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1. 防灾科技学院 灾害信息工程系, 北京 101601;
2. 信阳师范学院 计算机与信息技术学院，河南 信阳 464000

Surrogate model of radial basis function networks based on width factor sensitivity analysis
ZHANG Yanxia1, CHEN Danqi1, HAN Ying1, LIU Daohua2
1. Disaster Information Engineering Department, Institute of Disaster Prevention, Beijing 101601, China ;
2. School of Computer and Information Technology, Xinyang Normal University, Xinyang 464000, China
Abstract: In order to improve the approximation accuracy for the surrogate model of radial basis function networks and break through the traditional way that fixed the radial basis width factor, this paper adopts sensitivity analysis method on width factor to build the surrogate model of radial basis function networks, and gives the construction method of specific parameters for the surrogate model of radial basis function networks based on sensitivity analysis. The Benchmark test function is used to verify the accuracy of the surrogate model and the tested results are compared with the fixed radial basis width factor. The comparison results indicate that training time of the surrogate model obtained by this method is longer than that of other methods, because it needs sensitivity analysis when obtaining the width factor, while the obtained model accuracy is higher than other methods, besides, this method can obtainstable the surrogate model parameters without more training samples.
Key words: sensitivity analysis     radial basis function(RBF) networks     surrogate model     width factor

1 径向基神经网络结构

 (1)

 (2)

2 变基宽灵敏度分析的RBF代理模型 2.1 变基宽的灵敏度分析

 式中: ，其是初始基宽在基宽扰动后产生的基宽值，在扰动下的连接权值为。由于该径向基高斯函数的中心、基宽以及连接权在微小扰动下对整个网络的输出均产生影响，由于文献[13]已分析过基中心以及连接权的扰动对整个网络性能的影响，在此仅分析基宽以及连接权的扰动对整个网络性能的影响。第i个隐层神经元与第j个输出层神经元宽度以及连接权的扰动和能被具有零均值和偏差和的高斯分布所定义。 (3)
 (4)

 (5)

 (6)

 (7)
2.2 基宽灵敏度分析的RBF模型关键参数的获取

 (8)

H阵分解过程中，只有一个列能被正交化，且在第K次分解时，一个正交列能够被先前第K-1次正交列得到，具体的相关分解式为

 (9)

 (10)

 (11)

2.3 基宽灵敏度分析的RBF代理模型构建算法

1)通过L个训练样本数据点信息构建3层RBF网络初始结构，并对该网络结构的所有参数进行初始化，包括构建矩阵HK个隐层节点、每个隐层径向基函数中心、基宽以及隐层与输出层的所有连接权wij的初始化。

2)依据灵敏度分析式(7)计算H中的每一列值，并将该列最大灵敏度值赋给Q(1)，然后计算该训练样本的输出值与样本的期望值的差值E(1)，此时设置K=2。

3)依据式(9)计算正交阵H中的剩余Q(K－1)列。

4)对于每一个训练样本，侯选值σi是隐层第i个神经元的函数基宽值，其与正交阵Hqi列相关，计算前K-1个RBF宽度以便求出S(K)(σi)，在求解过程中，连接权将被式(10)所更新，而正交矩阵Q(K)的值将被式(11)所排序。从产生最大值的灵敏度分析的迭代步中获得正交阵Q(K)的第K列，计算该训练样本的输出值与样本的期望值的差值E(K)

5)判断, δ为事先设定的常数值，如果该式成立，则转7)；否则转6)。

6)计数器K=K+1，并转3)。

7)输出矩阵Q(K)中的第K列值，即为该径向基函数神经网络的所有隐层神经元的高斯基基宽。

3 实例测试

 (12)

 s 样本点数 本文方法 固定基宽的RBF F1 F2 F1 F2 80 23.45 17.09 12.62 10.51 200 67.02 55.45 44.00 31.09 500 91.89 79.73 68.30 45.36

 s 样本点数 本文方法 固定基宽的RBF F1 F2 F1 F2 80 0.005 71 0.007 43 0.093 5 0.082 4 200 0.002 19 0.004 73 0.065 5 0.044 7 500 0.004 28 0.008 63 0.084 1 0.065 2

 图 1 测试函数F1在数据样本点为80时获得的模型 Fig. 1 The obtained model for the testing function F1(R=80)
 图 2 测试函数F1在数据样本点为200时获得的模型 Fig. 2 The obtained model for the testing function F1(R=200)
 图 3 测试函数F1在数据样本点为500时获得的模型 Fig. 3 The obtained model for the testing function F1(R=500)
 图 4 测试函数F2在数据样本点为80时获得的模型 Fig. 4 The obtained model for the testing function F2(R=80)
 图 5 测试函数F2在数据样本点为200时获得的模型 Fig. 5 The obtained model for the testing function F2(R=200)
 图 6 测试函数F2在数据样本点为500时获得的模型 Fig. 6 The obtained model for the testing function F2(R=500)
4 结束语

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DOI: 10.3969/j.issn.1673-4785.201309009

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#### 文章信息

ZHANG Yanxia, CHEN Danqi, HAN Ying, LIU Daohua

Surrogate model of radial basis function networks based on width factor sensitivity analysis

CAAI Transactions on Intelligent Systems, 2014, 9(2): 259-264
http://dx.doi.org/10.3969/j.issn.1673-4785.201309009