Fuzzy C-Mean Clustering (FCM)
matlab R2026a 官网对FCM解释 ，非常清晰。
Fuzzy C-Means Clustering for Iris Data
matlab
clc; clear; close all;
% 加载水仙花
iris = load("iris.mat");
% 特征
features = iris.data;
% 标签
label = iris.label;
% 对应的逻辑索引
setosaIndex = label == 1;
versicolorIndex = label == 2;
virginicaIndex = label == 3;
% 根据逻辑索引获取对应的特征数据
setosa = features(setosaIndex,:);
versicolor = features(versicolorIndex,:);
virginica = features(virginicaIndex,:);
characteristics = ["sepal length","sepal width",...
"petal length","petal width"];
pairs = [1 2; 1 3; 1 4; 2 3; 2 4; 3 4];
% 绘制每两个维度组合的数据点
for i = 1:6
x = pairs(i,1);
y = pairs(i,2);
subplot(2,3,i)
plot([setosa(:,x) versicolor(:,x) virginica(:,x)],...
[setosa(:,y) versicolor(:,y) virginica(:,y)],".")
xlabel(characteristics(x))
ylabel(characteristics(y))
end
% fcm的options
options = fcmOptions(...
NumClusters=3,...
Exponent=2.0, ...
MaxNumIteration=100, ...
MinImprovement=1e-6);
% 禁用每次 FCM 迭代的目标函数值在命令窗口中的输出
options.Verbose = false;
% fcm聚类
[centers,U] = fcm(features,options);
%绘制结果
for i = 1:6
subplot(2,3,i)
for j = 1:options.NumClusters
x = pairs(i,1);
y = pairs(i,2);
text(centers(j,x),centers(j,y),int2str(j),...
FontWeight="bold");
end
end对应的聚类结果
对于 fcmOptions 描述
R2023a版本
matlab
opt = fcmOptions()
% 返回默认结果
opt =
FCMOptions - 属性:
NumClusters: 'auto'
Exponent: 2
MaxNumIteration: 100
MinImprovement: 1.0000e-05
DistanceMetric: 'euclidean'
Verbose: 1
matlab
opt = fcmOptions(Name=Value)
eg:
% 聚类个数3，模糊因子2，最大迭代次数100，终止阈值1e-6
options = fcmOptions(...
NumClusters=3,...
Exponent=2.0, ...
MaxNumIteration=100, ...
MinImprovement=1e-6);Properties
NumClusters：聚类个数，默认为"auto"，即2个类别；Exponent：模糊因子，默认为2；MaxNumIteration：最大迭代次数，默认为100；MinImprovement：两次连续迭代之间目标函数的最小改进值，默认为1.0000e-05；Verbose：信息显示标志，指示是否在每次迭代后显示目标函数值；true- 显示目标函数false- 不显示目标函数
DistanceMetric：计算数据点与聚类中心之间距离的方法。euclidean- 欧氏距离mahalanobis- 马氏距离
myfcm
根据matlab源码，我们自己来写一个fcm算法
matlab
function varargout = myfcm(data, options)
% MYFCM
% 数据
% 选项 options
% 可选参数
% return
% center: 聚类中心
% fuzzyPartMat: 模糊隶属度
% objFcn: 目标函数
dataSize = size(data, 1);
objFcn = zeros(options.MaxNumIteration, 1); % 目标函数数组
fuzzyMatrix = fuzzy.clustering.initfcm(options, dataSize); % 初始模糊隶属度
numClusters = options.NumClusters;
expo = options.Exponent;
% clusterVolume = options.ClusterVolume; % 马氏距离中体积/缩放系数
options.ClusterVolume = options.ClusterVolume(1,ones(1,options.NumClusters));
brkCond = struct('isTrue',false,'description','');
% Main Loop
for iterId = 1 : options.MaxNumIteration
memFcnMat = fuzzyMatrix.^expo;
center = memFcnMat * data ./ (sum(memFcnMat, 2) * ones(1, size(data, 2)));
% 是使用马氏距离还是欧氏距离
if strcmp(options.DistanceMetric, getString(message('fuzzy:general:lblFcm_mahalanobis')))
% dist 距离；covMat 协方差矩阵；brkCond 终止条件
[dist, covMat, brkCond] = fuzzy.clustering.mahalanobisdist(center, data, memFcnMat, options.ClusterVolume);
else
dist = fuzzy.clustering.euclideandist(center, data);
covMat = [];
end
tmp = (max(dist, eps)) .^ (-2 / (expo - 1)); % Calculate new Fuzzy Partition Matrix, suppose expo != 1
fuzzyMatrix = tmp ./ (ones(numClusters, 1) * sum(tmp));
objFcn(iterId) = sum(sum((dist .^ 2) .* memFcnMat));
% 判断终止条件，返回true/false
brkCond = checkBreakCondition(options, objFcn(iterId : -1 : max(1, iterId-1)), iterId, brkCond);
% Check verbose condition
% if options.Verbose
% fprintf(iterationProgressFormat, iterId, objFcn(iterId));
% if ~isempty(brkCond.description)
% fprintf('%s\n',brkCond.description);
% end
% end
% Break if early termination condition is true.
if brkCond.isTrue
objFcn(iterId+1:end) = [];
break
end
end
[varargout{1:nargout}] = assignOutputs(center,fuzzyMatrix,objFcn);
end
%% Local functions
function brkCond = checkBreakCondition(options, objFcn, iterId, stepBrkCond)
%%
if stepBrkCond.isTrue
brkCond = stepBrkCond;
return
end
brkCond = struct('isTrue',false,'description','');
improvement = diff(objFcn);
if ~isempty(improvement) && abs(improvement)<=options.MinImprovement
% 达到阈值终止
brkCond.isTrue = true;
brkCond.description = getString(message('fuzzy:general:msgFcm_minImprovementReached'));
return
end
if iterId==options.MaxNumIteration
% 达到最大迭代次数终止
brkCond.isTrue = true;
brkCond.description = getString(message('fuzzy:general:msgFcm_maxIterationReached'));
end
end
function varargout = assignOutputs(center,fuzzyPartMat,objFcn)
%% 返回参数
if nargout>2
varargout{3} = objFcn;
end
if nargout>1
varargout{2} = fuzzyPartMat;
end
if nargout>0
varargout{1} = center;
end
end测试
使用前面的水仙花数据测试
matlab
clc; clear; close all;
iris = load("iris.mat");
features = iris.data;
label = iris.label;
setosaIndex = label==1;
versicolorIndex = label==2;
virginicaIndex = label==3;
setosa = features(setosaIndex,:);
versicolor = features(versicolorIndex,:);
virginica = features(virginicaIndex,:);
characteristics = ["sepal length","sepal width",...
"petal length","petal width"];
pairs = [1 2; 1 3; 1 4; 2 3; 2 4; 3 4];
for i = 1:6
x = pairs(i,1);
y = pairs(i,2);
subplot(2,3,i)
plot([setosa(:,x) versicolor(:,x) virginica(:,x)],...
[setosa(:,y) versicolor(:,y) virginica(:,y)],".")
xlabel(characteristics(x))
ylabel(characteristics(y))
end
options = fcmOptions(...
NumClusters=3,...
Exponent=2.0, ...
MaxNumIteration=100, ...
MinImprovement=1e-6, Verbose=0);
% % options.Verbose = false;
[centers,U] = myfcm(features, options);
for i = 1:6
subplot(2,3,i)
for j = 1:options.NumClusters
x = pairs(i,1);
y = pairs(i,2);
text(centers(j,x),centers(j,y),int2str(j),...
FontWeight="bold");
end
end