基于最佳范化能力的BP网络隐节点数反比关系式的环境预测模型

网络结构和样本集复杂性是影响BP网络泛化能力的两个最重要因素.对一个给定的训练样本集,为了构造一个与样本集复杂性相匹配的网络结构,使BP网络具有最佳泛化能力,在分析BP网络的泛化能力（用检测误差E₂表示）与网络结构和样本集复杂性之间关系的基础上,建立了含参数的BP网络检测误差E₂与网络隐节点数H、样本的因子数n、样本数N和样本集的复相关系数R之间的一般关系表达式,并提出了用于定量描述样本集复杂性的"广义"复相关系数R_n的新概念.借助于222个复杂函数的模拟仿真实验,应用免疫进化算法,对表达式中的参数进行优化,得到参数优化后的网络检测误差E₂的定量解析表达式;依据误差理论和灵敏度概念,对优化得到的最小检测误差E₂₀的表达式进行了可靠性论证.在此基础上导出了具有最佳泛化能力的BP网络隐节点数H₀与"广义"复相关系数R_n之间满足的H₀-R_n反比关系式.分别用满足H₀-R_n反比关系式的隐节点数和6个经验公式的隐节点数构造的BP网络用于100个模拟检测函数进行仿真实验,发现前者构造的BP网络具有最佳泛化能力（即最小检测误差）的函数个数达到76个,远远多于后者构造的BP网络具有最佳泛化能力的函数个数;还将二者构造的BP网络用于环境预测的7个具体实例,进行预测效果比较,结果表明,前者预测的相对误差绝对值的平均值和最大值小于或远小于后者的相应值.从而验证了由H₀-R_n反比关系式得出的BP网络隐节点数计算公式的可行性和实用性,为具有最佳泛化能力的BP网络的结构设计指出了新途径.

An environment prediction model based on the inverse relation of hidden nodes of BP network with the best normalization ability

Network structure and sample set complexity are the two most important factors that affect the generalization ability of BP network. For a given training sample set, the purpose is to construct a network structure matching the given complexity of the sample set, so that the BP network has the best generalization ability. Based on the analysis of the relationship between the generalization ability of BP network (expressed by detection error E₂) and the network structure and sample set complexity, the general expression between the detection error E₂ of BP network with parameters and the number of hidden nodes H, the number of sample factors n, the number of samples N and the complex correlation coefficient R of sample set is established, and a new concept of "generalized" complex correlation coefficient R_n is proposed to quantitatively describe the complexity of samples set. With the help of 222 complex function simulation experiments, the parameters in the expression are optimized by using immune evolution algorithm, and the quantitative analytical expression of the network detection error E₂ is obtained. According to the error theory and the concept of sensitivity, the reliability of the expression E₂₀ of the minimum generalization error is demonstrated. On this basis, the H₀-R_n inverse relation between the hidden nodes H₀ of BP network and the "generalized" complex correlation coefficient R_n with the best generalization ability is derived. The BP network constructed by the number of hidden nodes satisfying H₀-R_n inverse relation and the number of hidden nodes satisfying 6 empirical formulas are used in 100 simulated detection functions for simulation experiments. The number of functions with the best generalization ability (i.e. the minimum detection error) of the BP network constructed by the former reaches 76, far more than the number of functions with the best generalization ability of the BP network constructed by the latter. In addition, the BP networks constructed by the two methods were applied to 7 specific cases of environmental prediction, and compared the prediction effects, The results show that the average value and the maximum value of the absolute value of the relative error predicted by the former are less than or far less than the corresponding value of the latter. Thus, the feasibility and practicability of the calculation formula of hidden node numbers of BP network derived from H₀-R_n inverse relation are verified, and a new way for the structural design of BP network with the best generalization ability is pointed out.