异常数据4种剔除方法分别是：1、“isolation forest”，孤立森林；2、DBSCAN；3、OneClassSVM；4、“Local Outlier Factor”，计算一个数值score来反映一个样本的异常程度。
本教程操作环境：windows7系统、Dell G3电脑。
outlier detection异常点识别方法1. isolation forest 孤立森林1.1 测试样本示例
文件 test.pkl
1.2 孤立森林 demo
孤立森林原理
通过对特征进行随机划分，建立随机森林，将经过较少次数进行划分就可以划分出来的点认为时异常点。
# 参考https://blog.csdn.net/ye1215172385/article/details/79762317 # 官方例子https://scikit-learn.org/stable/auto_examples/ensemble/plot_isolation_forest.html#sphx-glr-auto-examples-ensemble-plot-isolation-forest-pyimport numpy as npimport matplotlib.pyplot as pltfrom sklearn.ensemble import IsolationForest rng = np.random.RandomState(42) # 构造训练样本n_samples = 200 #样本总数outliers_fraction = 0.25 #异常样本比例n_inliers = int((1. - outliers_fraction) * n_samples)n_outliers = int(outliers_fraction * n_samples) X = 0.3 * rng.randn(n_inliers // 2, 2)X_train = np.r_[X + 2, X - 2] #正常样本X_train = np.r_[X_train, np.random.uniform(low=-6, high=6, size=(n_outliers, 2))] #正常样本加上异常样本 # 构造模型并拟合clf = IsolationForest(max_samples=n_samples, random_state=rng, contamination=outliers_fraction)clf.fit(X_train)# 计算得分并设置阈值scores_pred = clf.decision_function(X_train)threshold = np.percentile(scores_pred, 100 * outliers_fraction) #根据训练样本中异常样本比例，得到阈值，用于绘图 # plot the line, the samples, and the nearest vectors to the planexx, yy = np.meshgrid(np.linspace(-7, 7, 50), np.linspace(-7, 7, 50))Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape) plt.title("IsolationForest")# plt.contourf(xx, yy, Z, cmap=plt.cm.Blues_r)plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), threshold, 7), cmap=plt.cm.Blues_r) #绘制异常点区域，值从最小的到阈值的那部分a = plt.contour(xx, yy, Z, levels=[threshold], linewidths=2, colors='red') #绘制异常点区域和正常点区域的边界plt.contourf(xx, yy, Z, levels=[threshold, Z.max()], colors='palevioletred') #绘制正常点区域，值从阈值到最大的那部分 b = plt.scatter(X_train[:-n_outliers, 0], X_train[:-n_outliers, 1], c='white', s=20, edgecolor='k')c = plt.scatter(X_train[-n_outliers:, 0], X_train[-n_outliers:, 1], c='black', s=20, edgecolor='k')plt.axis('tight')plt.xlim((-7, 7))plt.ylim((-7, 7))plt.legend([a.collections[0], b, c], ['learned decision function', 'true inliers', 'true outliers'], loc="upper left")plt.show()
1.3 自己修改的，X_train能够改成自己需要的数据
此处没有进行标准化，可以先进行标准化再在标准化的基础上去除异常点， from sklearn.preprocessing import StandardScaler
import numpy as npimport matplotlib.pyplot as pltfrom sklearn.ensemble import IsolationForestfrom scipy import stats rng = np.random.RandomState(42) X_train = X_train_demo.valuesoutliers_fraction = 0.1n_samples = 500# 构造模型并拟合clf = IsolationForest(max_samples=n_samples, random_state=rng, contamination=outliers_fraction)clf.fit(X_train)# 计算得分并设置阈值scores_pred = clf.decision_function(X_train)threshold = stats.scoreatpercentile(scores_pred, 100 * outliers_fraction) #根据训练样本中异常样本比例，得到阈值，用于绘图 # plot the line, the samples, and the nearest vectors to