K-means

参考资料：https://github.com/fengdu78/Coursera-ML-AndrewNg-Notes
先看数据：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sb
from scipy.io import loadmat
data = loadmat('data/ex7data2.mat')
data2 = pd.DataFrame(data.get('X'), columns=['X1', 'X2'])
plt.scatter(data2['X1'],data2['X2'],c='b')
plt.show()

执行k-means算法：

# 聚类中心已知，根据数据点距离聚类中心的距离分类
def find_closest_centroids(X, centroids):m = X.shape[0]  # X.shape = (300,2)k = centroids.shape[0]  # centrids.shape = (3,2)idx = np.zeros(m)  # m = 300, 每个数据的标签，默认为0for i in range(m):  # m = 300 样本个数min_dist = 1000000for j in range(k):  # k = 3 聚类中心个数dist = np.sum((X[i, :] - centroids[j, :]) ** 2)if dist < min_dist:min_dist = distidx[i] = jreturn idx  # 返回数据点的标签# k个聚类中心重新计算均值，返回计算后的坐标
def compute_centroids(X, idx, k):m, n = X.shape  # (300,2)centroids = np.zeros((k, n))  # （3，2）for i in range(k):indices = np.where(idx == i)centroids[i, :] = (np.sum(X[indices, :], axis=1) / len(indices[0])).ravel()return centroidsdef run_k_means(X, centroids, max_iters):m, n = X.shape  # (300,2)k = centroids.shape[0]  # 3idx = np.zeros(m)  # 标签初始为0for i in range(max_iters):idx = find_closest_centroids(X, centroids)  # 给最近的数据做好标签centroids = compute_centroids(X, idx, k)  # 重新计算聚类坐标，重复max_iters次return idx, centroidsX = data['X']
initial_centroids = np.array([[3, 3], [6, 2], [8, 5]])  # 初始化聚类中心
idx, centroids = run_k_means(X, initial_centroids, 10)
cluster1 = X[np.where(idx == 0)[0],:]
cluster2 = X[np.where(idx == 1)[0],:]
cluster3 = X[np.where(idx == 2)[0],:]
plt.scatter(cluster1[:,0], cluster1[:,1], s=30, color='r', label='Cluster 1')
plt.scatter(cluster2[:,0], cluster2[:,1], s=30, color='g', label='Cluster 2')
plt.scatter(cluster3[:,0], cluster3[:,1], s=30, color='b', label='Cluster 3')
plt.legend()
plt.show()

效果如图：

在执行算法的过程中我们选择了手动初始化聚类中心，可以使算法个更快的收敛，当然也可以选择随机初始化，但要执行多次来选择效果最好的一个。

def init_centroids(X, k):m, n = X.shape  # (300,2)centroids = np.zeros((k, n))  # (3,2),三个聚类中心，每个中心有两个坐标来确定idx = np.random.randint(0, m, k)  # 产生k个0~m的数for i in range(k):centroids[i, :] = X[idx[i], :]  # 将随机选取的三个数据点作为聚类中心return centroids

k-means压缩图片

1，还是上述的思想不再赘述

from PIL import Image
import numpy as np
import matplotlib.pyplot as plt
from k_means import find_closest_centroids, init_centroids, run_k_means
filename = "data/bird_small.png"
im = np.array(Image.open(filename))/255
im2 = np.reshape(im, (im.shape[0]*im.shape[1], im.shape[2]))
initial_centroids = init_centroids(im2, 16)  # 随机选取16个数据点作为聚类中心idx, centroids = run_k_means(im2, initial_centroids, 10)  # 执行10次k-means算法
idx = find_closest_centroids(im2, centroids)X_recovered = centroids[idx.astype(int),:]  # X.shape = (16384,3)X_recovered = np.reshape(X_recovered, (im.shape[0], im.shape[1], im.shape[2]))  # 返回最初的维度plt.imshow(X_recovered)
plt.show()

2，用scikit-learn来实现K-means

from skimage import io
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans#导入kmeans库
# cast to float, you need to do this otherwise the color would be weird after clustring
pic = io.imread('data/bird_small.png') / 255.
data = pic.reshape(128*128, 3)
model = KMeans(n_clusters=16, n_init=100, n_jobs=-1)
model.fit(data)
centroids = model.cluster_centers_
C = model.predict(data)
compressed_pic = centroids[C].reshape((128,128,3))
fig, ax = plt.subplots(1, 2)
ax[0].imshow(pic)
ax[1].imshow(compressed_pic)
plt.show()

PCA

先看数据：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.io import loadmat
data = loadmat('data/ex7data1.mat')
X = data['X']
plt.scatter(X[:, 0], X[:, 1])
plt.savefig("PCA.png")
plt.show()

def pca(X):# normalize the featuresX = (X - X.mean()) / X.std()# compute the covariance matrixX = np.matrix(X)cov = (X.T * X) / X.shape[0]# perform SVDU, S, V = np.linalg.svd(cov)return U, S, Vdef project_data(X, U, k):U_reduced = U[:,:k]return np.dot(X, U_reduced)def recover_data(Z, U, k):U_reduced = U[:,:k]return np.dot(Z, U_reduced.T)U, S, V = pca(X)
Z = project_data(X, U, 1)
X_recovered = recover_data(Z, U, 1)
fig, ax = plt.subplots(figsize=(12,8))
ax.scatter(list(X_recovered[:, 0]), list(X_recovered[:, 1]))
plt.savefig("PCA2.png")
plt.show()