Prediction of chaotic time series based on Nyström Cauchy kernel conjugate gradient algorithm

Qi Le-Tian; Wang Shi-Yuan; Shen Ming-Lin; Huang Gang-Yi

doi:10.7498/aps.71.20212274

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.

Keywords:

Authors and contacts

1.
College of Electronic Information Engineering, Southwest University, Chongqing 400715, China
2.
Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing 400715, China

Corresponding author: Wang Shi-Yuan, wsy@swu.edu.cn

Funds: Project supported by the National Nature Science Foundation of China (Grant No. 62071391), Natural Science Foundation of Chongqing (Grant No. cstc2020jcyj-msxmX0234), and Fundamental Research Funds for the Central Universities (Grant No. 2020jd001).

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References

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Cited By

图 1 示波器示意图

Figure 1. Schematic diagram of oscilloscope.

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图 2 蔡氏电路原理图

Figure 2. Schematic diagram of the Chua’s circuit.

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图 3 不同采样点个数$ m $对NCKCG-PRQ算法的稳态MSE值和平均运算时间的影响　(a) MG混沌时间序列; (b) 蔡氏混沌时间序列

Figure 3. Influence of different number of sampling points on steady-state MSE value and average operation time of NCKCG-PRQ algorithm: (a) MG chaotic time series; (b) chaotic time series based on Chua’s circuit.

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图 4 在脉冲噪声环境下不同算法的测试MSE学习曲线　(a) MG时间序列 ; (b)蔡氏混沌时间序列

Figure 4. Testing MSE learning curves of different algorithms in impulsive noise environment: (a) MG chaotic time series; (b) chaotic time series based on Chua’s circuit.

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图 5 NCKCG-PRQ算法对测试数据的最终预测结果　(a) MG混沌时间序列; (b) 蔡氏混沌时间序列

Figure 5. Final predicted results of NCKCG-PRQ algorithm for the test sets: (a) MG chaotic time series; (b) chaotic time series based on Chua’s circuit.

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表 1 NCKCG-PRQ算法

Table 1. NCKCG-PRQ algorithm.

输入: 输入输出对$\left\{ { {{\boldsymbol{u}}_k}, {d_k} } \right\}, k{\text{ = } }1, 2, \cdot \cdot \cdot$
