径向基（RBF）神经网络

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网络参考这里。下面是网上找的一个比较好的Python的RBF网络实现。

代码：

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()

最后提供Github上的一个C++实现的RBF，供日后参考。