Mathematical Problems in Engineering

Volume 2016, Article ID 8087178, 10 pages

http://dx.doi.org/10.1155/2016/8087178

## The Chaotic Attractor Analysis of DJIA Based on Manifold Embedding and Laplacian Eigenmaps

School of Economics and Management, North China Electric Power University, Beijing 102206, China

Received 31 January 2016; Accepted 3 May 2016

Academic Editor: Fazal M. Mahomed

Copyright © 2016 Xiaohua Song 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

By using the techniques of Manifold Embedding and Laplacian Eigenmaps, a novel strategy has been proposed in this paper to detect the chaos of Dow Jones Industrial Average. Firstly, the chaotic attractor of financial time series is assumed to lie on a low-dimensional manifold that is embedded into a high-dimensional Euclidean space. Then, an improved phase space reconstruction method and a nonlinear dimensionality reduction method are introduced to help reveal the structure of the chaotic attractor. Next, the empirical study on the financial time series of Dow Jones Industrial Average shows that there exists an attractor which lies on a manifold constructed by the time sequence of Moving average convergence divergence; finally, Determinism Test, Poincaré section, and translation analysis are used as test approaches to prove both whether it is a chaos and how it works.

#### 1. Introduction

Chaos is a concept defined as a deterministic dynamical system that is sensitive to initial conditions and gives rise to an unpredictable behavior in the long term. Although it seems like a paradox, this concept of chaos has been widely accepted in fields such as economics and finance for almost three decades, since the first chaotic phenomenon was reported in economics [1]. Chaos theory promotes the search for a mechanism that allows economists to interpret the observed movements in real economic and financial data and helps to distinguish whether it is exogenous or endogenous. In this sense, it represents a deeper thinking about the essential characteristics of the evolutionary process of an economic and finance system, namely, whether the fluctuation of a system is caused by stochastic phenomena or effected by strong nonlinearities. If the outlier events are caused by some sort of random walk which is unpredictable, we may just neglect them in empirical study. However, if they are caused by internal factors of the system, their characteristics are informative for revealing their generating mechanism, which will be an important issue for forecasting the system.

Chaos and its implications for system-forecasting have been enthusiastically and hotly debated in financial markets for a long time [2]. Takens’ theorem has proved that a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system [3], thus making financial time series a potentially simple and convenient measurement for chaos detecting. Despite this, it remains difficult to detect an obviously chaotic behavior in financial time series. This is because, from an empirical point of view, it is difficult to distinguish which part of the sequence is provoked by randomness and which part is determined by nonlinearity [4].

In view of this, some indirect methods have been introduced to help investigate the natural characteristics of financial chaotic phenomena. For example, the correlation dimension method is a straightforward and fast implement to calculate the fractal dimension of a possibly underlying strange attractor [5, 6]; the Largest Lyapunov Exponents (LLE) proposed by Wolf et al. and Rosenestein et al. is a more appropriate tool to detect chaotic phenomena [7]. Besides, based on the correlation integral, Brock et al. developed a statistical test for independence known as the BDS test to test the serial dependence and nonlinear structures of financial time series. Notably, unlike the correlation dimension method, it assumes the delay time equals one [8].

The above described applications for chaos detecting require large, stable, and clean sample data. However, the empirical data from financial markets are often small and noisy, which make it hard to detect the chaos. Thus, to overcome this limit, some topological methods have been introduced to detect the financial chaotic behaviors. These methods are characterized by studying the organization of the strange attractor [9] and exploiting an essential property of chaotic systems, through which the relationship between the empirical data that are not possible to be discovered in the original time sequence can be revealed [10].

Although studies on chaotic financial systems are fruitful [11–13], there are still various disputes. Among them, two most controversial views are as follows: Could the financial system be characterized by a low-dimensional chaos [14]? And if there is a chaotic phenomenon, is it able to prove that it is generated from a deterministic system [4]?

