### 类别可分性准则函数 \( J \) 值的 Python 实现 类别可分性准则函数 \( J \) 通常用于评估类间差异相对于类内差异的程度。在 Fisher 判别分析（LDA）中，该准则函数被定义为类间散度矩阵 \( S_b \) 和类内散度矩阵 \( S_w \) 的比例关系。具体形式如下： \[ J(W) = \frac{\text{Tr}(W^T S_b W)}{\text{Tr}(W^T S_w W)}, \] 其中： - \( W \) 是投影方向矩阵； - \( S_b \) 是类间散度矩阵； - \( S_w \) 是类内散度矩阵； - \( \text{Tr}(\cdot) \) 表示矩阵的迹运算。 以下是计算类别可分性准则函数 \( J \) 值的具体 Python 实现代码： ```python import numpy as np def compute_between_class_scatter(X, y): """ Compute the between-class scatter matrix. Parameters: X : array-like of shape (n_samples, n_features) Feature data. y : array-like of shape (n_samples,) Target labels corresponding to each sample. Returns: Sb : ndarray of shape (n_features, n_features) Between-class scatter matrix. """ unique_classes = np.unique(y) overall_mean = np.mean(X, axis=0).reshape(-1, 1) Sb = np.zeros((X.shape[1], X.shape[1])) for c in unique_classes: Xi = X[y == c] class_mean = np.mean(Xi, axis=0).reshape(-1, 1) ni = Xi.shape[0] Sb += ni * (class_mean - overall_mean).dot((class_mean - overall_mean).T) return Sb def compute_within_class_scatter(X, y): """ Compute the within-class scatter matrix. Parameters: X : array-like of shape (n_samples, n_features) Feature data. y : array-like of shape (n_samples,) Target labels corresponding to each sample. Returns: Sw : ndarray of shape (n_features, n_features) Within-class scatter matrix. """ unique_classes = np.unique(y) Sw = np.zeros((X.shape[1], X.shape[1])) for c in unique_classes: Xi = X[y == c] class_mean = np.mean(Xi, axis=0).reshape(-1, 1) Si = np.sum( [(x.reshape(-1, 1) - class_mean).dot((x.reshape(-1, 1) - class_mean).T) for x in Xi], axis=0 ) Sw += Si return Sw def compute_discriminability_criterion(Sw, Sb): """ Compute the discriminability criterion function value J. Parameters: Sw : ndarray of shape (n_features, n_features) Within-class scatter matrix. Sb : ndarray of shape (n_features, n_features) Between-class scatter matrix. Returns: J_value : float Discriminability criterion function value. """ eigenvalues, _ = np.linalg.eig(np.linalg.inv(Sw).dot(Sb)) J_value = np.sum(eigenvalues) return J_value # Example Usage if __name__ == "__main__": # Generate synthetic dataset with two classes from sklearn.datasets import make_classification X, y = make_classification(n_samples=100, n_features=2, n_informative=2, n_redundant=0, n_clusters_per_class=1, random_state=42) # Compute matrices Sb = compute_between_class_scatter(X, y) Sw = compute_within_class_scatter(X, y) # Compute J value J_value = compute_discriminability_criterion(Sw, Sb) print(f"Discriminability Criterion Function Value J: {J_value}") ``` --- #### 关键点解析 1. **类间散度矩阵 \( S_b \)** 类间散度矩阵反映的是不同类别中心之间的距离平方和。它的计算依赖于总体均值向量以及各类别的均值向量[^1]。 2. **类内散度矩阵 \( S_w \)** 类内散度矩阵描述了同类别内部样本的变化程度。对于每一个类别，分别计算其协方差矩阵并进行累加得到最终结果[^1]。 3. **准则函数 \( J \)** 准则函数的目标是最大化类间的分离程度同时最小化类内的紧凑程度。这通过求解广义特征值问题实现，即找到使 \( J \) 达到最大的投影方向[^2]。 ---