给出以下6个五维样本X1,X2,X3,X4,X5和X6。 X1=[0,3,1,2,0]T,X2=[1,3,0,1,0]T,X3=[3,3,0,0,1]T, X4=[1,1,0,2,0]T,X5=[3,2,1,2,1]T,X6=[4,1,1,1,0]T 本次实验对上面6个样本按照最小距离准则进行聚类分析,编

### 使用最小距离准则对五维样本数据进行聚类分析 #### 背景介绍 层次聚类是一种基于距离度量的聚类方法，其核心思想是通过计算样本之间的相似性（通常采用欧氏距离），逐步将距离最近的两个簇合并为一个新的簇，直至满足特定条件停止[^1]。对于五维样本数据而言，可以通过定义合适的距离函数来衡量不同样本之间的差异。 #### 数据准备 假设我们有如下一组五维样本数据 \( X \)，其中每一行表示一个样本： \[ X = \begin{bmatrix} x_{1,1} & x_{1,2} & x_{1,3} & x_{1,4} & x_{1,5}\\ x_{2,1} & x_{2,2} & x_{2,3} & x_{2,4} & x_{2,5}\\ ...\\ x_{n,1} & x_{n,2} & x_{n,3} & x_{n,4} & x_{n,5} \end{bmatrix} \] 每条记录是一个长度为 5 的向量，代表五个维度上的数值。 #### 计算欧氏距离矩阵 为了应用最小距离准则，首先需要构建样本两两之间的欧氏距离矩阵 \( D \)。设任意两点 \( A(x_1, y_1, z_1, w_1, v_1) \) 和 \( B(x_2, y_2, z_2, w_2, v_2) \)，它们之间的欧氏距离可由下式给出： \[ d(A,B) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2 + (w_1-w_2)^2 + (v_1-v_2)^2}. \] 利用此公式逐一遍历所有可能的点对组合即可得到完整的距离矩阵 \( D \)[^1]。 #### 实现过程 以下是具体的实现流程以及对应的 Python 示例代码展示如何操作这些步骤: ```python import numpy as np from scipy.spatial.distance import pdist, squareform def hierarchical_clustering_min_distance(data, threshold): """ Perform agglomerative clustering using the minimum distance criterion. Parameters: data (numpy.ndarray): The dataset with shape (n_samples, n_features). threshold (float): Distance threshold to stop merging clusters. Returns: list of lists: Each sublist represents a cluster containing indices from original data array. """ # Step 1: Compute pairwise distances between all points dist_matrix_condensed = pdist(data, metric='euclidean') dist_matrix_square = squareform(dist_matrix_condensed) # Initialize each point as its own cluster represented by index sets current_clusters = [[i] for i in range(len(data))] while True: min_dist = float('inf') merge_pair = (-1,-1) # Find two closest clusters based on their minimal inter-cluster distance num_clusters = len(current_clusters) for idx_i in range(num_clusters): for idx_j in range(idx_i+1, num_clusters): ci_indices_set = set(current_clusters[idx_i]) cj_indices_set = set(current_clusters[idx_j]) pair_distances = [ dist_matrix_square[i][j] for i in ci_indices_set for j in cj_indices_set ] curr_min_between_ci_cj = min(pair_distances) if curr_min_between_ci_cj < min_dist: min_dist = curr_min_between_ci_cj merge_pair = (idx_i,idx_j) # If no pairs have smaller than given threshold then terminate procedure if min_dist >= threshold or not merge_pair[0]>=0 : break # Merge found nearest clusters into one new single entity merged_cluster_idx = max(merge_pair)+1 combined_memberships = ( current_clusters.pop(max(merge_pair))+current_clusters.pop(min(merge_pair)) ) current_clusters.insert(merged_cluster_idx ,combined_memberships ) return current_clusters # Example usage if __name__ == "__main__": five_dimensional_data = np.array([ [1.0, 2.0, 3.0, 4.0, 5.0], [9.0, 8.0, 7.0, 6.0, 5.0], [1.1, 2.1, 3.1, 4.1, 5.1], [8.9, 7.9, 6.9, 5.9, 4.9] ]) result_clusters = hierarchical_clustering_min_distance(five_dimensional_data, threshold=2.0) print("Resulting Clusters:",result_clusters) ``` 上述程序实现了基于最小距离标准的凝聚型分层聚类功能，并允许指定终止聚合的距离门限值 `threshold` 参数控制最终形成的类别数量。 #### 结果解释 运行以上脚本后会输出若干组索引列表形式的结果集合，每一个子列表即对应着发现的一个独立群组成员编号位置关系说明文档[^1]。