逻辑回归python sigmoid(z)_python实现吴恩达机器学习练习2（逻辑回归）-data1

python实现吴恩达机器学习练习2(逻辑回归)-data1

这篇是第一个数据集：这部分练习中，你将建立一个预测学生是否被大学录取的逻辑回归模型。

假如一所大学会每个报名学生进行两项入学考试，根据两项考试成绩决定是否录取。我们的任务是根据以往100名学生的考试成绩和录取与否的数据进行训练，编码一个学生是否录取的分类器模型。

参考链接：https://blog.csdn.net/Cowry5/article/details/80247569

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

%matplotlib inline

data1 = pd.read_csv('D:/Python/exercise/samples/AndrewNg/ex2/ex2data1.csv', names = ['x1','x2','y'])

len(data1)

100

1 Logistic Regression

1.1 visualizing the data

#散点图做法，有两种点，所以把每类点的数据分开，分别同一张图上画

positive = data1[data1['y'] == 1]

negative = data1[data1['y'] == 0]

positive_x1 = positive.loc[:,'x1']

negative_x2 = negative.loc[:,'x2']

positive_x2 = positive.loc[:,'x2']

negative_x1 = negative.loc[:,'x1']

plt.figure(figsize= (7,7))

plt.scatter(x = positive_x1, y = positive_x2, marker = 'x', color = 'r', label = 'positive')

plt.scatter(x = negative_x1, y = negative_x2, marker = 'o', color = 'b', label = 'negative')

plt.legend()

data1['intercept'] = 1 # 插入一列x0=1

data1 = data1.reindex(columns = ['intercept', 'x1', 'x2', 'y'])

X_ori = data1.iloc[: ,: 3].values

y_ori = data1.iloc[: ,-1].values

1.2 implementation

1.2.1 sigmoid function

定义 Sigmoid函数：

hθ(x)=g(θTx)hθ(x)=g(θTx)h_{\theta}(x)=g(\theta^{T}x)hθ​(x)=g(θTx)

令z=θTx令z=θTx令z=\theta^{T}x令z=θTx

∴sigmoid(x)=11+e−θTx⟹g(z)=11+e−z∴sigmoid(x)=11+e−θTx⟹g(z)=11+e−z\therefore sigmoid(x)=\frac{1}{1+e^{-\theta^{T}x}}\Longrightarrow g(z)=\frac{1}{1+e^{-z}}∴sigmoid(x)=1+e−θTx1​⟹g(z)=1+e−z1​

def sigmoid(z):

g = 1 / (1 + np.exp(-z))

return g

1.2.2 cost function and gradient

定义J(θ)J(θ)J(\theta)J(θ)函数，J(θ)=−1m[∑mi=1y(i)lnhθ(x(i))+(1−y(i))ln(1−hθ(x(i)))]J(θ)=−1m[∑i=1my(i)lnhθ(x(i))+(1−y(i))ln(1−hθ(x(i)))]J(\theta)=-\frac{1}{m}[\sum_{i=1}^my^{(i)}lnh_{\theta}(x^{(i)})+(1-y^{(i)})ln(1-h_{\theta}(x^{(i)})) ]J(θ)=−m1​[i=1∑m​y(i)lnhθ​(x(i))+(1−y(i))ln(1−hθ​(x(i)))]

def J_func(theta, x, y):

cost = -y * np.log(sigmoid(x.dot(theta.T))) - (1-y) * np.log(1-sigmoid(x.dot(theta.T)))

J = cost.mean()

return J

θθ\thetaθ初始值为零

theta_ori = np.zeros(3)

J_func(theta_ori, X_ori, y_ori)

0.6931471805599453

定义梯度gradient,即∂∂θjJ(θ)=1m∑mi=1(hθ(x(i))−y(i))x(i)j∂∂θjJ(θ)=1m∑i=1m(hθ(x(i))−y(i))xj(i)\frac{\partial}{\partial{\theta_j}}J(\theta) = \frac{1}{m}\sum_{i=1}^m(h_{\theta}(x^{(i)})-y^{(i)})x_j^{(i)}∂θj​∂​J(θ)=m1​i=1∑m​(hθ​(x(i))−y(i))xj(i)​

