python实现径向基核函数

1、生成数据集（双月数据集）

class moon_data_class(object):
    def __init__(self,N,d,r,w):
        self.N=N
        self.w=w
        self.d=d
        self.r=r
    def sgn(self,x):
        if(x>0):
            return 1;
        else:
            return -1;
        
    def sig(self,x):
        return 1.0/(1+np.exp(x))
    
        
    def dbmoon(self):
        N1 = 10*self.N
        N = self.N
        r = self.r
        w2 = self.w/2
        d = self.d
        done = True
        data = np.empty(0)
        while done:
            #generate Rectangular data
            tmp_x = 2*(r+w2)*(np.random.random([N1, 1])-0.5)
            tmp_y = (r+w2)*np.random.random([N1, 1])
            tmp = np.concatenate((tmp_x, tmp_y), axis=1)
            tmp_ds = np.sqrt(tmp_x*tmp_x + tmp_y*tmp_y)
            #generate double moon data ---upper
            idx = np.logical_and(tmp_ds > (r-w2), tmp_ds < (r+w2))
            idx = (idx.nonzero())[0]
     
            if data.shape[0] == 0:
                data = tmp.take(idx, axis=0)
            else:
                data = np.concatenate((data, tmp.take(idx, axis=0)), axis=0)
            if data.shape[0] >= N:
                done = False
        #print (data)
        db_moon = data[0:N, :]
        #print (db_moon)
        #generate double moon data ----down
        data_t = np.empty([N, 2])
        data_t[:, 0] = data[0:N, 0] + r
        data_t[:, 1] = -data[0:N, 1] - d
        db_moon = np.concatenate((db_moon, data_t), axis=0)
        return db_moon

2、k均值聚类

def k_means(input_cells, k_count):
    count = len(input_cells)      #点的个数
    x = input_cells[0:count, 0]
    y = input_cells[0:count, 1]
    #随机选择K个点
    k = rd.sample(range(count), k_count)
    
    k_point = [[x[i], [y[i]]] for i in k]   #保证有序
    k_point.sort()

    global frames
    #global step
    while True:
        km = [[] for i in range(k_count)]      #存储每个簇的索引
        #遍历所有点
        for i in range(count):
            cp = [x[i], y[i]]                   #当前点
            #计算cp点到所有质心的距离
            _sse = [distance(k_point[j], cp) for j in range(k_count)]
            #cp点到那个质心最近
            min_index = _sse.index(min(_sse))   
            #把cp点并入第i簇
            km[min_index].append(i)
        #更换质心
       
        k_new = []
        for i in range(k_count):
            _x = sum([x[j] for j in km[i]]) / len(km[i])
            _y = sum([y[j] for j in km[i]]) / len(km[i])
            k_new.append([_x, _y])
        k_new.sort()        #排序
      

        if (k_new != k_point):#一直循环直到聚类中心没有变化
            k_point = k_new
        else:
            return k_point,km

3、高斯核函数
高斯核函数，主要的作用是衡量两个对象的相似度，当两个对象越接近，即a与b的距离趋近于0，则高斯核函数的值趋近于1，反之则趋近于0，换言之：

两个对象越相似，高斯核函数值就越大

作用:
用于分类时，衡量各个类别的相似度，其中sigma参数用于调整过拟合的情况，sigma参数较小时，即要求分类器，加差距很小的类别也分类出来，因此会出现过拟合的问题；
用于模糊控制时，用于模糊集的隶属度。

def gaussian (a,b, sigma):
    return np.exp(-norm(a-b)**2 / (2 * sigma**2))

4、求高斯核函数的方差

 Sigma_Array = []
    for j in range(k_count):
        Sigma = []
        for i in range(len(center_array[j][0])):
            temp =  Phi(np.array([center_array[j][0][i],center_array[j][1][i]]),np.array(center[j]))
            Sigma.append(temp)
        Sigma = np.array(Sigma)
        Sigma_Array.append(np.cov(Sigma))

5、显示高斯核函数计算结果

gaussian_kernel_array = []
    fig = plt.figure()
    ax = Axes3D(fig)
    
    for j in range(k_count):
        gaussian_kernel = []
        for i in range(len(center_array[j][0])):
            temp =  Phi(np.array([center_array[j][0][i],center_array[j][1][i]]),np.array(center[j]))
            temp1 = gaussian(temp,Sigma_Array[0])
            gaussian_kernel.append(temp1)
        
        gaussian_kernel_array.append(gaussian_kernel)
 
        ax.scatter(center_array[j][0], center_array[j][1], gaussian_kernel_array[j],s=20)
    plt.show()

6、运行结果

7、完整代码

# coding:utf-8
import numpy as np
import pylab as pl
import random as rd
import imageio
import math
import random
import matplotlib.pyplot as plt
import numpy as np
import mpl_toolkits.mplot3d
from mpl_toolkits.mplot3d import Axes3D

from scipy import *
from scipy.linalg import norm, pinv
 
from matplotlib import pyplot as plt
random.seed(0)

