基于高斯过程的混沌时间序列单步与多步预测

Single-step and multiple-step prediction of chaotic time series using Gaussian process model

针对混沌时间序列单步和多步预测,提出基于复合协方差函数的高斯过程(GP)模型方法.GP模型的确立由协方差函数决定,通过对训练数据集的学习,在证据最大化框架内,利用矩阵运算和优化算法自适应地确定协方差函数和均值函数中的超参数.GP模型与神经网络、模糊模型相比,其可调整参数很少.将不同复合协方差函数的GP模型应用在混沌时间序列单步及多步提前预测中,并与单一协方差函数的GP、支持向量机、最小二乘支持向量机、径向基函数神经网络等方法进行了比较.仿真结果表明,基于不同复合协方差函数的GP方法能精确地预测混沌时间序列,具有稳健的特性.因此,它是研究复杂非线性动力系统辨识和控制的一种有效方法.

For the chaotic time series single-step and multi-step prediction, Gaussian processes （GPs） method based on composite covariance function is proposed. GP priors over functions are determined mainly by covariance functions, and through learning from training data sets, all hyperparameters that define the covariance function and mean function can be estimated by using matrix operations and optimal algorithms within evidence maximum bayesian framework. As a probabilistic kernel machine, the number of tunable parameters for a GP model is greatly reduced compared with those for neural networks and fuzzy models. GP models with different composite covariance functions are applied to chaotic time series single-step and multi-step ahead prediction and compared with other models such as standard GP model with single covariance function, standard support vector machines, least square support vector machine, radial basis functional （RBF） neural networks, etc. Simulation results reveal that GP method with using different composite covariance functions can be used to accurately predict the chaotic time series and show stable performance with robustness. Hence, it provides an effective approach to studying the properties of complex nonlinear system modeling and control.