为了帮朋友搞定个作业, 没学过现在只能临时抱佛脚, 做个笔记方便后面回查.
完整的例子:
print(__doc__) # Code source: Gaël Varoquaux # Modified for documentation by Jaques Grobler # License: BSD 3 clause import numpy as np import matplotlib.pyplot as plt # Though the following import is not directly being used, it is required # for 3D projection to work from mpl_toolkits.mplot3d import Axes3D from sklearn.cluster import KMeans from sklearn import datasets np.random.seed(5) iris = datasets.load_iris() X = iris.data y = iris.target estimators = [('k_means_iris_8', KMeans(n_clusters=8)), ('k_means_iris_3', KMeans(n_clusters=3)), ('k_means_iris_bad_init', KMeans(n_clusters=3, n_init=1, init='random'))] fignum = 1 titles = ['8 clusters', '3 clusters', '3 clusters, bad initialization'] for name, est in estimators: fig = plt.figure(fignum, figsize=(4, 3)) ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=48, azim=134) est.fit(X) labels = est.labels_ ax.scatter(X[:, 3], X[:, 0], X[:, 2], c=labels.astype(np.float), edgecolor='k') ax.w_xaxis.set_ticklabels([]) ax.w_yaxis.set_ticklabels([]) ax.w_zaxis.set_ticklabels([]) ax.set_xlabel('Petal width') ax.set_ylabel('Sepal length') ax.set_zlabel('Petal length') ax.set_title(titles[fignum - 1]) ax.dist = 12 fignum = fignum + 1 # Plot the ground truth fig = plt.figure(fignum, figsize=(4, 3)) ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=48, azim=134) for name, label in [('Setosa', 0), ('Versicolour', 1), ('Virginica', 2)]: ax.text3D(X[y == label, 3].mean(), X[y == label, 0].mean(), X[y == label, 2].mean() + 2, name, horizontalalignment='center', bbox=dict(alpha=.2, edgecolor='w', facecolor='w')) # Reorder the labels to have colors matching the cluster results y = np.choose(y, [1, 2, 0]).astype(np.float) ax.scatter(X[:, 3], X[:, 0], X[:, 2], c=y, edgecolor='k') ax.w_xaxis.set_ticklabels([]) ax.w_yaxis.set_ticklabels([]) ax.w_zaxis.set_ticklabels([]) ax.set_xlabel('Petal width') ax.set_ylabel('Sepal length') ax.set_zlabel('Petal length') ax.set_title('Ground Truth') ax.dist = 12 fig.show()这里有个提醒, 就是不管怎么调都不可能保证分类出来的结果就百分百对, 类聚从另一方面来说就是选少量的样例来压缩信息, 解决的问题也叫向量量化, 一般可以用来分离色彩.
这里是个例子:
print(__doc__) # Code source: Gaël Varoquaux # Modified for documentation by Jaques Grobler # License: BSD 3 clause import numpy as np import scipy as sp import matplotlib.pyplot as plt from sklearn import cluster try: # SciPy >= 0.16 have face in misc from scipy.misc import face face = face(gray=True) except ImportError: face = sp.face(gray=True) n_clusters = 5 np.random.seed(0) X = face.reshape((-1, 1)) # We need an (n_sample, n_feature) array k_means = cluster.KMeans(n_clusters=n_clusters, n_init=4) k_means.fit(X) values = k_means.cluster_centers_.squeeze() labels = k_means.labels_ # create an array from labels and values face_compressed = np.choose(labels, values) face_compressed.shape = face.shape vmin = face.min() vmax = face.max() # original face plt.figure(1, figsize=(3, 2.2)) plt.imshow(face, cmap=plt.cm.gray, vmin=vmin, vmax=256) # compressed face plt.figure(2, figsize=(3, 2.2)) plt.imshow(face_compressed, cmap=plt.cm.gray, vmin=vmin, vmax=vmax) # equal bins face regular_values = np.linspace(0, 256, n_clusters + 1) regular_labels = np.searchsorted(regular_values, face) - 1 regular_values = .5 * (regular_values[1:] + regular_values[:-1]) # mean regular_face = np.choose(regular_labels.ravel(), regular_values, mode="clip") regular_face.shape = face.shape plt.figure(3, figsize=(3, 2.2)) plt.imshow(regular_face, cmap=plt.cm.gray, vmin=vmin, vmax=vmax) # histogram plt.figure(4, figsize=(3, 2.2)) plt.clf() plt.axes([.01, .01, .98, .98]) plt.hist(X, bins=256, color='.5', edgecolor='.5') plt.yticks(()) plt.xticks(regular_values) values = np.sort(values) for center_1, center_2 in zip(values[:-1], values[1:]): plt.axvline(.5 * (center_1 + center_2), color='b') for center_1, center_2 in zip(regular_values[:-1], regular_values[1:]): plt.axvline(.5 * (center_1 + center_2), color='b', linestyle='--') plt.show()一般的类聚其实无非两种思路(很多时候神经网络里设计loss函数也是这么两个思路):
又下至上的思路: 每一个样本从它们自身的簇开始, 然后慢慢合并去最小化同类簇之间的间隔, 这种情况一般适合用在簇类少的情况, 当簇的种类数量多的时候, 计算量就会爆炸