#定义sigmoid函数和它的导数
def sigmoid(x):
    return 1.0/(1.0+np.exp(-x))
def sigmoid_derivate(x):
    return x*(1-x) #sigmoid函数的导数


class moon_data_class(object):
    def __init__(self,N,d,r,w):
        self.N=N
        self.w=w
      
        self.d=d
        self.r=r
    
   
    def sgn(self,x):
        if(x>0):
            return 1;
        else:
            return -1;
        
    def sig(self,x):
        return 1.0/(1+np.exp(x))
    
        
    def dbmoon(self):
        N1 = 10*self.N
        N = self.N
        r = self.r
        w2 = self.w/2
        d = self.d
        done = True
        data = np.empty(0)
        while done:
            #generate Rectangular data
            tmp_x = 2*(r+w2)*(np.random.random([N1, 1])-0.5)
            tmp_y = (r+w2)*np.random.random([N1, 1])
            tmp = np.concatenate((tmp_x, tmp_y), axis=1)
            tmp_ds = np.sqrt(tmp_x*tmp_x + tmp_y*tmp_y)
            #generate double moon data ---upper
            idx = np.logical_and(tmp_ds > (r-w2), tmp_ds < (r+w2))
            idx = (idx.nonzero())[0]
     
            if data.shape[0] == 0:
                data = tmp.take(idx, axis=0)
            else:
                data = np.concatenate((data, tmp.take(idx, axis=0)), axis=0)
            if data.shape[0] >= N:
                done = False
        #print (data)
        db_moon = data[0:N, :]
        #print (db_moon)
        #generate double moon data ----down
        data_t = np.empty([N, 2])
        data_t[:, 0] = data[0:N, 0] + r
        data_t[:, 1] = -data[0:N, 1] - d
        db_moon = np.concatenate((db_moon, data_t), axis=0)
        return db_moon

def distance(a, b):
    return (a[0]- b[0]) ** 2 + (a[1] - b[1]) ** 2
#K均值算法
def k_means(input_cells, k_count):
    count = len(input_cells)      #点的个数
    x = input_cells[0:count, 0]
    y = input_cells[0:count, 1]
    #随机选择K个点
    k = rd.sample(range(count), k_count)
    
    k_point = [[x[i], [y[i]]] for i in k]   #保证有序
    k_point.sort()

    global frames
    #global step
    while True:
        km = [[] for i in range(k_count)]      #存储每个簇的索引
        #遍历所有点
        for i in range(count):
            cp = [x[i], y[i]]                   #当前点
            #计算cp点到所有质心的距离
            _sse = [distance(k_point[j], cp) for j in range(k_count)]
            #cp点到那个质心最近
            min_index = _sse.index(min(_sse))   
            #把cp点并入第i簇
            km[min_index].append(i)
        #更换质心
       
        k_new = []
        for i in range(k_count):
            _x = sum([x[j] for j in km[i]]) / len(km[i])
            _y = sum([y[j] for j in km[i]]) / len(km[i])
            k_new.append([_x, _y])
        k_new.sort()        #排序
    
        if (k_new != k_point):#一直循环直到聚类中心没有变化
            k_point = k_new
        else:
            pl.figure()
            pl.title("N=%d,k=%d  iteration"%(count,k_count))
            for j in range(k_count):
                pl.plot([x[i] for i in km[j]], [y[i] for i in km[j]], color[j%4])
                pl.plot(k_point[j][0], k_point[j][1], dcolor[j%4])
            return k_point,km
    
def Phi(a,b):
    return norm(a-b)

def gaussian (x, sigma):
    return np.exp(-x**2 / (2 * sigma**2))
        
if __name__ == '__main__':
    
    #计算平面两点的欧氏距离
    step=0
    color=['.r','.g','.b','.y']#颜色种类
    dcolor=['*r','*g','*b','*y']#颜色种类
    frames = []
    
    N = 200
    d = -4
    r = 10
    width = 6
        
    data_source = moon_data_class(N, d, r, width)
    data = data_source.dbmoon()
       # x0 = [1 for x in range(1,401)]
    input_cells = np.array([np.reshape(data[0:2*N, 0], len(data)), np.reshape(data[0:2*N, 1], len(data))]).transpose()
        
    labels_pre = [[1] for y in range(1, 201)]
    labels_pos = [[0] for y in range(1, 201)]
    labels=labels_pre+labels_pos
    
    
    k_count = 2 
    center,km = k_means(input_cells, k_count)
    test = Phi(input_cells[1],np.array(center[0]))
    print(test)
    test = distance(input_cells[1],np.array(center[0]))
    print(np.sqrt(test))
    count = len(input_cells)  
    x = input_cells[0:count, 0]
    y = input_cells[0:count, 1]
    center_array = []

    for j in range(k_count):
       
           center_array.append([[x[i] for i in km[j]], [y[i] for i in km[j]]])
    Sigma_Array = []
    for j in range(k_count):
        Sigma = []
        for i in range(len(center_array[j][0])):
            temp =  Phi(np.array([center_array[j][0][i],center_array[j][1][i]]),np.array(center[j]))
            Sigma.append(temp)
      
        Sigma = np.array(Sigma)
        Sigma_Array.append(np.cov(Sigma))
    
    gaussian_kernel_array = []
    fig = plt.figure()
    ax = Axes3D(fig)
    
    for j in range(k_count):
        gaussian_kernel = []
        for i in range(len(center_array[j][0])):
            temp =  Phi(np.array([center_array[j][0][i],center_array[j][1][i]]),np.array(center[j]))
            temp1 = gaussian(temp,Sigma_Array[0])
            gaussian_kernel.append(temp1)
        
        gaussian_kernel_array.append(gaussian_kernel)
        
        ax.scatter(center_array[j][0], center_array[j][1], gaussian_kernel_array[j],s=20)
    plt.show()