Abstract
Chaotic time series can well reflect the nonlinearity and non-stationarity of real environment changes. The traditional kernel adaptive filter (KAF) with second-order statistical characteristics suffers performance degeneration dramatically for predicting chaotic time series containing noises and outliers. In order to improve the robustness of adaptive filters in the presence of impulsive noise, a nonlinear similarity measure named Cauchy kernel loss (CKL) is proposed, and the global convexity of CKL is guaranteed by the half-quadratic (HQ) method. To improve the convergence rate of stochastic gradient descent and avoid a local optimum simultaneously, the conjugate gradient (CG) method is used to optimize CKL. Furthermore, to address the issue of kernel matrix network growth, the Nyström sparse strategy is adopted to approximate the kernel matrix and then the probability density rank-based quantization (PRQ) is used to improve the approximation accuracy. To this end, a novel Nyström Cauchy kernel conjugate gradient with PRQ (NCKCG-PRQ) algorithm is proposed for the prediction of chaotic time series in this paper. Simulations on prediction of synthetic and real-world chaotic time series validate the advantages of the proposed algorithm in terms of filtering accuracy, robustness, and computational storage complexity.
Highlights
(Nyström kernel conjugate gradient based on k-means sampling,NKCG-KM) [28]算法。其中, CKCG 算法是采用具有鲁棒特性的柯西核函数作为代价函数,通过共轭梯度法(conjugate gradient,CG)有效优化该代价函数,但随着输入数据的不断加入,核矩阵规模不断增大, 计算存储负担加重。CKCG 算法因其鲁棒性强、滤波精度高和运算时间长的特点,是未经 过稀疏处理的典型代表。RFFCCG 算法是采用鲁棒柯西函数作为代价函数,通过 CG 方法优 化该代价函数,利用随机傅里叶特征和均匀随机数来近似核函数,因此具备低计算复杂度 且与传统核算法相当的性能,是固定维度稀疏算法的代表。NKRGMC-probability density rank-based quantization (PRQ) 算法采用信息论 学习(information theoretic learning, ITL)准则中广泛采用的广义最大相关熵准则作为代价函 数,通过经典的递归更新方式、基于 Nyström 低秩近似的核矩阵以及 PRQ 采样方法进一步 提高了滤波精度。NKCG-KM 因采用传统二阶误差准则作为代价函数在非高斯环境下不具 备鲁棒性。因此,为了全面体现所提 NCKCG-PRQ 算法的优越性,本文是从代价函数的鲁 棒性、优化方法、稀疏策略以及采样方法选择比较算法。为了体现比较的公平性,对所有 参与比较的算法参数进行设置,使其都达到理想的滤波精度。
本文将基于 probability density rank-based quantization (PRQ) 采样的 Nyström 方法作为稀疏策略应用于核共轭梯度优化法,有效地 抑制了核共轭梯度法的核矩阵网络规模增长。进一步,采用半平方法转化柯西核函数生成 全局凸的鲁棒函数作为核共轭梯度法的代价函数,提出了一种新的具有鲁棒性的基于 PRQ 采样的 Nyström 柯西核共轭梯度(NCKCG-PRQ)算法。NCKCG-PRQ 在存储需求较低的情况 下,与没有采取稀疏策略的柯西核共轭梯度(Cauchy kernel conjugate gradient,CKCG)算法拥 有近乎相当的滤波精度。在数值仿真的 MG 混沌时间序列和由蔡氏电路真实产生的混沌时 间序列的预测中,结果表明在存在脉冲噪声的情况下,NCKCG-PRQ 与其他固定维度的自 适应滤波器相比,具有较低计算复杂度和较高的预测精度,有效证实了所提 NCKCG-PRQ 算法在混沌时间序列预测中的性能优势。
Summary
1)(西南大学电子信息工程学院,重庆 400715) 2) (非线性电路与智能信息处理重庆市重点实验室, 重庆 400715) 混沌时间序列能够较好反映真实环境的非线性和非平稳性特性,然而具有二阶统计特 性的核自适应滤波器(kernel adaptive filter, KAF)在处理含噪声和异常值的混沌时间序列时其 预测性能显著下降。为提高核自适应滤波器的鲁棒性,本文提出了一种用于测量非线性相 似度的柯西核损失(Cauchy kernel loss,CKL),并采用半平方(half-quadratic, HQ)方法保证了 CKL的全局凸性。为改善随机梯度下降法收敛速度较慢且容易陷入局部最优的不足,采用 共轭梯度(conjugate gradient,CG)方法优化CKL。进一步,为解决核矩阵网络增长的问题, 采取Nyström稀疏策略近似核矩阵,并利用概率密度秩量化(probability density rank-based quantization, PRQ)提高逼近精度。基于此,本文提出了一种新的基于Nyström和PRQ的柯西 核共轭梯度(Nyström Cauchy kernel conjugate gradient with PRQ,NCKCG-PRQ)算法有效实现 了混沌时间序列的预测。基于合成和真实两类混沌时间序列验证了所提NCKCG-PRQ算法在 稳态性能,鲁棒性和计算存储复杂度上的优势。 受启发于半平方(half-quadratic, HQ) [24]方法转化CKL后代价函数的全局凸性和鲁棒性, 本文根据混沌时间序列的短期可预测性在CG优化方法上引入了基于PRQ采样的Nyström方 法, 提出了 一种新的基于PRQ采样的 Nyström 柯西核共轭梯度 (Cauchy kernel conjugate gradient, CKCG)算法(NCKCG-PRQ)。所提的NCKCG-PRQ算法在预测麦克格拉斯(MackeyGlass,MG)混沌时间序列和实际电路采集的蔡氏混沌时间序列的两类实例中,展现了在非 高斯噪声环境下预测性能的强鲁棒性、低稳态误差及低计算复杂度的特点。 V 0 v = − exp (−si ) 时达到上确界。利用熵半二次优化法[24]可得: N 当 vi = − (1+ siexp(−si ))−1 时达到上确界。此命题证毕。
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