初始化参数: $ {\boldsymbol{\hat u}}(i) $为PRQ采样后样本点; $ {\boldsymbol{\varLambda }} $和W分别为有关$ {\boldsymbol{\hat U}} $核矩阵的特征值降序排列的对角矩阵和对应特征向量构成列正交矩阵; ${ {\boldsymbol{K} }_c}(1) = [\kappa ({\boldsymbol{u} }(1), {\boldsymbol{\hat u} }(1)), \cdots , \kappa ({\boldsymbol{u} }(1), {\boldsymbol{\hat u} }(m))]$为初始核向量; 映射输入$ {{\boldsymbol{z}}_1} = {{\boldsymbol{\varLambda }}^{ - 1/2}}{{\boldsymbol{W}}^{\text{T}}}{{\boldsymbol{K}}_c}{(1)^{\text{T}}} $; 权重$ {\boldsymbol{\varOmega }}_1^z = 0 $; 加权函数$ {\zeta _1} = {\exp}( - {s_1})/\left\{ {{\delta ^2}\left[ {{\eta ^{ - 1}} + {s_1}{\exp}( - {s_1})} \right]} \right\} $; 期望$ {d_1}{\text{ = }}{({{\boldsymbol{\varOmega }}_1}^z)^{\text{T}}}{{\boldsymbol{z}}_1} $; 相关矩阵$ R_1^z = {\zeta _1}{{\boldsymbol{z}}_1}{\boldsymbol{z}}_1^{\rm T} $; 互相关向量${\boldsymbol{c}}_1^z={\zeta _1}{d_1}{{\bf{z}}_1}$; 冗余向量${\boldsymbol{r}}_1^z ={\boldsymbol{ c}}_1^z - {\boldsymbol{R}}_1^z{\boldsymbol{\varOmega } }_1^z$; 方向向量${\boldsymbol{p}}_1^z = {\boldsymbol{r}}_1^z$, 遗忘因子$ \lambda {\text{ = }}0.999 $ 　　循环$ \left( {k{\text{ = 2}}, 3, \cdot \cdot \cdot } \right) $:
1.输入核向量${ {\boldsymbol{K} }_c}(i) = [\kappa ({\boldsymbol{u} }(i), {\boldsymbol{\hat u} }(1)), \cdots , \kappa ({\boldsymbol{u} }(i), {\boldsymbol{\hat u} }(m)]$; 　　2.映射输入${\boldsymbol{z} }( \cdot ) = { {\boldsymbol{\varLambda } }^{ - 1/2} }{ {\boldsymbol{W} }^{\text{T} } }{[\kappa ( \cdot , {\boldsymbol{\hat u} }(1)), \cdots , \kappa ( \cdot , {\boldsymbol{\hat u} }(m))]^{\text{T} } }$; 　　3.误差更新$ {e_{k + 1}} = {d_{k + 1}} - {\left( {{\boldsymbol{\varOmega }}_k^z} \right)^{\text{T}}}{{\boldsymbol{z}}_{k + 1}} $, 加权函数$ {\zeta _k} = {\exp}( - {s_k})/\left\{ {{\delta ^2}\left[ {{\eta ^{ - 1}} + {s_k}{\exp}( - {s_k})} \right]} \right\} $ 　　4.自相关矩阵更新${\boldsymbol{R}}_{k + 1}^z = \lambda {\boldsymbol{R}}_k^z + {\zeta _{k + 1} }{ {\boldsymbol{z} }_{k + 1} }{\boldsymbol{z} }_{k + 1}^{\rm T}$, 计算步长${\alpha _k} = \frac{ { { {\left( {{\boldsymbol{p}}_k^z} \right)}^{\text{T} } }{\boldsymbol{r}}_k^z} }{ { { {\left( {{\boldsymbol{p}}_k^z} \right)}^{\text{T} } }{\boldsymbol{R}}_{k + 1}^z{\boldsymbol{p}}_k^z} }$; 　　5.权重更新${\boldsymbol{\varOmega } }_{k + 1}^z = {\boldsymbol{\varOmega } }_k^z + {\alpha _k}{\boldsymbol{p}}_k^z$, 残差向量更新${\boldsymbol{r}}_{k + 1}^z = \lambda {\boldsymbol{r}}_k^z - {\alpha _k}{\boldsymbol{R}}_{k + 1}^z{\boldsymbol{p}}_k^z + {\zeta _{k + 1} }{{\bf{z}}_{k + 1} }{{\boldsymbol{e}}_{k + 1} }$; 　　6.计算步长 ${\beta _k} = \frac{ { { {\left( {{\boldsymbol{r}}_{k + 1}^z} \right)}^{\text{T} } }\left( {{\boldsymbol{r}}_{k + 1}^z - {\boldsymbol{r}}_k^z} \right)} }{ { { {\left( {{\boldsymbol{r}}_k^z} \right)}^{\text{T} } }{\boldsymbol{r}}_k^z} }$, 共轭方向更新${\boldsymbol{p}}_{k + 1}^z = {\boldsymbol{r}}_{k + 1}^z + {\beta _k}r_k^z$ 　　循环终止

