python 画出决策边界_Python3入门机器学习

python 画出决策边界_Python3入门机器学习 - 逻辑回归与决策边界

logistic回归又称logistic回归分析，是一种广义的线性回归分析模型,以胃癌病情分析为例，选择两组人群，一组是胃癌组，一组是非胃癌组，两组人群必定具有不同的体征与生活方式等。因此因变量就为是否胃癌，值为“是”或“否”，自变量就可以包括很多了，如年龄、性别、饮食习惯、幽门螺杆菌感染等。

逻辑回归需要将原本线性回归结果的值域置于(0，1)之间，概率大于0.5看作结果为1

常使用sigmoid函数将结果变为(0，1)之间的值域

绘制sigmoid曲线def sigmoid(t):

return 1/(1+np.exp(-t))

x = np.linspace(-10,10,100)

y = sigmoid(x)

plt.plot(x,y)

逻辑回归的损失函数

该损失函数没有公式解，可以用梯度下降法求最优解

损失函数求导

封装Logistics模型我使用Logistics模型及鸢尾花数据集的前两列，可得到1.0准确率的预测精准度。# 代码与线性回归及其相似，只是推导公式不同# _*_ encoding:utf-8 _*_import numpy as np

from sklearn.metrics import r2_score

from metrics import accuracy_scoreclass LinearRegression:

def __init__(self): self.coef_ = None self.interception_ = None self._theta = None def _sigmoid(self,t): return 1./(1.+ np.exp(-t)) def fit(self,X_train,y_train,eta=0.01,n_iters=1e6): def J(theta,X_b,y):

y_hat = self._sigmoid(X_b.dot(theta)) try:

return -np.sum(y*np.log(y_hat)+(1-y)*np.log(1-y_hat))/len(y) except:

return float("inf") def dJ(theta,X_b,y): # res = np.empty()

# res[0] = np.sum(X_b.dot(theta)-y)

# for i in range(1,len(theta)):

# res[i] = (X_b.dot(theta)-y).dot(X_b[:,i])

# return res * 2 / len(X_b)

return X_b.T.dot(self._sigmoid(X_b.dot(theta))-y)/len(X_b) def gradient_descent(X_b,y,initial_theta,eta,n_iters=1e6,epsilon=1e-8):

theta = initial_theta

cur_iter = 0

while cur_iter

gradient = dJ(theta,X_b,y)

last_theta = theta

theta = theta - eta * gradient if (abs(J(theta,X_b,y) - J(last_theta,X_b,y))

cur_iter+=1

return theta

X_b = np.hstack([np.ones((len(X_train),1)),X_train])

initial_theta = np.zeros(X_b.shape[1]) self._theta = gradient_descent(X_b,y_train,initial_theta,eta,n_iters) self.interception_ = self._theta[0] self.coef_ = self._theta[1: return self

def predict_proba(self,X_predict):

X_b = np.hstack([np.ones((len(X_predict),1)),X_predict]) return self._sigmoid(X_b.dot(self._theta)) def predict(self,X_predict):

proba = self.predict_proba(X_predict) return proba>=0.5

def score(self,X_test,y_test): return accuracy_score(y_test,self.predict(X_test)) def __repr__(self): return "LogisticRegreesion()"

决策边界在逻辑回归中，易得决策边界为 theta*X_b=0 的直线

如果只有两个特征值，则很容易通过公式画出逻辑回归的决策边界

蓝线即为鸢尾花数据前两列特征值通过逻辑回归得到的决策边界

不规则的决策边界的绘制

一种绘制思路def plot_decision_boundary(model,axis):

x0,x1 = np.meshgrid(

np.linspace(axis[0],axis[1],int((axis[1]-axis[0])*100)),

np.linspace(axis[2],axis[3],int((axis[3]-axis[2])*100))

)