the planerange_max_min0 = (X_train[:,0].max()-X_train[:,0].min())*0.2range_max_min1 = (X_train[:,1].max()-X_train[:,1].min())*0.2xx, yy = np.meshgrid(np.linspace(X_train[:,0].min()-range_max_min0, X_train[:,0].max()+range_max_min0, 500), np.linspace(X_train[:,1].min()-range_max_min1, X_train[:,1].max()+range_max_min1, 500))Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape) plt.title("IsolationForest")# plt.contourf(xx, yy, Z, cmap=plt.cm.Blues_r)plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), threshold, 7), cmap=plt.cm.Blues_r) #绘制异常点区域，值从最小的到阈值的那部分a = plt.contour(xx, yy, Z, levels=[threshold], linewidths=2, colors='red') #绘制异常点区域和正常点区域的边界plt.contourf(xx, yy, Z, levels=[threshold, Z.max()], colors='palevioletred') #绘制正常点区域，值从阈值到最大的那部分 is_in = clf.predict(X_train)>0b = plt.scatter(X_train[is_in, 0], X_train[is_in, 1], c='white', s=20, edgecolor='k')c = plt.scatter(X_train[~is_in, 0], X_train[~is_in, 1], c='black', s=20, edgecolor='k')plt.axis('tight')plt.xlim((X_train[:,0].min()-range_max_min0, X_train[:,0].max()+range_max_min0,))plt.ylim((X_train[:,1].min()-range_max_min1, X_train[:,1].max()+range_max_min1,))plt.legend([a.collections[0], b, c], ['learned decision function', 'inliers', 'outliers'], loc="upper left")plt.show()
1.4 核心代码1.4.1 示例样本
import numpy as np# 构造训练样本n_samples = 200 #样本总数outliers_fraction = 0.25 #异常样本比例n_inliers = int((1. - outliers_fraction) * n_samples)n_outliers = int(outliers_fraction * n_samples) X = 0.3 * rng.randn(n_inliers // 2, 2)X_train = np.r_[X + 2, X - 2] #正常样本X_train = np.r_[X_train, np.random.uniform(low=-6, high=6, size=(n_outliers, 2))] #正常样本加上异常样本
1.4.2 核心代码实现
clf = IsolationForest(max_samples=0.8, contamination=0.25)
from sklearn.ensemble import IsolationForest# fit the model# max_samples 构造一棵树使用的样本数，输入大于1的整数则使用该数字作为构造的最大样本数目，# 如果数字属于(0,1]则使用该比例的数字作为构造iforest# outliers_fraction 多少比例的样本可以作为异常值clf = IsolationForest(max_samples=0.8, contamination=0.25)clf.fit(X_train)# y_pred_train = clf.predict(X_train)scores_pred = clf.decision_function(X_train)threshold = np.percentile(scores_pred, 100 * outliers_fraction) #根据训练样本中异常样本比例，得到阈值，用于绘图## 以下两种方法的筛选结果，完全相同X_train_predict1 = X_train[clf.predict(X_train)==1]X_train_predict2 = X_train[scores_pred>=threshold,:]# 其中，1的表示非异常点，-1的表示为异常点clf.predict(X_train)array([ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1])
2. DBSCAN
DBSCAN(Density-Based Spatial Clustering of Applications with Noise) 原理
以每个点为中心，设定邻域及邻域内需要有多少个点，如果样本点大于指定要求，则认为该点与邻域内的点属于同一类，如果小于指定值，若该点位于其它点的邻域内，则属于边界点。
2.1 DBSCAN demo
