模糊核方法：概述及C++代码示例

基于模糊核的方法包括模糊C均值聚类（FCM）、模糊神经网络（FNN）、模糊决策树（FDT）、模糊关联规则（FAR）等。以下是一个基于模糊C均值聚类的代码示例：

#include <iostream>
#include <vector>
#include <cmath>
#include <algorithm>

using namespace std;

// 模糊C均值聚类函数
vector<vector<double>> fuzzy_c_means(vector<vector<double>> data, int k, double m, double eps) {
    int n = data.size(); // 样本数
    int d = data[0].size(); // 维数
    vector<vector<double>> centers(k, vector<double>(d)); // 聚类中心
    vector<vector<double>> membership(n, vector<double>(k)); // 每个样本对每个聚类中心的隶属度

    // 随机初始化聚类中心
    for (int i = 0; i < k; i++) {
        int idx = rand() % n;
        centers[i] = data[idx];
    }

    // 迭代计算聚类中心
    while (true) {
        // 计算每个样本对每个聚类中心的隶属度
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < k; j++) {
                double dist = 0.0;
                for (int l = 0; l < d; l++) {
                    dist += pow(data[i][l] - centers[j][l], 2.0);
                }
                membership[i][j] = 1.0 / accumulate(membership[i].begin(), membership[i].end(), 0.0, [&](double a, double b) { return a + pow(dist / b, 1.0 / (m - 1.0)); });
            }
        }

        // 计算新的聚类中心
        vector<vector<double>> new_centers(k, vector<double>(d));
        for (int j = 0; j < k; j++) {
            for (int l = 0; l < d; l++) {
                new_centers[j][l] = accumulate(membership.begin(), membership.end(), 0.0, [&](double a, vector<double> b) { return a + pow(b[j], m) * data[&b - &membership[0]][l]; }) / accumulate(membership.begin(), membership.end(), 0.0, [&](double a, vector<double> b) { return a + pow(b[j], m); });
            }
        }

        // 判断是否收敛
        double delta = 0.0;
        for (int j = 0; j < k; j++) {
            delta += accumulate(new_centers[j].begin(), new_centers[j].end(), 0.0, [&](double a, double b) { return a + pow(b - centers[j][&b - &new_centers[j][0]], 2.0); });
        }
        if (delta < eps) {
            break;
        }

        centers = new_centers;
    }

    return membership;
}

int main() {
    // 生成随机数据
    vector<vector<double>> data = {{1.0, 2.0}, {2.0, 3.0}, {3.0, 4.0}, {4.0, 5.0}, {5.0, 6.0}, {6.0, 7.0}};

    // 聚类
    int k = 2; // 聚类数
    double m = 2.0; // 模糊度
    double eps = 1e-6; // 收敛误差
    vector<vector<double>> membership = fuzzy_c_means(data, k, m, eps);

    // 输出结果
    for (int i = 0; i < data.size(); i++) {
        for (int j = 0; j < k; j++) {
            cout << membership[i][j] << " ";
        }
        cout << endl;
    }

    return 0;
}