In this paper, Dow Jones Industrial Average (Dow Jones Industrial Average, US) is analyzed as representative financial time series. Before the studying, a specific topological structure of financial system has been assumed, in which the complicated high-dimensional phase space of financial system can be represented by a few independent degrees of freedom embedded in a low-dimensional nonlinear manifold. Under this assumption, the rest of this paper is organized as follows. In Section 2, the theorems of time series reconstruction and dimensionality reduction are probed. Firstly, an improved embedding theorem is adopted to construct a compact but redundant phase space of financial system. Then, Laplacian Eigenmaps is used to denoise and map this space to a low-dimensional manifold, for the sake of extracting the strange attractor hidden behind the complicated chaotic system. In Section 3, empirical data selection and processing are discussed, and the time series of Moving average convergence divergence is applied to substitute the traditional price series to reconstruct the chaotic attractor of financial system. In addition, the wavelet denoising tool is also introduced to smooth the noise of MACD sequence. In Section 4, empirical data of financial market DJIA are studied to find out whether it truly has a chaotic attractor, by virtue of some measures such as Determinism Test and Poincaré section and then translation analysis of chaotic attractor will perform to help understand the internal mechanisms of chaos generation. Finally, some conclusions are drawn in Section 5.

#### 2. Phase Space Reconstruction and Manifold Dimensionality Reduction

##### 2.1. Phase Space Reconstruction

For a dynamical system, phase space volumes are contracted by the time evolution. The trajectory of such a system typically settles on a subset of , which is called an attractor [15]. However, in practice, it is usually impossible to measure all the components of the dimensional vector space. Fortunately, the embedding theorem proved the following.

If is a dimensional manifold, , is a smooth diffeomorphism, and is a twice-differentiable function, let , where ; then is an embedding from to .

Thus, the phase space of dynamical system may be reconstructed by one of the embedding features of and can be formalized aswhere denotes the embedding dimension, is the sample time, is an appropriate integer, and the delay time is . A diffeomorphism between the reconstructed and original phase spaces exists ifwhere is the dimension of the compact manifold containing the attractor. If an appropriate parameter of delay time and embedding dimension [16] are selected, the original dynamical system can be characterized by the reconstructed phase space.

In this paper, different from traditional methods of phase space reconstruction, the time delay is fixed to 1 so that a minimum correlation dimension of embedding space with respect to can be decided by the G-P algorithm.

The formula follows:where if , or ; is the size of the data set, denotes the number of embedded points in -dimensional space, and represents the sup-norm.

measures the fraction of the pairs of points , whose sup-norm separation is not bigger than . If the limit of as exists for each , then the fraction of all state vector points within can be denoted by . In particular, when , the correlation dimension is defined as

Thus, the G-P algorithm is simplified to a single parameter estimation with respect to . However, the simplification may cause another problem; that is, although the reconstruction of the phase space can preserve the properties of the original dynamical system, there will be a significant redundancy within the reconstructed phase space because of . This makes it hard to reveal the true structure of the chaotic attractor embedded in the phase space. To overcome this limit, some ideas of nonlinear dimensionality reduction [17] can be introduced. One approach is to assume the chaotic attractor of the dynamical system lies on an embedded nonlinear manifold within a higher-dimensional Euclidean space.

##### 2.2. Manifold Dimensionality Reduction

In this paper, the manifold dimensionality reduction methodology called Laplacian Eigenmaps [18] is used to extract the chaotic attractor from the redundant reconstructed phase space of financial system. Suppose there is a Euclidean space , and the Laplacian Eigenmaps is defined as , where is the adjacency matrix of edge weights and is the diagonal matrix with . The eigenvalues and eigenvectors of the Laplacian reveal a wealth of information about the phase space . The algorithmic procedure can be illustrated as follows.

Firstly, construct the adjacency distance between and , where by distinguishing whether , where the norm is the usual Euclidean norm in .

Then, determine the weights of the adjacency matrix by heat kernel, in which the weight is set as if there is an edge between node and .

Finally, compute Eigenmaps under the assumption that the weighted graph is connected, and map it into a lower dimensional space, where . This embedding is given by the matrix where the th row provides the embedding coordinates of the th vertex. Thus, the objective function can be formalized aswhere is the dimensional representation of the th vertex. In this way, this algorithm can be reduced to finding

The entire process of this section can be shown in Figure 1 (the Lorenz system as an example).