计算机应用   2016, Vol. 36 Issue (11): 3229-3233, 3238  DOI: 10.11772/j.issn.1001-9081.2016.11.3229 0

### 引用本文

LI Junshan, TONG Qi, YE Xia, XU Yuan. Adaptive residual error correction support vector regression prediction algorithm based on phase space reconstruction[J]. Journal of Computer Applications, 2016, 36(11): 3229-3233, 3238. DOI: 10.11772/j.issn.1001-9081.2016.11.3229.

### 文章历史

1. 东莞理工学院 城市学院, 广东 东莞 523419 ;
2. 火箭军工程大学 信息工程系, 西安 710025 ;
3. 中国人民解放军96261部队, 河南 灵宝 471700

Adaptive residual error correction support vector regression prediction algorithm based on phase space reconstruction
LI Junshan1,2, TONG Qi3, YE Xia2, XU Yuan1
1. City College, Dongguan University of Technology, Dongguan Guangdong 523419, China ;
2. Department of Information Engineering, Rocket Force University of Engineering, Xi'an Shaanxi 710025, China ;
3. 96261 Unit of PLA, Lingbao Henan 471700, China
Background: This work is partially supported by the Scientific Research and Reform project of Equipment Maintenance.
LI Junshan, born in 1956, Ph.D., professor. His research interests include fault diagnosis and maintenance support technology, image processing, computer vision.
TONG Qi, born in 1988, M.S., assistant engineer. His research interests include electronic equipment fault diagnosis.
YE Xia, born in 1977, Ph.D., associate professor. Her research interests include command and information system, network security.
XU Yuan, born in 1986, M.S., assistant. His research interests include intelligent detection.
Abstract: Focusing on the problem of nonlinear time series prediction in the field of analog circuit fault prediction and the problem of error accumulation in traditional Support Vector Regression (SVR) multi-step prediction, a new adaptive SVR prediction algorithm based on phase space reconstruction was proposed. Firstly, the significance of SVR multi-step prediction method for time series trend prediction and the error accumulation problem caused by multi-step prediction were analyzed. Secondly, phase space reconstruction technique was introduced into SVR prediction, the phase space of the time series of the analog circuit state was reconstructed, and then the SVR prediction was carried out. Thirdly, on the basis of the two SVR prediction of the error accumulated sequence generated in the multi-step prediction process, the adaptive correction of the initial prediction error was realized. Finally, the proposed algorithm was simulated and verified. The simulation verification results and experimental results of the health degree prediction of the analog circuit show that the proposed algorithm can effectively reduce the error accumulation caused by multi-step prediction, and significantly improve the accuracy of regression estimation, and better predict the change trend of analog circuit state.
Key words: Support Vector Regression (SVR)    multi-step prediction    error accumulation    phase space reconstruction    residual
0 引言

1 支持向量回归 1.1 ε-支持向量回归

 $f(\mathit{\boldsymbol{x}}) = \langle \mathit{\boldsymbol{w}}, \varphi (\mathit{\boldsymbol{x}})\rangle + b$ (1)

 $Q(\mathit{\boldsymbol{w}}, {\xi _i}, \xi _i^*) = \frac{1}{2}{\left\| \mathit{\boldsymbol{w}} \right\|^2} + C\sum\limits_{i = 1}^N {({\xi _i} + \xi _i^*)}$ (2)
 $\begin{array}{*{20}{c}} {s.t.}&{\left\{ \begin{array}{l} {y_i}-f({\mathit{\boldsymbol{x}}_i})-\varepsilon \le {\xi _i}, i = 1, \cdots, N\\ f({\mathit{\boldsymbol{x}}_i})-{y_i} - \varepsilon \le \xi _i^ *, i = 1, \cdots, N\\ {\xi _i}, \xi _i^ * \ge 0, i = 1, \cdots, N \end{array} \right.} \end{array}$ (3)

 $\begin{array}{l} L(\mathit{\boldsymbol{w}}, \xi, {\xi ^ * }, \varepsilon, b) = \frac{1}{2}{\left\| \mathit{\boldsymbol{w}} \right\|^2} + C\sum\limits_{i = 1}^N {({\xi _i} + \xi _i^*)}-\\ \begin{array}{*{20}{c}} {}&{}&{}&{\begin{array}{*{20}{c}} {}&{\sum\limits_{i = 1}^N {{\alpha _i}({\xi _i} + \varepsilon-{y_i} + \langle \mathit{\boldsymbol{w}} \cdot \varphi ({\mathit{\boldsymbol{x}}_i})\rangle + b)} } \end{array}} \end{array}\\ \begin{array}{*{20}{c}} {}&{}&{}&{\begin{array}{*{20}{c}} {}&{-\sum\limits_{i = 1}^N {\alpha _i^ * (\xi _i^ * + \varepsilon + {y_i} - \langle \mathit{\boldsymbol{w}} \cdot \varphi ({\mathit{\boldsymbol{x}}_i})\rangle - b)} } \end{array}} \end{array}\\ \begin{array}{*{20}{c}} {}&{}&{}&{\begin{array}{*{20}{c}} {}&{ - \sum\limits_{i = 1}^N {({\eta _i}{\xi _i} + \eta _i^ * \xi _i^ * )} } \end{array}} \end{array} \end{array}$ (4)

