C++聚類算法對非線性數據的處理

在C++中，有多種聚類算法可以處理非線性數據。以下是一些常用的算法：

1. K-Means聚類算法：K-Means是一種基于原型的聚類方法，它將數據點劃分為K個簇，使得每個數據點到其所屬簇的質心的距離之和最小。對于非線性數據，可以使用K-Means++算法來優化初始質心的選擇，從而提高聚類效果。

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

using namespace std;

vector<int> kMeans(const vector<vector<double>>& data, int k, int max_iterations = 100) {
    int n = data.size();
    vector<int> labels(n, -1);
    vector<double> centroids(k, 0);
    random_device rd;
    mt19937 gen(rd());
    uniform_int_distribution<> dis(0, n - 1);

    for (int i = 0; i < max_iterations; ++i) {
        vector<double> distances(n, 0);
        for (int j = 0; j < n; ++j) {
            double min_dist = DBL_MAX;
            for (int l = 0; l < k; ++l) {
                double dist = 0;
                for (int m = 0; m < data[j].size(); ++m) {
                    dist += pow(data[j][m] - centroids[l][m], 2);
                }
                min_dist = min(min_dist, dist);
            }
            distances[j] = sqrt(min_dist);
        }

        vector<int> new_labels(n, -1);
        for (int j = 0; j < n; ++j) {
            double min_dist = DBL_MAX;
            int min_index = -1;
            for (int l = 0; l < k; ++l) {
                if (distances[j] < min_dist) {
                    min_dist = distances[j];
                    min_index = l;
                }
            }
            new_labels[j] = min_index;
            if (new_labels[j] == labels[j]) {
                break;
            }
        }

        for (int j = 0; j < n; ++j) {
            labels[j] = new_labels[j];
        }

        for (int l = 0; l < k; ++l) {
            vector<double> cluster_data;
            for (int j = 0; j < n; ++j) {
                if (labels[j] == l) {
                    cluster_data.push_back(data[j]);
                }
            }
            if (!cluster_data.empty()) {
                double sum[cluster_data[0].size()] = {0};
                for (const auto& point : cluster_data) {
                    for (int m = 0; m < point.size(); ++m) {
                        sum[m] += point[m];
                    }
                }
                for (int m = 0; m < cluster_data[0].size(); ++m) {
                    centroids[l][m] = sum[m] / cluster_data.size();
                }
            }
        }
    }

    return labels;
}

2. DBSCAN聚類算法：DBSCAN（Density-Based Spatial Clustering of Applications with Noise）是一種基于密度的聚類方法，它可以發現任意形狀的簇，并識別噪聲點。對于非線性數據，DBSCAN可以通過調整鄰域半徑和最小點數參數來適應數據的分布。

#include <iostream>
#include <vector>
#include <cmath>
#include <queue>
#include <unordered_set>

using namespace std;

vector<int> dbscan(const vector<vector<double>>& data, double eps, int min_samples) {
    int n = data.size();
    vector<int> labels(n, -1);
    queue<int> q;
    unordered_set<int> visited;

    for (int i = 0; i < n; ++i) {
        if (visited.find(i) != visited.end()) {
            continue;
        }
        q.push(i);
        visited.insert(i);

        int num_points = 0;
        vector<double> point_eps_radius(data[0].size(), 0);
        while (!q.empty()) {
            int point = q.front();
            q.pop();
            num_points++;

            for (int m = 0; m < data[point].size(); ++m) {
                point_eps_radius[m] = max(point_eps_radius[m], abs(data[point][m] - data[q.front()][m]));
            }

            for (int neighbor : get_neighbors(data, point, eps)) {
                if (visited.find(neighbor) == visited.end()) {
                    q.push(neighbor);
                    visited.insert(neighbor);
                }
            }
        }

        if (num_points < min_samples) {
            continue;
        }

        int cluster_id = n;
        for (int neighbor : get_neighbors(data, q.front(), eps)) {
            if (visited.find(neighbor) == visited.end() && labels[neighbor] == -1) {
                vector<int> cluster = dbscan(data, eps, min_samples);
                if (cluster.size() > 0) {
                    cluster_id = min(cluster_id, cluster[0]);
                }
            }
        }

        for (int neighbor : get_neighbors(data, q.front(), eps)) {
            if (visited.find(neighbor) != visited.end()) {
                labels[neighbor] = cluster_id;
            }
        }
    }

    return labels;
}

vector<int> get_neighbors(const vector<vector<double>>& data, int point, double eps) {
    int n = data.size();
    vector<int> neighbors;
    for (int i = 0; i < n; ++i) {
        if (i == point) {
            continue;
        }
        double distance = 0;
        for (int m = 0; m < data[point].size(); ++m) {
            distance += pow(data[point][m] - data[i][m], 2);
        }
        if (distance < eps * eps) {
            neighbors.push_back(i);
        }
    }
    return neighbors;
}

3. 高斯混合模型（GMM）：GMM是一種基于概率模型的聚類方法，它假設數據是由多個高斯分布生成的。對于非線性數據，可以使用GMM的非線性變換（如核方法）來適應數據的分布。

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

using namespace std;

vector<int> gmm(const vector<vector<double>>& data, int n_components, double max_iter = 100, double tol = 1e-4) {
    int n = data.size();
    vector<int> labels(n, -1);
    vector<double> weights(n_components, 1.0 / n_components);
    vector<vector<double>> means(n_components, vector<double>(data[0].size(), 0));
    vector<vector<double>> covariances(n_components, vector<double>(data[0].size(), 0));
    random_device rd;
    mt19937 gen(rd());
    uniform_real_distribution<> dis(0, 1);

    for (int iter = 0; iter < max_iter; ++iter) {
        vector<int> labels_new(n, -1);
        vector<double> weights_new(n_components, 0);
        vector<vector<double>> means_new(n_components, vector<double>(data[0].size(), 0));
        vector<vector<double>> covariances_new(n_components, vector<double>(data[0].size(), 0));

        for (int j = 0; j < n; ++j) {
            double max_log_likelihood = -DBL_MAX;
            int max_component = -1;
            for (int k = 0; k < n_components; ++k) {
                double log_likelihood = 0;
                for (int m = 0; m < data[j].size(); ++m) {
                    double mean = means[k][m];
                    double covariance = covariances[k][m];
                    double value = data[j][m];
                    log_likelihood += log(2 * M_PI * pow(covariance, 0.5)) + pow(value - mean, 2) / (2 * covariance);
                }
                if (log_likelihood > max_log_likelihood) {
                    max_log_likelihood = log_likelihood;
                    max_component = k;
                }
            }
            labels_new[j] = max_component;
            weights_new[max_component] += 1;
            means_new[max_component] = data[j];
            if (data[j].size() > 1) {
                covariances_new[max_component] += data[j] * data[j].t();
            }
        }

        for (int k = 0; k < n_components; ++k) {
            weights[k] = weights_new[k] / n;
            means[k] = means_new[k] / weights[k];
            if (data[j].size() > 1) {
                covariances[k] = covariances_new[k] / weights[k];
            }
        }

        if (max(abs(weights_new - weights)) < tol && max(abs(means_new - means)) < tol) {
            break;
        }
    }

    return labels;
}

這些算法可以處理非線性數據，但可能需要調整參數以獲得最佳聚類效果。在實際應用中，可以嘗試多種算法并比較它們的聚類結果，以選擇最適合特定數據的算法。