基于不连续回归树的最大李雅谱诺夫指数计算方法

DISCONTINUOUS REGRESSION TREE ESTIMATION OF LARGEST LYAPUNOV EXPONENT

最大李雅谱诺夫指数是判断动力系统稳定性和检验混沌的主要依据。运用回归树的随机梯度Boosting拟合非线性函数,提出一种从时间序列计算最大李雅谱诺夫指数的新方法。由于回归树不连续,其雅可比矩阵不存在,传统雅可比方法不能运用。直接从回归树计算最大李雅谱诺夫指数,不考虑拟合函数的雅可比函数。随机模拟结果表明该方法能很好逼近真值,且对噪声和嵌入维数稳健。最后计算移动通话和短信总量两个实测数据的李雅谱诺夫指数,结果表明本文方法和人工神经网络具有同样的结论。

Largest Lyapunov exponent is a useful measure of the stability of a dynamics system and successful method to test for chaos. A new method to estimate largest Lyapunov exponent from observed time series is proposed. It fits nonlinear function by stochastic gradient Boosting of regression tree. Since regression trees are discontinuous, their Jacobin matrix doesn＇t exist and regular methods based on function estimator fails. It directly calculate Lyapunov exponent from them without using Jacobin matrix of the fitted function. A simulation study shows that the new method approximates the true value excellently and has great robustness to noise and embedding dimeusion. The Lyapunov exponents of two observed daily series, mobile telephonometry and total number of short message, are calculated. The result manifests that this method and artificial neural network deduce the same conclusion.