The evolution of the betweenness centrality of multiscale conformations over time.
An example of multiscale conformations with high betweenness centrality and low weighted outdegree. The multiscale conformations
Based on weighted outdegree and betweenness centrality, we find that transmissions among some multiscale conformations are difficult. Without some conformations as medium, all multiscale conformations will be separated into several distinct groups. We ask whether there are a number of multiscale conformations clustered closely that could easily transmit to each other. Therefore, we use a modularity index to measure the partition of communities in the evolution network of multiscale conformations, with higher values of the modularity demonstrating a better partition of a complex network. The definition of modularity can be written as follows [
The modularity partitions algorithm is divided into two phases repeated iteratively. First, each node is considered to be a community. The number of communities is therefore equal to that of the nodes. We then evaluate the gain of the modularity (
In the second phase, taking the communities found in the first phase to be nodes, we can build up a new network. The weight of the edges of the new network can be obtained from the sum of the weights of edges between nodes in two corresponding communities. Edges between nodes of the same community lead to selfloops in the new network. When the second phase is finished, the first phase is repeated, thus resulting in a new network. These two phases are repeated until there are no more changes (details in [
We apply the algorithm to the evolution complex network of multiscale conformations. The modularity of the evolution complex network is 0.809 and the whole complex network is separated into 20 communities. The number of each community ranges from 8 to 32. Inside each community, multiscale conformations transmit to each other continently. Among these communities, some cannot transmit to another directly. In this case, conformations with high betweenness centrality work as pivots to connect isolated communities.
The evolution complex network of multiscale conformations is characterized by a clustering effect. Furthermore, we divided the evolution process of multiscale conformations into various chains that consist of different numbers of multiscale conformations. The number of multiscale conformations inside the chain is defined as the size of the chain. We then explore the transmission features of these chains with size changes. We define the chain size as
First, the number of multiscale conformation chains increases with increasing size, meaning that the diversity of chains is growing when their components increase. Simultaneously, the percentage of chains with the highest attendance accounts in the total number of chains decreases as the size of the chain grows. As mentioned above, it is obvious that there is a turning point at which the repetition of chains is scarce. For a multiscale complex network, the turning point appears when the size of the chain becomes 14.
Another point deserving notice is the status of chains with the highest attendance. Generally, the selftransmission of multiscale conformations with high weighted outdegree constitutes the chain of the multiscale fluctuation with high attendance. Hence, when the multiscale conformation with high weighted outdegree appears for several days, this conformation may last for a period in the near future (Figure
The number of chains of multiscale conformations and percentage of the chains with the highest attendance accounts in the total number of chains of multiscale conformations.
The nonlinearity of financial time series caused by the hidden multiscale information complicates the exploration of underlying mechanisms. We present a multiscale algorithm combining wavelet transform and the complex network, offering meticulous insight into characterizing the evolution process of the multiscale conformation that drives the fluctuation of observed values within the original time series. We first implement a continuous wavelet to obtain the wavelet power spectrum of the original time series that depicts its fluctuation as changes with time and fluctuation. We then define the multiscale conformations of each time point based on the construction of an evolution complex network. Finally, we characterize the evolution complex network through analyses of the major conformations, the transmission capacity of the conformations, the clustering effect, and the transmission of the chains of multiscale conformations.
We use the SHCI daily data from January 1990 to December 2014 as an example and construct the evolution complex network of multiscale conformations. More specifically, the weighted outdegree and betweenness centrality of the multiscale conformations follow the power law distribution. We could therefore identify major conformations that are the main contributors in the whole network and major transmit conformations which serve as pivots connecting separated conformations. Such information could be considered to be a reference to changes within the SHCI. We then find that the whole evolution complex network can be partitioned into 20 communities inside of which transmissions among conformations are easily made. Among some communities the major transmit conformations only transmit to each other. In this case, the appearance of transmit conformations could be a signal that depicts structural changes characteristic of SHCI. Finally, through the analysis of transmissions of multiscale fluctuation chains, the turning point of SHCI appears when chain size is 14.
In addition, to prove the validation and effectiveness of the algorithm, we apply it to other stock indices from four major countries, namely, UK (FTSE100), Germany (Dax 30), US (DJIA), and Japan (NIKKEI225), from January of 1991 to December of 2014 in daily frequency. They contain 6084, 6254, 6079, and 5912 data points, respectively. We found that the weighted outdegrees of these four stock indices also follow the power law distribution and their exponents of the distributions of the weighted outdegree are approximately 1. Moreover, it is obvious that nearly 80% transmissions happen among roughly 30% of the multiscale conformations. These results prove that the multiscale evolution features of the four major stock indices are very similar to that of the SHIC (Figure
Distributions of the weighted outdegree of node. Cumulative distribution of the weighted outdegree of the node of (a) DAX, (c) FTSE100, (e) DJIA, and (g) NIKKEI225 (sorted by the value of the weighted outdegree of the nodes in descending order). Doublelogarithmic plot between the weighted outdegree of node and the probability of the weighted outdegree of (b) DAX, (d) FTSE100, (f) DJIA, and (h) NIKKEI225.
DAX 30
DAX 30
FTSE100
FTSE100
DJIA
DJIA
NIKKEI225
NIKKEI225
In this study, we try to put forward an approach to uncovering the underlying mechanism of nonlinearity expressed by a financial time series, which can help us to understand financial markets in greater detail. However, the nonlinearity of financial prices may be affected by other factors that we try to involve in a future study.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was partly supported by the National Natural Science Foundation of China (Grant no. 71173199) and the Fundamental Research Funds for the Central Universities (Grant no. 35732015060).