Mathematical Problems in Engineering

Volume 2015, Article ID 563145, 9 pages

http://dx.doi.org/10.1155/2015/563145

## The Multiscale Conformation Evolution of the Financial 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}Lab of Resources and Environmental Management, China University of Geosciences, Beijing 100083, China

Received 29 May 2015; Accepted 22 July 2015

Academic Editor: Michael Small

Copyright © 2015 Shupei Huang 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

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.

#### 1. Introduction

Detecting the dynamical features of a time-dependent complex system mainly depends on time series analysis. This problem is complicated by the nonlinear characteristic of the original time series [1]. Since Lacasa et al. proposed their famous visibility graph algorithm to transfer the time series into networks [2], the last decade has witnessed the success and effectiveness of the complex network in solving nonlinear problems for time series analysis in multiple disciplines, including financial markets [3–5], engineering [6–8], medicine [9–11], and geophysics [12]. Based on these existing contributions, there is yet another trigger of nonlinearity to be concerned with the hidden multiscale information in the frequency domain. It is still a challenge to determine how to transfer a complex network involving multiscale information from the original time series and to explore the underlying evolution features with time and frequency changes simultaneously.

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 [13–15]. Such frequency components form a multiscale conformation behind the original time series and changes within multiscale conformations drive fluctuations of the observed time series values [16, 17]. In other words, at one time point, a multiscale conformation decides the corresponding observed value. Hence, exploring the dynamic features of the multiscale conformational evolution progress will offer new insight into understanding fluctuations in financial time series from a meticulous perspective.

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 [18]. Hence, the implementation of wavelet transform enables us to detect the evolution of different frequency components for different time [19]. In other words, a wavelet working as a “microscope” could observe an original time series using a different “resolution.” A finer resolution is better at detecting the details of an original signal, and a low resolution is well-suited for trend analysis [20, 21].

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 time-frequency 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.

#### 2. Algorithm and Data Description

##### 2.1. Decomposition in Time-Frequency Domains

First, we use the continuous wavelet transform to obtain the wavelet power spectrum of an original financial time series [22] which could depict the fluctuation of the time series for different frequencies and time [23]. At the heart of continuous wavelet transform is the idea that an original time series should be represented as a function of frequency and time through a wavelet, while the original time series is considered as a function of time alone [18]. A wavelet is a square integral function with real value and zero mean in which there are two parameters: namely, location () and scale (). The location parameter could determine the wavelet’s position in time by shifting the wavelet, while the scale parameter could stretch or dilate the wavelet to localize different frequencies:

According to the Heisenberg uncertainty principle, there is always a trade-off between the localization of time and scale. For the purpose of extracting features, the Morlet wavelet with is a good choice because it provides a good balance between time and frequency localization [24]:

The continuous wavelet transform could be obtained by projecting the original time series onto the specific wavelet as characterized by location and scale parameters. It could thus be represented as the following equation:

From the continuous wavelet transform, we can obtain further information about the time series: namely, amplitude. The square of the amplitude is defined as the wavelet power spectrum, which indicates the power distribution of different frequency components of the original time series evolving over time, a large power corresponding to high fluctuation and vice versa. In actuality, the wavelet power result is an matrix , where represents a different time point and represents a different frequency band. We can thus represent the wavelet power matrix as a visible wavelet power spectrum.

##### 2.2. Constructing the Multiscale Evolution Complex Network

*Step 1 (discretization of the frequency band). *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 .

*Step 2 (symbolization of the fluctuation level). *According to the actual value of , defining the fluctuation level as four types by discretizing a subsequence of continuous observations from one frequency band into four equal zones, the symbolization of the four fluctuation levels is defined as (very high), (high), (weak), and (very weak).

*Step 3 (definition of a multiscale conformation). *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 .

*Step 4 (construction of the multiscale evolution complex network). *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.

##### 2.3. Data Description

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 1). There are 5870 data points, and the data are extracted from the wind database. We transform the original SHCI time series into a logarithmic return rate as .