 $\begin{array}{l} {L_p} =-\varepsilon \sum\limits_{i = 0}^N {(\alpha _i^* + {\alpha _i})} + \sum\limits_{i = 0}^N {{y_i}(\alpha _i^*-{\alpha _i})} \\ \begin{array}{*{20}{c}} {}&{-\frac{1}{2}} \end{array}\sum\limits_{i = 0}^N {\sum\limits_{j = 0}^N {(\alpha _i^* - {\alpha _i})} } (\alpha _j^* - {\alpha _j})K({{\boldsymbol{x}}_i}, {{\boldsymbol{x}}_j}), \\ \begin{array}{*{20}{c}} {}&{s.t.}&{\sum\limits_{i = 0}^N {(\alpha _i^* - {\alpha _i})} } \end{array} = 0, 0 \le {\alpha _i}, \alpha _i^* \le C \end{array}$ (5)

 $f(x) = \sum\limits_{i = 1}^N {({\alpha _i}-\alpha _i^ * )} K(\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{x}}_i}) + b$ (6)

 $K(\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{x}}_i}) = \exp \left( {\frac{{-{{\left\| {{\mathit{\boldsymbol{x}}_i}-\mathit{\boldsymbol{x}}} \right\|}^2}}}{{2{\sigma ^2}}}} \right)\begin{array}{*{20}{c}} , &\sigma \end{array} > 0$ (7)
1.2 SVR性能分析

 $\begin{array}{*{20}{c}} {y = 0.4\sin t + 0.3\sin (\sqrt 2 t) + }\\ {0.2\sin (\sqrt 3 t) + 0.1{\rm{N}}(0, 1)} \end{array}$ (8)

 图 1 SVR回归估计

1.3 多步预测的误差累积

 图 2 SVR多步预测结果

2 相空间重构的自适应残差修正SVR预测

2.1 相空间重构技术

 $\begin{array}{l} {\mathit{\boldsymbol{X}}_i}(t) = (x({t_i}), x({t_{i + \tau }}), \cdots, x({t_{i + (m-1)\tau }}))\\ \begin{array}{*{20}{c}} {}&{}&{i = 1, 2, \cdots, N-(m-1)\tau } \end{array} \end{array}$ (9)

2.1.1 时间延迟τ的求解

 $I(\tau ) = \sum\limits_{k = 1}^N {p({x_k}, {x_{k + \tau }})In\frac{{p({x_k}, {x_{k + \tau }})}}{{p({x_k})p({x_{k + \tau }})}}}$ (10)

2.1.2 嵌入维数m的求解

 $a(i, m) = \frac{{\left\| {{\mathit{\boldsymbol{X}}_i}(m + 1)-{\mathit{\boldsymbol{X}}_{n(i, m)}}(m + 1)} \right\|}}{{\left\| {\mathit{\boldsymbol{X}}(m)-{\mathit{\boldsymbol{X}}_{n(i, m)}}(m)} \right\|}}$ (11)

 $E(m) = \frac{1}{{N-m\tau }}\sum\limits_{i = 1}^{N-m\tau } {a(i, m)}$ (12)

E(m)实际上是一个期望值，然后将mm+1维的变化定义为：

 $F(m) = E(m + 1)/E(m)$ (13)

2.2 本文算法

 图 3 自适应残差修正SVR预测算法示意图

3 实验验证与结果分析 3.1 仿真验证与分析

 图 4 自适应残差修正SVR多步预测结果

3.2 实例验证与分析

 $r(\mathit{\boldsymbol{x}}, {\mathit{\boldsymbol{y}}_i}) = \frac{{\sum\limits_{j = 1}^N {({x_j}-\bar x)} ({y_{ij}}-{{\bar y}_i})}}{{\sqrt {\sum\limits_{j = 1}^N {{{({x_j}-\bar x)}^2}\sum\limits_{j = 1}^n {{{({y_{ij}} - {{\bar y}_i})}^2}} } } }}$ (14)

PPMCC与其他距离度量方式相比，无需对特征量进行预处理，可直接计算健康度，步骤简单，对实现模拟电路故障预测非常有利。若元器件无故障，参数值与标称值相等，则PPMCC的值为1，即元器件健康度也为1；反之，参数值偏离标称值，健康度会慢慢降低。

 图 5 CTSV滤波器电路

 图 6 R3的健康度轨迹

 图 7 R3-200仿真实验的预测结果

4 结语

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