​ Fuzzy C-Means 模糊c均值聚类，它的一大优势就是引入了一个隶属度的概念，没有对样本进行非黑即白的分类，而是分类的时候乘上隶属度，直白点说就是他和某个中心有多像，到底是40%像还是70%像。

​ 参考：在众多模糊聚类算法中，模糊C-均值（ FCM） 算法应用最广泛且较成功，它通过优化目标函数得到每个样本点对所有类中心的隶属度，从而决定样本点的类属以达到自动对样本数据进行分类的目的。

目标函数

隶属度为u_ij，表示第i个样本对第j类的隶属度，其中每个数据x_i对于所有类别的隶属度和要为1。u_ij所有值求和要为1。m为聚类的簇数。x_i表示第i个样本，c_j表示第j个聚类中心

最小化目标函数，先将u_ij所有值求和要为1作为约束条件利用拉格朗日数乘法引入

再分别对u_ij，c_j求偏导令导数等于0解得

利用这个式子进行迭代，就能得到最小化的目标函数。迭代的方式有两种，一种是设置迭代次数，另一种是设置误差阈值，当误差小于某个值的时候停止迭代。

具体步骤如下：

1. 初始化聚类中心或隶属度举证

2. 利用公式，不断更新隶属度矩阵和聚类中心

3. 满足条件后停止迭代，输出聚类结果。

iris.data数据下载

python代码实现

#!/usr/bin/python3
# -*- coding: utf-8 -*-'''
@Date    : 2019/9/11
@Author  : Rezero
'''import numpy as np
import pandas as pddef loadData(datapath):data = pd.read_csv(datapath, sep=',', header=None)data = data.sample(frac=1.0)   # 打乱数据顺序dataX = data.iloc[:, :-1].values # 特征labels = data.iloc[:, -1].values # 标签# 将标签类别用 0, 1, 2表示labels[np.where(labels == "Iris-setosa")] = 0labels[np.where(labels == "Iris-versicolor")] = 1labels[np.where(labels == "Iris-virginica")] = 2return dataX, labelsdef initialize_U(samples, classes):U = np.random.rand(samples, classes)  # 先生成随机矩阵sumU = 1 / np.sum(U, axis=1)   # 求每行的和U = np.multiply(U.T, sumU)   # 使隶属度矩阵每一行和为1return U.T# 计算样本和簇中心的距离，这里使用欧氏距离
def distance(X, centroid):return np.sqrt(np.sum((X-centroid)**2, axis=1))def computeU(X, centroids, m=2):sampleNumber = X.shape[0]  # 样本数classes = len(centroids)U = np.zeros((sampleNumber, classes))# 更新隶属度矩阵for i in range(classes):for k in range(classes):U[:, i] += (distance(X, centroids[i]) / distance(X, centroids[k])) ** (2 / (m - 1))U = 1 / Ureturn Udef ajustCentroid(centroids, U, labels):newCentroids = [[], [], []]curr = np.argmax(U, axis=1)  # 当前中心顺序得到的标签for i in range(len(centroids)):index = np.where(curr == i)   # 建立中心和类别的映射trueLabel = list(labels[index])  # 获取labels[index]出现次数最多的元素，就是真实类别trueLabel = max(set(trueLabel), key=trueLabel.count)newCentroids[trueLabel] = centroids[i]return newCentroidsdef cluster(data, labels, m, classes, EPS):""":param data: 数据集:param m: 模糊系数(fuzziness coefficient):param classes: 类别数:return: 聚类中心"""sampleNumber = data.shape[0]  # 样本数cNumber = data.shape[1]       # 特征数U = initialize_U(sampleNumber, classes)   # 初始化隶属度矩阵U_old = np.zeros((sampleNumber, classes))while True:centroids = []# 更新簇中心for i in range(classes):centroid = np.dot(U[:, i]**m, data) / (np.sum(U[:, i]**m))centroids.append(centroid)U_old = U.copy()U = computeU(data, centroids, m)  # 计算新的隶属度矩阵if np.max(np.abs(U - U_old)) < EPS:# 这里的类别和数据标签并不是一一对应的, 调整使得第i个中心表示第i类centroids = ajustCentroid(centroids, U, labels)return centroids, U# 预测所属的类别
def predict(X, centroids):labels = np.zeros(X.shape[0])U = computeU(X, centroids)  # 计算隶属度矩阵labels = np.argmax(U, axis=1)  # 找到隶属度矩阵中每行的最大值，即该样本最大可能所属类别return labelsdef main():datapath = "iris.data"dataX, labels = loadData(datapath)  # 读取数据# 划分训练集和测试集ratio = 0.6  # 训练集的比例trainLength = int(dataX.shape[0] * ratio)  # 训练集长度trainX = dataX[:trainLength, :]trainLabels = labels[:trainLength]testX = dataX[trainLength:, :]testLabels = labels[trainLength:]EPS = 1e-6   # 停止误差条件m = 2        # 模糊因子classes = 3  # 类别数# 得到各类别的中心centroids, U = cluster(trainX, trainLabels, m, classes, EPS)trainLabels_prediction = predict(trainX, centroids)testLabels_prediction = predict(testX, centroids)train_error = 1 - np.sum(np.abs(trainLabels_prediction - trainLabels)) / trainLengthtest_error = 1 - np.sum(np.abs(testLabels_prediction - testLabels)) / (dataX.shape[0] - trainLength)print("Clustering on traintset is %.2f%%" % (train_error*100))print("Clustering on testset is %.2f%%" % (test_error*100))if __name__ == "__main__":main()

