像聚类算法一样，降低维度算法试图分析数据的内在结构，不过降低维度算法是以非监督学习的方式试图利用较少的信息来归纳或者解释数据。这类算法可以用于高维数据的可视化或者用来简化数据以便监督式学习使用。
常见的算法包括：主成份分析，偏最小二乘回归， Sammon映射，多维尺度, 投影追踪等。
from __future__ import print_function from sklearn import datasets import matplotlib.pyplot as plt import matplotlib.cm as cmx import matplotlib.colors as colors import numpy as np 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] # 正规化数据集 X def 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) # 标准化数据集 X def standardize(X): X_std = np.zeros(X.shape) mean = X.mean(axis=0) std = X.std(axis=0) # 做除法运算时请永远记住分母不能等于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 # 划分数据集为训练集和测试集 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 # 计算矩阵X的协方差矩阵 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) # 计算数据集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 # 计算数据集X每列的标准差 def calculate_std_dev(X): std_dev = np.sqrt(calculate_variance(X)) return std_dev # 计算相关系数矩阵 def calculate_correlation_matrix(X, Y=np.empty([0])): # 先计算协方差矩阵 covariance_matrix = calculate_covariance_matrix(X, Y) # 计算X, Y的标准差 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，非监督学习算法. """ def __init__(self): self.eigen_values = None self.eigen_vectors = None self.k = 2 def transform(self, X): """ 将原始数据集X通过PCA进行降维 """ 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) # 将特征值从大到小进行排序，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个特征值对应的特征向量 idx = self.eigen_values.argsort()[::-1] eigenvalues = self.eigen_values[idx][:self.k] eigenvectors = self.eigen_vectors[:, idx][:, :self.k] # 将原始数据集X映射到低维空间 X_transformed = X.dot(eigenvectors) return X_transformed def main(): # Load the dataset data = datasets.load_iris() X = data.data y = data.target # 将数据集X映射到低维空间 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()