基于 Python 的 11 種經典數據降維算法

from __future__ import print_functionfrom sklearn import datasetsimport matplotlib.pyplot as pltimport matplotlib.cm as cmximport matplotlib.colors as colorsimport numpy as np%matplotlib inline
def shuffle_data(X, y, seed=None):   if seed:     np.random.seed(seed)
   idx = np.arange(X.shape[0])   np.random.shuffle(idx)
   return X[idx], y[idx]
# 正規(guī)化數據集 Xdef normalize(X, axis=-1, p=2):   lp_norm = np.atleast_1d(np.linalg.norm(X, p, axis))   lp_norm[lp_norm == 0] = 1   return X / np.expand_dims(lp_norm, axis)
# 標準化數據集 Xdef standardize(X):   X_std = np.zeros(X.shape)   mean = X.mean(axis=0)   std = X.std(axis=0)
   # 做除法運算時請永遠記住分母不能等于 0 的情形   # X_std = (X - X.mean(axis=0)) / X.std(axis=0)    for col in range(np.shape(X)[1]):     if std[col]:       X_std[:, col] = (X_std[:, col] - mean[col]) / std[col]   return X_std
# 劃分數據集為訓練集和測試集def train_test_split(X, y, test_size=0.2, shuffle=True, seed=None):   if shuffle:     X, y = shuffle_data(X, y, seed)   n_train_samples = int(X.shape[0] * (1-test_size))   x_train, x_test = X[:n_train_samples], X[n_train_samples:]   y_train, y_test = y[:n_train_samples], y[n_train_samples:]     return x_train, x_test, y_train, y_test
# 計算矩陣 X 的協(xié)方差矩陣def calculate_covariance_matrix(X, Y=np.empty((0,0))):   if not Y.any():      Y = X   n_samples = np.shape(X)[0]   covariance_matrix = (1 / (n_samples-1)) * (X - X.mean(axis=0)).T.dot(Y - Y.mean(axis=0))   return np.array(covariance_matrix, dtype=float)
# 計算數據集 X 每列的方差def calculate_variance(X):   n_samples = np.shape(X)[0]   variance = (1 / n_samples) * np.diag((X - X.mean(axis=0)).T.dot(X - X.mean(axis=0)))   return variance
# 計算數據集 X 每列的標準差def calculate_std_dev(X):   std_dev = np.sqrt(calculate_variance(X))   return std_dev
# 計算相關系數矩陣def calculate_correlation_matrix(X, Y=np.empty([0])):   # 先計算協(xié)方差矩陣   covariance_matrix = calculate_covariance_matrix(X, Y)   # 計算 X, Y 的標準差   std_dev_X = np.expand_dims(calculate_std_dev(X), 1)   std_dev_y = np.expand_dims(calculate_std_dev(Y), 1)   correlation_matrix = np.divide(covariance_matrix, std_dev_X.dot(std_dev_y.T))     return np.array(correlation_matrix, dtype=float)
class PCA():   """   主成份分析算法 PCA，非監(jiān)督學習算法.   """   def __init__(self):     self.eigen_values = None     self.eigen_vectors = None     self.k = 2
   def transform(self, X):     """      將原始數據集 X 通過 PCA 進行降維     """     covariance = calculate_covariance_matrix(X)
     # 求解特征值和特征向量     self.eigen_values, self.eigen_vectors = np.linalg.eig(covariance)
     # 將特征值從大到小進行排序，注意特征向量是按列排的，即 self.eigen_vectors 第 k 列是 self.eigen_values 中第 k 個特征值對應的特征向量     idx = self.eigen_values.argsort()[::-1]     eigenvalues = self.eigen_values[idx][:self.k]     eigenvectors = self.eigen_vectors[:, idx][:, :self.k]
     # 將原始數據集 X 映射到低維空間     X_transformed = X.dot(eigenvectors)        return X_transformed
def main():   # Load the dataset   data = datasets.load_iris()   X = data.data   y = data.target
   # 將數據集 X 映射到低維空間   X_trans = PCA().transform(X)
   x1 = X_trans[:, 0]   x2 = X_trans[:, 1]
   cmap = plt.get_cmap('viridis')   colors = [cmap(i) for i in np.linspace(0, 1, len(np.unique(y)))]
   class_distr = []   # Plot the different class distributions   for i, l in enumerate(np.unique(y)):       _x1 = x1[y == l]       _x2 = x2[y == l]       _y = y[y == l]       class_distr.append(plt.scatter(_x1, _x2, color=colors[i]))
   # Add a legend   plt.legend(class_distr, y, loc=1)
   # Axis labels   plt.xlabel('Principal Component 1')   plt.ylabel('Principal Component 2')   plt.show()
if __name__ == "__main__":   main()