另一个代码

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 27 10:51:45 2019
模糊c聚类:https://blog.csdn.net/lyxleft/article/details/88964494
@author: youxinlin
"""
import copy
import math
import random
import timeglobal MAX  # 用于初始化隶属度矩阵U
MAX = 10000.0global Epsilon  # 结束条件
Epsilon = 0.0000001def print_matrix(list):"""以可重复的方式打印矩阵"""for i in range(0, len(list)):print(list[i])def initialize_U(data, cluster_number):"""这个函数是隶属度矩阵U的每行加起来都为1. 此处需要一个全局变量MAX."""global MAXU = []for i in range(0, len(data)):current = []rand_sum = 0.0for j in range(0, cluster_number):dummy = random.randint(1, int(MAX))current.append(dummy)rand_sum += dummyfor j in range(0, cluster_number):current[j] = current[j] / rand_sumU.append(current)return Udef distance(point, center):"""该函数计算2点之间的距离（作为列表）。我们指欧几里德距离。闵可夫斯基距离"""if len(point) != len(center):return -1dummy = 0.0for i in range(0, len(point)):dummy += abs(point[i] - center[i]) ** 2return math.sqrt(dummy)def end_conditon(U, U_old):"""结束条件。当U矩阵随着连续迭代停止变化时，触发结束"""global Epsilonfor i in range(0, len(U)):for j in range(0, len(U[0])):if abs(U[i][j] - U_old[i][j]) > Epsilon:return Falsereturn Truedef normalise_U(U):"""在聚类结束时使U模糊化。每个样本的隶属度最大的为1，其余为0"""for i in range(0, len(U)):maximum = max(U[i])for j in range(0, len(U[0])):if U[i][j] != maximum:U[i][j] = 0else:U[i][j] = 1return Udef fuzzy(data, cluster_number, m):"""这是主函数，它将计算所需的聚类中心，并返回最终的归一化隶属矩阵U.输入参数：簇数(cluster_number)、隶属度的因子(m)的最佳取值范围为[1.5，2.5]"""# 初始化隶属度矩阵UU = initialize_U(data, cluster_number)# print_matrix(U)# 循环更新Uwhile (True):# 创建它的副本，以检查结束条件U_old = copy.deepcopy(U)# 计算聚类中心C = []for j in range(0, cluster_number):current_cluster_center = []for i in range(0, len(data[0])):dummy_sum_num = 0.0dummy_sum_dum = 0.0for k in range(0, len(data)):# 分子dummy_sum_num += (U[k][j] ** m) * data[k][i]# 分母dummy_sum_dum += (U[k][j] ** m)# 第i列的聚类中心current_cluster_center.append(dummy_sum_num / dummy_sum_dum)# 第j簇的所有聚类中心C.append(current_cluster_center)# 创建一个距离向量, 用于计算U矩阵。