这个是从上往下的思路: 所有样本从同一个簇出发, 然后慢慢分隔成下一级的几个簇然后又是下一级的几个簇, 对于簇数量多的情况,这玩意儿也很慢也很吃酸梨
这里有两个可视化的例子:
# Author : Vincent Michel, 2010 # Alexandre Gramfort, 2011 # License: BSD 3 clause print(__doc__) import time as time import numpy as np from scipy.ndimage.filters import gaussian_filter import matplotlib.pyplot as plt import skimage from skimage.data import coins from skimage.transform import rescale from sklearn.feature_extraction.image import grid_to_graph from sklearn.cluster import AgglomerativeClustering from sklearn.utils.fixes import parse_version # these were introduced in skimage-0.14 if parse_version(skimage.__version__) >= parse_version('0.14'): rescale_params = {'anti_aliasing': False, 'multichannel': False} else: rescale_params = {} # ############################################################################# # Generate data orig_coins = coins() # Resize it to 20% of the original size to speed up the processing # Applying a Gaussian filter for smoothing prior to down-scaling # reduces aliasing artifacts. smoothened_coins = gaussian_filter(orig_coins, sigma=2) rescaled_coins = rescale(smoothened_coins, 0.2, mode="reflect", **rescale_params) X = np.reshape(rescaled_coins, (-1, 1)) # ############################################################################# # Define the structure A of the data. Pixels connected to their neighbors. connectivity = grid_to_graph(*rescaled_coins.shape) # ############################################################################# # Compute clustering print("Compute structured hierarchical clustering...") st = time.time() n_clusters = 27 # number of regions ward = AgglomerativeClustering(n_clusters=n_clusters, linkage='ward', connectivity=connectivity) ward.fit(X) label = np.reshape(ward.labels_, rescaled_coins.shape) print("Elapsed time: ", time.time() - st) print("Number of pixels: ", label.size) print("Number of clusters: ", np.unique(label).size) # ############################################################################# # Plot the results on an image plt.figure(figsize=(5, 5)) plt.imshow(rescaled_coins, cmap=plt.cm.gray) for l in range(n_clusters): plt.contour(label == l, colors=[plt.cm.nipy_spectral(l / float(n_clusters)), ]) plt.xticks(()) plt.yticks(()) plt.show()我们可以减少维度来避免维度灾难, 特征块的作用就是合并相同的特征: This approach can be implemented by clustering in the feature direction, in other words clustering the transposed data.
>>> digits = datasets.load_digits() >>> images = digits.images >>> X = np.reshape(images, (len(images), -1)) >>> connectivity = grid_to_graph(*images[0].shape) >>> agglo = cluster.FeatureAgglomeration(connectivity=connectivity, ... n_clusters=32) >>> agglo.fit(X) FeatureAgglomeration(connectivity=..., n_clusters=32) >>> X_reduced = agglo.transform(X) >>> X_approx = agglo.inverse_transform(X_reduced) >>> images_approx = np.reshape(X_approx, images.shape)下面是个例子:
print(__doc__) # Code source: Gaël Varoquaux # Modified for documentation by Jaques Grobler # License: BSD 3 clause import numpy as np import matplotlib.pyplot as plt from sklearn import datasets, cluster from sklearn.feature_extraction.image import grid_to_graph digits = datasets.load_digits() images = digits.images X = np.reshape(images, (len(images), -1)) connectivity = grid_to_graph(*images[0].shape) agglo = cluster.FeatureAgglomeration(connectivity=connectivity, n_clusters=32) agglo.fit(X) X_reduced = agglo.transform(X) X_restored = agglo.inverse_transform(X_reduced) images_restored = np.reshape(X_restored, images.shape) plt.figure(1, figsize=(4, 3.5)) plt.clf() plt.subplots_adjust(left=.01, right=.99, bottom=.01, top=.91) for i in range(4): plt.subplot(3, 4, i + 1) plt.imshow(images[i], cmap=plt.cm.gray, vmax=16, interpolation='nearest') plt.xticks(()) plt.yticks(()) if i == 1: plt.title('Original data') plt.subplot(3, 4, 4 + i + 1) plt.imshow(images_restored[i], cmap=plt.cm.gray, vmax=16, interpolation='nearest') if i == 1: plt.title('Agglomerated data') plt.xticks(()) plt.yticks(()) plt.subplot(3, 4, 10) plt.imshow(np.reshape(agglo.labels_, images[0].shape), interpolation='nearest', cmap=plt.cm.nipy_spectral) plt.xticks(()) plt.yticks(()) plt.title('Labels') plt.show()有些模型有transform 方法, 这个方法可以用来给数据集降维
