python不使用sklearn实现LDA降维

LDA（线性判别分析）是一种经典的有监督降维方法，它可以将高维数据映射到一个低维空间中，以便于分类和可视化。下面是一个简单的 Python 实现：

1.首先，我们需要计算类内散度矩阵 Sw 和类间散度矩阵 Sb。

import numpy as np

def compute_scatter_matrices(X, y):
    # 计算均值向量
    class_labels = np.unique(y)
    n_classes = len(class_labels)
    n_features = X.shape[1]
    mean_vectors = []
    for cl in class_labels:
        mean_vectors.append(np.mean(X[y==cl], axis=0))
    # 计算类内散度矩阵
    Sw = np.zeros((n_features, n_features))
    for cl,mv in zip(class_labels, mean_vectors):
        class_sc_mat = np.zeros((n_features, n_features))                  # scatter matrix for every class
        for row in X[y == cl]:
            row, mv = row.reshape(n_features,1), mv.reshape(n_features,1) # make column vectors
            class_sc_mat += (row-mv).dot((row-mv).T)
        Sw += class_sc_mat                             # sum class scatter matrices
    # 计算类间散度矩阵
    overall_mean = np.mean(X, axis=0)
    Sb = np.zeros((n_features, n_features))
    for i,mean_vec in enumerate(mean_vectors):
        n = X[y==class_labels[i]].shape[0]
        mean_vec = mean_vec.reshape(n_features,1) # make column vector
        overall_mean = overall_mean.reshape(n_features,1) # make column vector
        Sb += n * (mean_vec - overall_mean).dot((mean_vec - overall_mean).T)
    return Sw, Sb

2.然后，我们需要计算 Sw 的逆矩阵和 Sw 和 Sb 的乘积。

def lda(X, y, n_components):
    Sw, Sb = compute_scatter_matrices(X, y)
    eig_vals, eig_vecs = np.linalg.eig(np.linalg.inv(Sw).dot(Sb))
    eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))]
    eig_pairs = sorted(eig_pairs, key=lambda k: k[0], reverse=True)
    W = np.hstack([eig_pairs[i][1].reshape(len(X[0]),1) for i in range(n_components)])
    return X.dot(W)

这个函数将返回一个降维后的特征矩阵，其中 n_components 是我们想要的输出维度数。