distance_matrix = []for i in range(0, len(data)):current = []for j in range(0, cluster_number):current.append(distance(data[i], C[j]))distance_matrix.append(current)# 更新Ufor j in range(0, cluster_number):for i in range(0, len(data)):dummy = 0.0for k in range(0, cluster_number):# 分母dummy += (distance_matrix[i][j] / distance_matrix[i][k]) ** (2 / (m - 1))U[i][j] = 1 / dummyif end_conditon(U, U_old):print("已完成聚类")breakU = normalise_U(U)return Uif __name__ == '__main__':data = [[6.1, 2.8, 4.7, 1.2], [5.1, 3.4, 1.5, 0.2], [6.0, 3.4, 4.5, 1.6], [4.6, 3.1, 1.5, 0.2],[6.7, 3.3, 5.7, 2.1], [7.2, 3.0, 5.8, 1.6], [6.7, 3.1, 4.4, 1.4], [6.4, 2.7, 5.3, 1.9],[4.8, 3.0, 1.4, 0.3], [7.9, 3.8, 6.4, 2.0], [5.2, 3.5, 1.5, 0.2], [5.9, 3.0, 5.1, 1.8],[5.7, 2.8, 4.1, 1.3], [6.8, 3.2, 5.9, 2.3], [5.4, 3.4, 1.5, 0.4], [5.4, 3.7, 1.5, 0.2],[6.6, 3.0, 4.4, 1.4], [5.1, 3.5, 1.4, 0.2], [6.0, 2.2, 4.0, 1.0], [7.7, 2.8, 6.7, 2.0],[6.3, 2.8, 5.1, 1.5], [7.4, 2.8, 6.1, 1.9], [5.5, 4.2, 1.4, 0.2], [5.7, 3.0, 4.2, 1.2],[5.5, 2.6, 4.4, 1.2], [5.2, 3.4, 1.4, 0.2], [4.9, 3.1, 1.5, 0.1], [4.6, 3.6, 1.0, 0.2],[4.6, 3.2, 1.4, 0.2], [5.8, 2.7, 3.9, 1.2], [5.0, 3.4, 1.5, 0.2], [6.1, 3.0, 4.6, 1.4],[4.7, 3.2, 1.6, 0.2], [6.7, 3.3, 5.7, 2.5], [6.5, 3.0, 5.8, 2.2], [5.4, 3.4, 1.7, 0.2],[5.8, 2.7, 5.1, 1.9], [5.4, 3.9, 1.3, 0.4], [5.3, 3.7, 1.5, 0.2], [6.1, 3.0, 4.9, 1.8],[7.2, 3.2, 6.0, 1.8], [5.5, 2.3, 4.0, 1.3], [5.7, 2.8, 4.5, 1.3], [4.9, 2.4, 3.3, 1.0],[5.4, 3.0, 4.5, 1.5], [5.0, 3.5, 1.6, 0.6], [5.2, 4.1, 1.5, 0.1], [5.8, 4.0, 1.2, 0.2],[5.4, 3.9, 1.7, 0.4], [6.5, 3.2, 5.1, 2.0], [5.5, 2.4, 3.7, 1.0], [5.0, 3.5, 1.3, 0.3],[6.3, 2.5, 5.0, 1.9], [6.9, 3.1, 4.9, 1.5], [6.2, 2.2, 4.5, 1.5], [6.3, 3.3, 4.7, 1.6],[6.4, 3.2, 4.5, 1.5], [4.7, 3.2, 1.3, 0.2], [5.5, 2.4, 3.8, 1.1], [5.0, 2.0, 3.5, 1.0],[4.4, 2.9, 1.4, 0.2], [4.8, 3.4, 1.9, 0.2], [6.3, 3.4, 5.6, 2.4], [5.5, 2.5, 4.0, 1.3],[5.7, 2.5, 5.0, 2.0], [6.5, 3.0, 5.2, 