Low dimensional mid-term chaotic time series prediction by delay parameterized method

Abstract

How to predict the future behavior of complex systems with insufficient information, i.e., low dimensional mid-term chaotic time series prediction in mathematical terms, is not only a significant theoretical problem, but a more intricate practical problem. To address this issue, a Delay Parameterized Method (DPM) for low dimensional mid-term chaotic time series forecasting is presented. The correlation function, which immerses the low dimensional information into reconstructed space, is introduced to bridge time series and hidden order of system in DPM. Traversal algorithm and intelligent algorithm including particle swarm optimization or genetic algorithm, are used to obtain the optimal parameters for prediction. In addition, the applications of the proposed method on Lorenz chaotic time series, stress-strain signals and stock K-line maps show that it produces high quality predictions.

Introduction

Predicting the future behavior of complex systems from observations is one of the most important and classical problems in various scientific fields, such as earthquake prediction [1], temperature prediction [2], wind speed forecasting [3], stock prediction [4], [5], estimating of agriculture production [6], quality of service prediction [7], [8], etc. Generally, the predictability of time series is believed to come from various kinds of information hidden in the system’s output. That is to say, linear or nonlinear correlations and statistical characteristics in time series provide a source for predicting future behavior. A challenging task is to make accurate prediction based on time series data sets [9].

In the past decades, advanced methods have been proposed for prediction based on those sources. In 1987, Farmer and Sidorowich pioneered work in this field, focusing on the prediction of a single variable chaotic time series [10]. The time series, x(t), were reconstructed into a state space, $X\left(t\right)=\left(x\right(t),x(t-\tau ),x(t-2\tau ),\cdots ),$ by delay coordinates technique [11], [12], where τ is time delay. The predictor, $f_T:X(t+T)=f_T\left(X\left(t\right)\right),$ is then approximated by the global or local neighbor information of past data in reconstructed phase space. Later work by Sugihara and May [13] fixed the number of neighbour points for prediction, and compared the standard correlation coefficient between predicted and real value to distinguish chaos from measurement error in time series.

For the multi-dimensional time series prediction, Ye and Sugihara developed a new idea and proposed the Multi-View Embedding (MVE) method [14], based on the generalized embedding theorem [15]. In this MVE method, $C_{nk}^m-C_{n(k-1)}^m$ numbers of $m-$dimensional phase spaces were reconstructed with a given k lags for each of n variables, where $C_{nk}^m$ is the number of possible combinations picking m out of nk coordinates. This method is computationally intensive if the dimension is relatively high. For example, with $k=3,m=3$ for a 6-variables system, the number of phase spaces would be 164. Recently, Ma et al. proposed an Inverse Embedding Method (IEM) for multi-dimensional time series prediction, which reduced the computational complexity [16]. The intertwined dynamic (hidden in the high dimensional data) was transformed into a phase space reconstructed by one-dimensional signal. The predictor was obtained from the correlation between non-delay reconstructed state space and delay reconstructed phase space.

Meanwhile, with the development of computer science and technology, machine learning methods [17], [18], including deep belief network [19], [20], wireless/wired networks [21], multi-input-multi-output network [22], [23], [24], neural network ensemble [25], [26], [27], long-short memory [28], reservoir computing [29], [30], cloud computing [31], [32] have been intensively studied and applied to achieve dynamical prediction of the system. The machine learning methods generally extract statistical or dynamical characteristics in a black-box manner for prediction. Moreover, artificial intelligence algorithms, such as Particle Swarm Optimization (PSO) [33], and Genetic Algorithm (GA) [34], employ different strategies to optimize the objective function during the forecasting process. It should be noted that the performance of the artificial neural networks crucially relies on the length of the available training data. The accurate prediction needs the training set contain sufficiently large amount of training data.