2022年11月30日 Fuzzy C-Means学习笔记

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

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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

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再分别对u_ij，c_j求偏导令导数等于0解得

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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。

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具体步骤如下：

初始化聚类中心或隶属度举证

利用公式，不断更新隶属度矩阵和聚类中心

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 pd

def 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")] = 0
    labels[np.where(labels == "Iris-versicolor")] = 1
    labels[np.where(labels == "Iris-virginica")] = 2

    return dataX, labels


def 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1

    return 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 / U

    return U


def 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 newCentroids

def 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 labels


def 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)) / trainLength
    test_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 time

global MAX  # 用于初始化隶属度矩阵U
MAX = 10000.0

global Epsilon  # 结束条件
Epsilon = 0.0000001


def 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 MAX
    U = []
    for i in range(0, len(data)):
        current = []
        rand_sum = 0.0
        for j in range(0, cluster_number):
            dummy = random.randint(1, int(MAX))
            current.append(dummy)
            rand_sum += dummy
        for j in range(0, cluster_number):
            current[j] = current[j] / rand_sum
        U.append(current)
    return U


def 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 -1
    dummy = 0.0
    for i in range(0, len(point)):
        dummy += abs(point[i] - center[i]) ** 2
    return 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 Epsilon
    for 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 False
    return True


def 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] = 0
            else:
                U[i][j] = 1
    return U


def 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U
    U = initialize_U(data, cluster_number)
    # print_matrix(U)
    # 循环更新U
    while (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 = 0.0
                dummy_sum_dum = 0.0
                for 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U
        for j in range(0, cluster_number):
            for i in range(0, len(data)):
                dummy = 0.0
                for k in range(0, cluster_number):
                    # 分母
                    dummy += (distance_matrix[i][j] / distance_matrix[i][k]) ** (2 / (m - 1))
                U[i][j] = 1 / dummy

        if end_conditon(U, U_old):
            print("已完成聚类")
            break

    U = normalise_U(U)
    return U


if __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))