# 参考https://blog.csdn.net/hb707934728/article/details/71515160## 官方示例 https://scikit-learn.org/stable/auto_examples/cluster/plot_dbscan.html#sphx-glr-auto-examples-cluster-plot-dbscan-pyimport numpy as npimport matplotlib.pyplot as pltimport matplotlib.colorsimport sklearn.datasets as dsfrom sklearn.cluster import DBSCANfrom sklearn.preprocessing import StandardScalerdef expand(a, b): d = (b - a) * 0.1 return a-d, b+dif __name__ == "__main__": N = 1000 centers = [[1, 2], [-1, -1], [1, -1], [-1, 1]] #scikit中的make_blobs方法常被用来生成聚类算法的测试数据，直观地说，make_blobs会根据用户指定的特征数量、 # 中心点数量、范围等来生成几类数据，这些数据可用于测试聚类算法的效果。 #函数原型：sklearn.datasets.make_blobs(n_samples=100, n_features=2, # centers=3, cluster_std=1.0, center_box=(-10.0, 10.0), shuffle=True, random_state=None)[source] #参数解析： # n_samples是待生成的样本的总数。 # # n_features是每个样本的特征数。 # # centers表示类别数。 # # cluster_std表示每个类别的方差，例如我们希望生成2类数据，其中一类比另一类具有更大的方差，可以将cluster_std设置为[1.0, 3.0]。 data, y = ds.make_blobs(N, n_features=2, centers=centers, cluster_std=[0.5, 0.25, 0.7, 0.5], random_state=0) data = StandardScaler().fit_transform(data) # 数据1的参数：(epsilon, min_sample) params = ((0.2, 5), (0.2, 10), (0.2, 15), (0.3, 5), (0.3, 10), (0.3, 15)) plt.figure(figsize=(12, 8), facecolor='w') plt.suptitle(u'DBSCAN clustering', fontsize=20) for i in range(6): eps, min_samples = params[i] #参数含义： #eps:半径，表示以给定点P为中心的圆形邻域的范围 #min_samples:以点P为中心的邻域内最少点的数量 #如果满足,以点P为中心,半径为EPS的邻域内点的个数不少于MinPts,则称点P为核心点 model = DBSCAN(eps=eps, min_samples=min_samples) model.fit(data) y_hat = model.labels_ core_indices = np.zeros_like(y_hat, dtype=bool) # 生成数据类型和数据shape和指定array一致的变量 core_indices[model.core_sample_indices_] = True # model.core_sample_indices_ border point位于y_hat中的下标 # 统计总共有积累，其中为-1的为未分类样本 y_unique = np.unique(y_hat) n_clusters = y_unique.size - (1 if -1 in y_hat else 0) print (y_unique, '聚类簇的个数为：', n_clusters) plt.subplot(2, 3, i+1) # 对第几个图绘制，2行3列，绘制第i+1个图 # plt.cm.spectral https://blog.csdn.net/robin_Xu_shuai/article/details/79178857 clrs = plt.cm.Spectral(np.linspace(0, 0.8, y_unique.size)) #用于给画图灰色 for k, clr in zip(y_unique, clrs): cur = (y_hat == k) if k == -1: # 用于绘制未分类样本 plt.scatter(data[cur, 0], data[cur, 1], s=20, c='k') continue # 绘制正常节点 plt.scatter(data[cur, 0], data[cur, 1], s=30, c=clr, edgecolors='k') # 绘制边缘点 plt.scatter(data[cur & core_indices][:, 0], data[cur & core_indices][:, 1], s=60, c=clr, marker='o', edgecolors='k') x1_min, x2_min = np.min(data, axis=0) x1_max, x2_max = np.max(data, axis=0) x1_min, x1_max = expand(x1_min, x1_max) x2_min, x2_max = expand(x2_min, x2_max) plt.xlim((x1_min, x1_max)) plt.ylim((x2_min, x2_max)) plt.grid(True) plt.title(u'$epsilon$ = %.1f m = %d clustering num %d'%(eps, min_samples, n_clusters), fontsize=16) plt.tight_layout() plt.subplots_adjust(top=0.9) plt.show()[-1 0 1 2 3] 聚类簇的个数为： 4[-1 0 1 2 3] 聚类簇的个数为： 4[-1 0 1 2 3 4] 聚类簇的个数为： 5[-1 0] 聚类簇的个数为： 1[-1 0 1] 聚类簇的个数为： 2[-1 0 1 2 3] 聚类簇的个数为： 4
2.2 使用自定义测试样例
