使用幂法技术获得第二主特征值的错误输出

Question

当我尝试从头开始编写PCA，使用幂法技术来计算特征值时，只有第一个是正确的，其余的都是错误的。我做错了什么？

使用 numpy ，协方差矩阵的特征值为 [9.49506917 8.11919641 5.27687829] 而我的为 [9.495069168715409, 485.7201377095837, 25631.534012572032]。我也在分享我的代码结构。

def normalize_data(train_x, test_x=None):
    scaler = StandardScaler()

    train_normalized = pd.DataFrame(scaler.fit_transform(train_x), columns=train_x.columns)
    if test_x is not None:
        test_normalized = pd.DataFrame(scaler.transform(test_x), columns=test_x.columns)
    else:
        test_normalized = None

    return train_normalized, test_normalized

def extract_PCA(X, n_components=3):
    n = X.shape[0]
    cov_matrix = (X.T @ X) / (n-1)    
        
    eigen_vectors = []
    eigen_values = []
        
    for _ in range(n_components):
        # Find dominant eigen value and its corresponding eigen vector
        eigen_value, eigen_vector = using_power_method_find_dominant_eigen_value(cov_matrix)
# eigen_vectors.append(eigen_vector)
        eigen_values.append(eigen_value)
        
        eigen_vector = eigen_vector.reshape(256, 1)

        cov_matrix = cov_matrix - ( eigen_value * (eigen_vector @ eigen_vector.T) ) 
    
    
def using_power_method_find_dominant_eigen_value(mat_A, tolrence = 1e-10):
    n = mat_A.shape[0]
    x = np.ones(n)
    
    prev_value = 0
    prev_eigen_vector = x

    # while True:
    for _ in range(1000):
        x = np.dot(mat_A, x)
        eigen_value, x = vector_norm(x)
        print(eigen_value, np.linalg.norm(prev_eigen_vector - x, 2))
                
        if ( abs(eigen_value - prev_value) < tolrence ) and ( np.linalg.norm(prev_eigen_vector - x, 2) < tolrence):
            break
            
        prev_value = eigen_value
        prev_eigen_vector = x
        
    return eigen_value, x

def vector_norm(v):
    norm_coeff = max(v)
    
    # Step 2: Normalize the vector
    normalized_v = [vi / norm_coeff for vi in v]
    
    return norm_coeff, np.array(normalized_v)

if __name__ == "__main__":
    
    train_df = pd.read_csv("train.csv")
    test_df = pd.read_csv("test.csv")
    
    X_train, y_train = train_df.drop(columns= ['y'], axis=1), train_df['y']
    X_test, y_test = test_df.drop(columns = ['y'], axis=1), test_df['y']
    
    X_train, X_test = normalize_data(X_train, X_test)
    
    extract_PCA(X_train)
    `

Answer 1

幂方法是一种特征值方法，只能找到单个最大特征值（在您的情况下为 9.495....）及其相应的特征向量。

如果您想要所有特征值，您将需要不同的方法。

[ https://www.sciencedirect.com/science/article/pii/B9780123944351000223/pdfft ]