RBF神经网络及Python实现（附源码）

作者：ACdreamers

http://blog.csdn.net/acdreamers/article/details/46327761

RBF网络能够逼近任意非线性的函数。可以处理系统内难以解析的规律性，具有很好的泛化能力，并且具有较快的学习速度。当网络的一个或多个可调参数（权值或阈值）对任何一个输出都有影响时，这样的网络称为全局逼近网络。

由于对于每次输入，网络上的每一个权值都要调整，从而导致全局逼近网络的学习速度很慢，比如BP网络。如果对于输入空间的某个局部区域只有少数几个连接权值影响输出，则该网络称为局部逼近网络，比如RBF网络。接下来重点先介绍RBF网络的原理，然后给出其实现。先看如下图

正则化的RBF神经网路请参照博客园的文章(链接：http://www.cnblogs.com/zhangchaoyang/articles/2591663.html)。下面是一个比较好的Python的RBF网络实现。最后提供Github上的一个C++实现的RBF，供日后参考（点击阅读原文查看）。

from scipy import *  
from scipy.linalg import norm, pinv  
from matplotlib import pyplot as plt  
class RBF:  
    def __init__(self, indim, numCenters, outdim):  
        self.indim = indim  
        self.outdim = outdim  
        self.numCenters = numCenters  
        self.centers = [random.uniform(-1, 1, indim) for i in xrange(numCenters)]  
        self.beta = 8  
        self.W = random.random((self.numCenters, self.outdim))  
    def _basisfunc(self, c, d):  
        assert len(d) == self.indim  
        return exp(-self.beta * norm(c-d)**2)  
    def _calcAct(self, X):  
        # calculate activations of RBFs  
        G = zeros((X.shape[0], self.numCenters), float)  
        for ci, c in enumerate(self.centers):  
            for xi, x in enumerate(X):  
                G[xi,ci] = self._basisfunc(c, x)  
        return G  
    def train(self, X, Y):  
        """ X: matrix of dimensions n x indim  
            y: column vector of dimension n x 1 """  
        # choose random center vectors from training set  
        rnd_idx = random.permutation(X.shape[0])[:self.numCenters]  
        self.centers = [X[i,:] for i in rnd_idx]  
        print "center", self.centers  
        # calculate activations of RBFs  
        G = self._calcAct(X)  
        print G  
        # calculate output weights (pseudoinverse)  
        self.W = dot(pinv(G), Y)  
    def test(self, X):  
        """ X: matrix of dimensions n x indim """  
        G = self._calcAct(X)  
        Y = dot(G, self.W)  
        return Y  
if __name__ == '__main__':  
    n = 100  
    x = mgrid[-1:1:complex(0,n)].reshape(n, 1)  
    # set y and add random noise  
    y = sin(3*(x+0.5)**3 - 1)  
    # y += random.normal(0, 0.1, y.shape)  
    # rbf regression  
    rbf = RBF(1, 10, 1)  
    rbf.train(x, y)  
    z = rbf.test(x)  
    # plot original data  
    plt.figure(figsize=(12, 8))  
    plt.plot(x, y, 'k-')  
    # plot learned model  
    plt.plot(x, z, 'r-', linewidth=2)  
    # plot rbfs  
    plt.plot(rbf.centers, zeros(rbf.numCenters), 'gs')  
    for c in rbf.centers:  
        # RF prediction lines  
        cx = arange(c-0.7, c+0.7, 0.01)  
        cy = [rbf._basisfunc(array([cx_]), array([c])) for cx_ in cx]  
        plt.plot(cx, cy, '-', color='gray', linewidth=0.2)  
    plt.xlim(-1.2, 1.2)  
    plt.show()