## 参考https://blog.csdn.net/hb707934728/article/details/71515160## 官方示例 https://scikit-learn.org/stable/auto_examples/cluster/plot_dbscan.html#sphx-glr-auto-examples-cluster-plot-dbscan-pyimport numpy as npimport matplotlib.pyplot as pltimport matplotlib.colorsimport sklearn.datasets as dsfrom sklearn.cluster import DBSCANfrom sklearn.preprocessing import StandardScalerdef expand(a, b): d = (b - a) * 0.1 return a-d, b+dif __name__ == "__main__": N = 1000 data = X_train_demo.values # 数据1的参数：(epsilon, min_sample) params = ((0.2, 5), (0.2, 10), (0.2, 15), (0.2, 20), (0.2, 25), (0.2, 30)) plt.figure(figsize=(12, 8), facecolor='w') plt.suptitle(u'DBSCAN clustering', fontsize=20) for i in range(6): eps, min_samples = params[i] #参数含义： #eps:半径，表示以给定点P为中心的圆形邻域的范围 #min_samples:以点P为中心的邻域内最少点的数量 #如果满足,以点P为中心,半径为EPS的邻域内点的个数不少于MinPts,则称点P为核心点 model = DBSCAN(eps=eps, min_samples=min_samples) model.fit(data) y_hat = model.labels_ core_indices = np.zeros_like(y_hat, dtype=bool) # 生成数据类型和数据shape和指定array一致的变量 core_indices[model.core_sample_indices_] = True # model.core_sample_indices_ border point位于y_hat中的下标 # 统计总共有积累，其中为-1的为未分类样本 y_unique = np.unique(y_hat) n_clusters = y_unique.size - (1 if -1 in y_hat else 0) print (y_unique, '聚类簇的个数为：', n_clusters) plt.subplot(2, 3, i+1) # 对第几个图绘制，2行3列，绘制第i+1个图 # plt.cm.spectral https://blog.csdn.net/robin_Xu_shuai/article/details/79178857 clrs = plt.cm.Spectral(np.linspace(0, 0.8, y_unique.size)) #用于给画图灰色 for k, clr in zip(y_unique, clrs): cur = (y_hat == k) if k == -1: # 用于绘制未分类样本 plt.scatter(data[cur, 0], data[cur, 1], s=20, c='k') continue # 绘制正常节点 plt.scatter(data[cur, 0], data[cur, 1], s=30, c=clr, edgecolors='k') # 绘制边缘点 plt.scatter(data[cur & core_indices][:, 0], data[cur & core_indices][:, 1], s=60, c=clr, marker='o', edgecolors='k') x1_min, x2_min = np.min(data, axis=0) x1_max, x2_max = np.max(data, axis=0) x1_min, x1_max = expand(x1_min, x1_max) x2_min, x2_max = expand(x2_min, x2_max) plt.xlim((x1_min, x1_max)) plt.ylim((x2_min, x2_max)) plt.grid(True) plt.title(u'$epsilon$ = %.1f m = %d clustering num %d'%(eps, min_samples, n_clusters), fontsize=14) plt.tight_layout() plt.subplots_adjust(top=0.9) plt.show()
注意：可以看到在测试样例的两端，相比与孤立森林，DBSCAN能够很好地对“尖端”处的样本的分类。
2.3 核心代码
model = DBSCAN(eps=eps, min_samples=min_samples) # 构造分类器
from sklearn.cluster import DBSCANfrom sklearn import metricsdata = X_train_demo.valueseps, min_samples = 0.2, 10# eps为领域的大小，min_samples为领域内最小点的个数model = DBSCAN(eps=eps, min_samples=min_samples) # 构造分类器model.fit(data) # 拟合labels = model.labels_ # 获取类别标签，-1表示未分类# 获取其中的core pointscore_indices = np.zeros_like(labels, dtype=bool) # 生成数据类型和数据shape和指定array一致的变量core_indices[model.core_sample_indices_] = True # model.core_sample_indices_ border point位于labels中的下标core_point = data[core_indices]# 获取非异常点normal_point = data[labels>=0]# 绘制剔除了异常值后的图plt.scatter(normal_point[:,0],normal_point[:,1],edgecolors='k')plt.show()
2.4 构造过滤函数
该函数先进行了标准化，方便使用固定的参数进行分析
2.4.1 过滤函数