2.0], [6.7, 3.0, 5.0, 1.7], [5.2, 2.7, 3.9, 1.4],[6.9, 3.1, 5.1, 2.3], [7.2, 3.6, 6.1, 2.5], [4.8, 3.0, 1.4, 0.1], [6.3, 2.9, 5.6, 1.8],[5.1, 3.5, 1.4, 0.3], [6.9, 3.1, 5.4, 2.1], [5.6, 3.0, 4.1, 1.3], [7.7, 2.6, 6.9, 2.3],[6.4, 2.9, 4.3, 1.3], [5.8, 2.7, 4.1, 1.0], [6.1, 2.9, 4.7, 1.4], [5.7, 2.9, 4.2, 1.3],[6.2, 2.8, 4.8, 1.8], [4.8, 3.4, 1.6, 0.2], [5.6, 2.9, 3.6, 1.3], [6.7, 2.5, 5.8, 1.8],[5.0, 3.4, 1.6, 0.4], [6.3, 3.3, 6.0, 2.5], [5.1, 3.8, 1.9, 0.4], [6.6, 2.9, 4.6, 1.3],[5.1, 3.3, 1.7, 0.5], [6.3, 2.5, 4.9, 1.5], [6.4, 3.1, 5.5, 1.8], [6.2, 3.4, 5.4, 2.3],[6.7, 3.1, 5.6, 2.4], [4.6, 3.4, 1.4, 0.3], [5.5, 3.5, 1.3, 0.2], [5.6, 2.7, 4.2, 1.3],[5.6, 2.8, 4.9, 2.0], [6.2, 2.9, 4.3, 1.3], [7.0, 3.2, 4.7, 1.4], [5.0, 3.2, 1.2, 0.2],[4.3, 3.0, 1.1, 0.1], [7.7, 3.8, 6.7, 2.2], [5.6, 3.0, 4.5, 1.5], [5.8, 2.7, 5.1, 1.9],[5.8, 2.8, 5.1, 2.4], [4.9, 3.1, 1.5, 0.1], [5.7, 3.8, 1.7, 0.3], [7.1, 3.0, 5.9, 2.1],[5.1, 3.7, 1.5, 0.4], [6.3, 2.7, 4.9, 1.8], [6.7, 3.0, 5.2, 2.3], [5.1, 2.5, 3.0, 1.1],[7.6, 3.0, 6.6, 2.1], [4.5, 2.3, 1.3, 0.3], [4.9, 3.0, 1.4, 0.2], [6.5, 2.8, 4.6, 1.5],[5.7, 4.4, 1.5, 0.4], [6.8, 3.0, 5.5, 2.1], [4.9, 2.5, 4.5, 1.7], [5.1, 3.8, 1.5, 0.3],[6.5, 3.0, 5.5, 1.8], [5.7, 2.6, 3.5, 1.0], [5.1, 3.8, 1.6, 0.2], [5.9, 3.0, 4.2, 1.5],[6.4, 3.2, 5.3, 2.3], [4.4, 3.0, 1.3, 0.2], [6.1, 2.8, 4.0, 1.3], [6.3, 2.3, 4.4, 1.3],[5.0, 2.3, 3.3, 1.0], [5.0, 3.6, 1.4, 0.2], [5.9, 3.2, 4.8, 1.8], [6.4, 2.8, 5.6, 2.2],[6.1, 2.6, 5.6, 1.4], [5.6, 2.5, 3.9, 1.1], [6.0, 2.7, 5.1, 1.6], [6.0, 3.0, 4.8, 1.8],[6.4, 2.8, 5.6, 2.1], [6.0, 2.9, 4.5, 1.5], [5.8, 2.6, 4.0, 1.2], [7.7, 3.0, 6.1, 2.3],[5.0, 3.3, 1.4, 0.2], [6.9, 3.2, 5.7, 2.3], [6.8, 2.8, 4.8, 1.4], [4.8, 3.1, 1.6, 0.2],[6.7, 3.1, 4.7, 1.5], [4.9, 3.1, 1.5, 0.1], [7.3, 2.9, 6.3, 1.8], [4.4, 3.2, 1.3, 0.2],[6.0, 2.2, 5.0, 1.5], [5.0, 3.0, 1.6, 0.2]]start = time.time()# 调用模糊C均值函数res_U = fuzzy(data, 3, 2)# 计算准确率print("用时：{0}".format(time.time() - start))