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表 2 不同算法在MG混沌时间序列中的仿真结果

Table 2. Simulation results of different algorithms in MG chaotic time series.

实验模型	算法	字典数目	运算时间/s	稳态MSE/dB
MG混沌时间序列	CKCG	2000	41.486	–35.443
	RFFCCG	60	2.095	–32.006
	NKRGMC-PRQ	60	2.383	–33.068
	NKCG-KM	60	3.745	N/A
	NCKCG-PRQ	60	1.584	–34.446

DownLoad: CSV

表 3 不同算法在蔡氏电路混沌时间序列中的仿真结果

Table 3. Simulation results of different algorithms in chaotic time series based on Chua's circuit.

实验模型	算法	字典数目	运算时间/s	稳态MSE/dB
蔡氏混沌时间序列	CKCG	2000	42.618	–35.840
	RFFCCG	13	1.128	–34.926
	NKRGMC-PRQ	13	0.988	–35.819
	NKCG-KM	13	1.146	N/A
	NCKCG-PRQ	13	0.957	–35.865

DownLoad: CSV

[1]	林毅, 刘文波, 沈骞 2018 物理学报 67 230502 Google Scholar Lin Y, Liu W B, Shen Q 2018 Acta Phys. Sin. 67 230502 Google Scholar
[2]	王梦蛟, 吴中堂, 冯久超 2015 物理学报 64 040503 Google Scholar Wang M J, Wu Z T, Feng J C 2015 Acta Phys. Sin. 64 040503 Google Scholar
[3]	唐舟进, 任峰, 彭涛, 王文博 2014 物理学报 63 050505 Google Scholar Tang Z J, Ren F, Peng T, Wang W B 2014 Acta Phys. Sin. 63 050505 Google Scholar
[4]	梅英, 谭冠政, 刘振焘, 武鹤 2018 物理学报 67 080502 Google Scholar Mei Y, Tan G Z, Liu Z T, Wu H 2018 Acta Phys. Sin. 67 080502 Google Scholar
[5]	王新迎, 韩敏, 王亚楠 2013 物理学报 62 050504 Google Scholar Wang X Y, Han M, Wang Y N 2013 Acta Phys. Sin. 62 050504 Google Scholar
[6]	王世元, 史春芬, 钱国兵, 王万里 2018 物理学报 67 018401 Google Scholar Wang S Y, Shi C F, Qian G B, Wang W L 2018 Acta Phys. Sin. 67 018401 Google Scholar
[7]	Peng L B, Li X F, Bi D J, Xie Y L 2018 Signal Process. Lett. 25 1335 Google Scholar
[8]	赵永平, 张丽艳, 李德才, 王立峰, 蒋洪章 2013 物理学报 62 120511 Google Scholar Zhao Y P, Zhang L Y, Li D C, Wang L F, Jiang H Z 2013 Acta Phys. Sin. 62 120511 Google Scholar
[9]	张家树, 党建亮, 李恒超 2007 物理学报 56 67 Google Scholar Zhang J S, Dang J L, Li H C 2007 Acta Phys. Sin. 56 67 Google Scholar
[10]	张洪宾, 孙小端, 贺玉龙 2014 物理学报 63 040505 Google Scholar Zhang H B, Sun X D, He Y L 2014 Acta Phys. Sin. 63 040505 Google Scholar
[11]	唐舟进, 彭涛, 王文博 2014 物理学报 63 130504 Google Scholar Tang Z J, Peng T, Wang W B 2014 Acta Phys. Sin. 63 130504 Google Scholar
[12]	火元莲, 王丹凤, 龙小强, 连培君, 齐永锋 2021 物理学报 70 158401 Google Scholar Huo Y L, Wang D F, Long X Q, Lian P J, Qi Y F 2021 Acta Phys. Sin. 70 158401 Google Scholar
[13]	火元莲, 王丹凤, 龙小强, 连培君, 齐永锋 2021 物理学报 70 028401 Google Scholar Huo Y L, Wang D F, Long X Q, Lian P J, Qi Y F 2021 Acta Phys. Sin. 70 028401 Google Scholar
[14]	Wu Z, Shi J, Xie Z, Ma W 2015 Signal Process. 117 11 Google Scholar
[15]	Liu W F, Pokharel P P, Príncipe J C 2008 IEEE Trans. Signal Process. 56 543 Google Scholar
[16]	Engel Y, Mannor S, Meir R 2004 IEEE Trans. Signal Process. 52 2275 Google Scholar
[17]	Chen B D, Príncipe J C 2012 Signal Process. Lett. 19 491 Google Scholar
[18]	Li C G, Shen P C, Liu Y, Zhang Z Y 2013 IEEE Trans. Signal Process. 61 4011 Google Scholar
[19]	Chen B D, Xing L, Zhao H Q, Zheng N N, Príncipe J C 2016 IEEE Trans. Signal Process. 64 3376 Google Scholar
[20]	Li X L, Lu Q M, Dong Y S, Tao D C 2019 Trans. Neural Netw. Learn. Syst. 30 2067 Google Scholar
[21]	Shi W, Xiong K, Wang S Y 2019 IEEE Access. 7 120548 Google Scholar
[22]	Lei Y W, Hu T, Li G Y, Tang K 2020 Trans. Neural Netw. Learn. Syst. 31 4394 Google Scholar
[23]	Zhang M, Wang X J, Chen X M, Zhang A X 2018 IEEE Trans. Signal Process. 66 4377 Google Scholar
[24]	Xiong K, Herbert H C, Wang S Y 2021 IEEE Trans. Cybern. 51 5497 Google Scholar
[25]	Chen B D, Zhao S L, Zhu P P, Príncipe J C 2013 Trans. Neural Netw. Learn. Syst. 24 1484 Google Scholar
[26]	Zhang T, Wang S Y, Huang X W, Jia L 2020 Signal Process. Lett. 27 361 Google Scholar
[27]	Zhang T, He F L, Zheng Z, Wang S Y 2020 IEEE Trans. Circuits Syst. Express Briefs 67 2772 Google Scholar
[28]	He F L, Xiong K, Wang S Y 2020 IEEE Access. 8 18716 Google Scholar
[29]	Zhang T, Wang S Y 2020 Signal Process. Lett. 27 1535 Google Scholar
[30]	Qi L T, Shen M L, Wang D L, Wang S Y 2021 Signal Process. Lett. 28 1011 Google Scholar
[31]	Zhang H N, Yang B, Wang L, Wang S Y 2021 IEEE Trans. Signal Process. 69 1859 Google Scholar
[32]	Xiong K, Wang S Y 2019 Signal Process. Lett. 26 740 Google Scholar
[33]	Qin Z D, Chen B D, GU Y T, Zheng N N, Príncipe J C 2020 IEEE Trans. Neural Netw. Learn. Syst. 31 3100 Google Scholar
[34]	Zheng Y F, Wang S Y, Feng J C, Tse C K 2016 Digit. Signal Process. 48 130 Google Scholar
[35]	Weng B W, Barner K E 2005 IEEE Trans. Signal Process. 53 2588 Google Scholar
[36]	Huang X W, Wang S Y, Xiong K 2019 Symmetry 11 1323 Google Scholar

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