def filter_data(data0, params): from sklearn.cluster import DBSCAN from sklearn import metrics scaler = StandardScaler() scaler.fit(data0) data = scaler.transform(data0) eps, min_samples = params # eps为领域的大小，min_samples为领域内最小点的个数 model = DBSCAN(eps=eps, min_samples=min_samples) # 构造分类器 model.fit(data) # 拟合 labels = model.labels_ # 获取类别标签，-1表示未分类 # 获取其中的core points core_indices = np.zeros_like(labels, dtype=bool) # 生成数据类型和数据shape和指定array一致的变量 core_indices[model.core_sample_indices_] = True # model.core_sample_indices_ border point位于labels中的下标 core_point = data[core_indices] # 获取非异常点 normal_point = data0[labels>=0] return normal_point
2.4.2 衡量分类结果
（markdown格式懒得转，直接截图了::>_<::）
# 轮廓系数metrics.silhouette_score(data, labels, metric='euclidean')[out]0.13250260550638607# Calinski-Harabaz Index 系数metrics.calinski_harabaz_score(data, labels,)[out]16.414158842632794
3. OneClassSVM
# reference:https://scikit-learn.org/stable/auto_examples/svm/plot_oneclass.html#sphx-glr-auto-examples-svm-plot-oneclass-pyimport numpy as npimport matplotlib.pyplot as pltimport matplotlib.font_managerfrom sklearn import svmxx, yy = np.meshgrid(np.linspace(-5, 5, 500), np.linspace(-5, 5, 500))# Generate train dataX = 0.3 * np.random.randn(100, 2)X_train = np.r_[X + 2, X - 2]# Generate some regular novel observationsX = 0.3 * np.random.randn(20, 2)X_test = np.r_[X + 2, X - 2]# Generate some abnormal novel observationsX_outliers = np.random.uniform(low=-4, high=4, size=(20, 2))# fit the modelclf = svm.OneClassSVM(nu=0.1, kernel="rbf", gamma=0.1)clf.fit(X_train)y_pred_train = clf.predict(X_train)y_pred_test = clf.predict(X_test)y_pred_outliers = clf.predict(X_outliers)n_error_train = y_pred_train[y_pred_train == -1].sizen_error_test = y_pred_test[y_pred_test == -1].sizen_error_outliers = y_pred_outliers[y_pred_outliers == 1].size# plot the line, the points, and the nearest vectors to the planeZ = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape)plt.title("Novelty Detection")plt.contourf(xx, yy, Z, levels=np.linspace(Z.min(), 0, 7), cmap=plt.cm.PuBu)a = plt.contour(xx, yy, Z, levels=[0], linewidths=2, colors='darkred')plt.contourf(xx, yy, Z, levels=[0, Z.max()], colors='palevioletred')s = 40b1 = plt.scatter(X_train[:, 0], X_train[:, 1], c='white', s=s, edgecolors='k')b2 = plt.scatter(X_test[:, 0], X_test[:, 1], c='blueviolet', s=s, edgecolors='k')c = plt.scatter(X_outliers[:, 0], X_outliers[:, 1], c='gold', s=s, edgecolors='k')plt.axis('tight')plt.xlim((-5, 5))plt.ylim((-5, 5))plt.legend([a.collections[0], b1, b2, c], ["learned frontier", "training observations", "new regular observations", "new abnormal observations"], loc="upper left", prop=matplotlib.font_manager.FontProperties(size=11))plt.xlabel( "error train: %d/200 ; errors novel regular: %d/40 ; " "errors novel abnormal: %d/40" % (n_error_train, n_error_test, n_error_outliers))plt.show()
3.2 核心代码