Components and loadings
If X is our multivariate data, then the problem that we are trying to solve is to rewrite it on a different observational basis: we want to learn loadings L and a set of components C such that X = L C. Different criteria exist to choose the components
降维的一种方法
print(__doc__) # Authors: Gael Varoquaux # Jaques Grobler # Kevin Hughes # License: BSD 3 clause from sklearn.decomposition import PCA from mpl_toolkits.mplot3d import Axes3D import numpy as np import matplotlib.pyplot as plt from scipy import stats # ############################################################################# # Create the data e = np.exp(1) np.random.seed(4) def pdf(x): return 0.5 * (stats.norm(scale=0.25 / e).pdf(x) + stats.norm(scale=4 / e).pdf(x)) y = np.random.normal(scale=0.5, size=(30000)) x = np.random.normal(scale=0.5, size=(30000)) z = np.random.normal(scale=0.1, size=len(x)) density = pdf(x) * pdf(y) pdf_z = pdf(5 * z) density *= pdf_z a = x + y b = 2 * y c = a - b + z norm = np.sqrt(a.var() + b.var()) a /= norm b /= norm # ############################################################################# # Plot the figures def plot_figs(fig_num, elev, azim): fig = plt.figure(fig_num, figsize=(4, 3)) plt.clf() ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=elev, azim=azim) ax.scatter(a[::10], b[::10], c[::10], c=density[::10], marker='+', alpha=.4) Y = np.c_[a, b, c] # Using SciPy's SVD, this would be: # _, pca_score, V = scipy.linalg.svd(Y, full_matrices=False) pca = PCA(n_components=3) pca.fit(Y) pca_score = pca.explained_variance_ratio_ V = pca.components_ x_pca_axis, y_pca_axis, z_pca_axis = 3 * V.T x_pca_plane = np.r_[x_pca_axis[:2], - x_pca_axis[1::-1]] y_pca_plane = np.r_[y_pca_axis[:2], - y_pca_axis[1::-1]] z_pca_plane = np.r_[z_pca_axis[:2], - z_pca_axis[1::-1]] x_pca_plane.shape = (2, 2) y_pca_plane.shape = (2, 2) z_pca_plane.shape = (2, 2) ax.plot_surface(x_pca_plane, y_pca_plane, z_pca_plane) ax.w_xaxis.set_ticklabels([]) ax.w_yaxis.set_ticklabels([]) ax.w_zaxis.set_ticklabels([]) elev = -40 azim = -80 plot_figs(1, elev, azim) elev = 30 azim = 20 plot_figs(2, elev, azim) plt.show()仔细看这两张图片, 在一个方向上可以发现数据是非常平滑的, 在另一个方向上数据是不平滑的, pca找的就是不平滑的数据方向
就是找到独立信息最大的分布部分, 一般用来回复非高斯分布的独立信号:
print(__doc__) import numpy as np import matplotlib.pyplot as plt from scipy import signal from sklearn.decomposition import FastICA, PCA # ############################################################################# # Generate sample data np.random.seed(0) n_samples = 2000 time = np.linspace(0, 8, n_samples) s1 = np.sin(2 * time) # Signal 1 : sinusoidal signal s2 = np.sign(np.sin(3 * time)) # Signal 2 : square signal s3 = signal.sawtooth(2 * np.pi * time) # Signal 3: saw tooth signal S = np.c_[s1, s2, s3] S += 0.2 * np.random.normal(size=S.shape) # Add noise S /= S.std(axis=0) # Standardize data # Mix data A = np.array([[1, 1, 1], [0.5, 2, 1.0], [1.5, 1.0, 2.0]]) # Mixing matrix X = np.dot(S, A.T) # Generate observations # Compute ICA ica = FastICA(n_components=3) S_ = ica.fit_transform(X) # Reconstruct signals A_ = ica.mixing_ # Get estimated mixing matrix # We can `prove` that the ICA model applies by reverting the unmixing. assert np.allclose(X, np.dot(S_, A_.T) + ica.mean_) # For comparison, compute PCA pca = PCA(n_components=3) H = pca.fit_transform(X) # Reconstruct signals based on orthogonal components # ############################################################################# # Plot results plt.figure() models = [X, S, S_, H] names = ['Observations (mixed signal)', 'True Sources', 'ICA recovered signals', 'PCA recovered signals'] colors = ['red', 'steelblue', 'orange'] for ii, (model, name) in enumerate(zip(models, names), 1): plt.subplot(4, 1, ii) plt.title(name) for sig, color in zip(model.T, colors): plt.plot(sig, color=color) plt.tight_layout() plt.show()参考文献:
sklearn 官网教程