X_new = np.c_[x0.ravel(),x1.ravel()]

y_predict = model.predict(X_new)

zz = y_predict.reshape(x0.shape)

from matplotlib.colors import ListedColormap

custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])

plt.contourf(x0,x1,zz,linewidth=5,cmap=custom_cmap)plot_decision_boundary(log_reg,axis=[4,7.5,1.5,4.5])

plt.scatter(X[y==0,0],X[y==0,1])

plt.scatter(X[y==1,0],X[y==1,1])

plt.show()

knn算法的决策边界knn_clf = KNeighborsClassifier()

knn_clf.fit(iris.data[:,:2],iris.target)

plot_decision_boundary(knn_clf,axis=[4,7.5,1.5,4.5])

plt.scatter(X[y==0,0],X[y==0,1])

plt.scatter(X[y==1,0],X[y==1,1])

plt.show()

n_neighbors默认等于5时

n_neighbors等于50时

多项式特征应用于逻辑回归#准备数据X = np.random.normal(0,1,size=(200,2))

y = np.array(X[:,0]**2 + X[:,1]**2<1.5,dtype='int')def PolynomialLogisticRegression(degree):

return Pipeline([

('Poly',PolynomialFeatures(degree=degree)),

('std_scaler',StandardScaler()),

('Logistic',LogisticRegression())

])

Log_reg = PolynomialLogisticRegression(2)

Log_reg.fit(X,y)

逻辑回归的模型正则化

逻辑回归的模型正则化方式#准备数据import numpy as npimport matplotlib.pyplot as plt

X = np.random.normal(0,1,size=(200,2))

y = np.array(X[:,0]**2 + X[:,1]<1.5,dtype='int')for _ in range(20):

y[np.random.randint(200)] = 1 #噪音plt .scatter(X[y==0,0],X[y==0,1])

plt .scatter(X[y==1,0],X[y==1,1])

plt.show()

数据.pngfrom sklearn.linear_model import LogisticRegressionfrom sklearn.preprocessing import PolynomialFeaturesfrom sklearn.pipeline import Pipeline

log_reg = LogisticRegression()

log_reg.fit(X,y)LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,

intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,

penalty='l2', random_state=None, solver='liblinear', tol=0.0001,

verbose=0, warm_start=False)

其中模型正则化公式中的参数C默认为1,penalty默认为'l2'from sklearn.preprocessing import StandardScalerdef PolynomialLogisticRegression(degree,C=1.0,penalty='l2'):

return Pipeline([

('Poly',PolynomialFeatures(degree=degree)),

('std_scaler',StandardScaler()),

('Logistic',LogisticRegression(C=C,penalty=penalty))

])

poly_log_reg = PolynomialLogisticRegression(degree=20,C=0.1,penalty='l1')

poly_log_reg.fit(X,y)

曲线相对平滑

应用OVR和OVO使逻辑回归处理多分类问题OVR:One Vs Rest

OVO:One Vs One

OVR耗时较少，性能较高，但分类准确度略低

OVO耗时较多，分类准确度较高#为了数据可视化方便，我们只使用鸢尾花数据集的前两列特征from sklearn import datasets

iris = datasets.load_iris()

X = iris['data'][:,:2]

y = iris['target']

log_reg = LogisticRegression(multi_class='ovr') #传入multi_class参数可以指定使用ovr或ovo，默认ovrlog_reg.score(X_test,y_test)>>> 0.578 #由于只使用前两列特征，导致分类准确度较低log_reg = LogisticRegression(multi_class='ovr',solver='newton-cg')

log_reg.fit(X_train,y_train)

log_reg.score(X_test,y_test)>>> 0.7894736842105263

OVR分类决策边界

OVO分类决策边界

使用scikitlearn中的OVO及OVR类来进行多分类from sklearn.multiclass import OneVsOneClassifierfrom sklearn.multiclass import OneVsRestClassifier

ovr = OneVsRestClassifier(log_reg)

ovr.fit(X_train,y_train)

print(ovr.score(X_test,y_test))

ovo = OneVsOneClassifier(log_reg)

ovo.fit(X_train,y_train)

print(ovo.score(X_test,y_test))>>> 0.7894736842105263>>> 0.8157894736842105

作者：冰源_63ad