from sklearn import svmX_train = X_train_demo.values# 构造分类器clf = svm.OneClassSVM(nu=0.2, kernel="rbf", gamma=0.2)clf.fit(X_train)# 预测，结果为-1或者1labels = clf.predict(X_train)# 分类分数score = clf.decision_function(X_train) # 获取置信度# 获取正常点X_train_normal = X_train[labels>0]
进行剔除异常点之前
剔除异常点之后
plt.scatter(X_train_normal[:,0],X_train_normal[:,1])plt.show()
4. Local Outlier Factor（LOF）
LOF通过计算一个数值score来反映一个样本的异常程度。 这个数值的大致意思是：
一个样本点周围的样本点所处位置的平均密度比上该样本点所在位置的密度。比值越大于1，则该点所在位置的密度越小于其周围样本所在位置的密度。
## 参考https://blog.csdn.net/hb707934728/article/details/71515160## 官方示例 https://scikit-learn.org/stable/auto_examples/cluster/plot_dbscan.html#sphx-glr-auto-examples-cluster-plot-dbscan-pyimport numpy as npimport matplotlib.pyplot as pltimport matplotlib.colorsfrom sklearn.neighbors import LocalOutlierFactordef expand(a, b): d = (b - a) * 0.1 return a-d, b+dif __name__ == "__main__": N = 1000 data = X_train_demo.values # 数据1的参数：(epsilon, min_sample) params = ((0.01, 5), (0.05, 10), (0.1, 15), (0.15, 20), (0.2, 25), (0.25, 30)) plt.figure(figsize=(12, 8), facecolor='w') plt.suptitle(u'DBSCAN clustering', fontsize=20) for i in range(6): outliers_fraction, min_samples = params[i] #参数含义： #eps:半径，表示以给定点P为中心的圆形邻域的范围 #min_samples:以点P为中心的邻域内最少点的数量 #如果满足,以点P为中心,半径为EPS的邻域内点的个数不少于MinPts,则称点P为核心点 model = LocalOutlierFactor(n_neighbors=min_samples, contamination=outliers_fraction) y_hat = model.fit_predict(X_train) # 统计总共有积累，其中为-1的为未分类样本 y_unique = np.unique(y_hat) # clrs = [] # for c in np.linspace(16711680, 255, y_unique.size): # clrs.append('#%06x' % c) plt.subplot(2, 3, i+1) # 对第几个图绘制，2行3列，绘制第i+1个图 # plt.cm.spectral https://blog.csdn.net/robin_Xu_shuai/article/details/79178857 clrs = plt.cm.Spectral(np.linspace(0, 0.8, y_unique.size)) #用于给画图灰色 for k, clr in zip(y_unique, clrs): cur = (y_hat == k) if k == -1: # 用于绘制未分类样本 plt.scatter(data[cur, 0], data[cur, 1], s=20, c='k') continue # 绘制正常节点 plt.scatter(data[cur, 0], data[cur, 1], s=30, c=clr, edgecolors='k') x1_max, x2_max = np.max(data, axis=0) x1_min, x2_min = np.min(data, axis=0) x1_min, x1_max = expand(x1_min, x1_max) x2_min, x2_max = expand(x2_min, x2_max) plt.xlim((x1_min, x1_max)) plt.ylim((x2_min, x2_max)) plt.grid(True) plt.title(u'outliers_fraction = %.1f min_samples = %d'%(outliers_fraction, min_samples), fontsize=12) plt.tight_layout() plt.subplots_adjust(top=0.9) plt.show()
4.1 核心代码
from sklearn.neighbors import LocalOutlierFactorX_train = X_train_demo.values# 构造分类器## 25个样本点为一组，异常值点比例为0.2clf = LocalOutlierFactor(n_neighbors=25, contamination=0.2)# 预测，结果为-1或者1labels = clf.fit_predict(X_train)# 获取正常点X_train_normal = X_train[labels>0]
进行剔除异常点之前
plt.scatter(X_train[:,0],X_train[:,1])plt.show()
剔除异常点之后
plt.scatter(X_train_normal[:,0],X_train_normal[:,1])plt.show()
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酒般的思念，一饮就醉，醉时就用全部的热情读这忧伤的月色，