#参数顺序不能乱

def gradient(theta, x, y):

gra = x.T.dot(sigmoid(x.dot(theta.T))-y) / len(x)

return gra

gradient(theta_ori, X_ori, y_ori)

array([ -0.1 , -12.00921659, -11.26284221])

1.2.3 learning parameters using fmintnc

import scipy.optimize as opt

result_ori = opt.fmin_tnc(func = J_func, x0 = theta_ori, fprime = gradient, args = (X_ori, y_ori))

result_ori[0]

array([-25.16131858, 0.20623159, 0.20147149])

决策边界：θ0+θ1x1+θ2x2=0决策边界：θ0+θ1x1+θ2x2=0决策边界：\theta_0+\theta_1x_1+\theta_2x_2=0决策边界：θ0​+θ1​x1​+θ2​x2​=0

x2=−(θ0+θ1x1)θ2x2=−(θ0+θ1x1)θ2x_2=\frac{-(\theta_0+\theta_1x_1)}{\theta_2}x2​=θ2​−(θ0​+θ1​x1​)​

x1 = np.arange(20,110,1)

x2 = - (result_ori[0][0] + result_ori[0][1] * x1) / result_ori[0][2]

plt.figure(figsize = (7, 7))

plt.plot(x1, x2, label = 'decision boundary') # 决策边界线

plt.scatter(x = positive_x1, y = positive_x2, marker = 'x', color = 'r', label = 'positive')

plt.scatter(x = negative_x1, y = negative_x2, marker = 'o', color = 'b', label = 'negative')

plt.legend()

实验：增加多项式项使决策边界更加拟合

根据吴恩达课程-《the problem of overfitting》

这个公式是：g(θ0+θ1x1+θ2x2+θ3x21+θ4x22+θ5x1x2)g(θ0+θ1x1+θ2x2+θ3x12+θ4x22+θ5x1x2)g(\theta_{0}+\theta_{1}x_{1}+\theta_{2}x_{2}+\theta_{3}x_{1}^2+\theta_{4}x_{2}^2+\theta_{5}x_{1}x_{2})g(θ0​+θ1​x1​+θ2​x2​+θ3​x12​+θ4​x22​+θ5​x1​x2​)

data1['x1^2'] = data1['x1'] ** 2

data1['x2^2'] = data1['x2'] ** 2

data1['x1*x2'] = data1['x1'] * data1['x2']

data1 = data1.reindex(columns = ['intercept', 'x1', 'x2', 'x1^2', 'x2^2', 'x1*x2', 'y'])

data1.head()

intercept

x1

x2

x1^2

x2^2

x1*x2

y

1

34.623660

78.024693

1198.797805

6087.852690

2701.500406

1

1

30.286711

43.894998

917.284849

1926.770807

1329.435094

2

1

35.847409

72.902198

1285.036716

5314.730478

2613.354893

3

1

60.182599

86.308552

3621.945269

7449.166166

5194.273015

1

4

1

79.032736

75.344376

6246.173368

5676.775061

5954.672216

1

X = data1.iloc[:,:6]

Y = data1.iloc[:,-1]

X = X.values

Y = Y.values

θθ\thetaθ的初始值用零

theta = np.zeros(6)

theta

array([0., 0., 0., 0., 0., 0.])

def J_func(theta, x, y):

cost = -y * np.log(sigmoid(x.dot(theta.T))) - (1-y) * np.log(1-sigmoid(x.dot(theta.T)))

J = cost.mean()

return J

#这个数等于-ln(0.5)

J_func(theta, X, Y)

0.6931471805599453

gradient(theta, X, Y)

array([-1.00000000e-01, -1.20092166e+01, -1.12628422e+01, -1.13895134e+03,

-1.06939408e+03, -1.09872219e+03])

import scipy.optimize as opt

result = opt.fmin_tnc(func = J_func, x0 = theta, fprime = gradient, args = (X, Y))

D:\ProgramData\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: divide by zero encountered in log

D:\ProgramData\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: invalid value encountered in multiply

result

(array([-1.86643292e-02, -3.75077940e-01, -3.09429907e-01, 1.06962108e-03,

3.50023874e-04, 1.03079001e-02]), 80, 1)

result = result[0]

result_2 = opt.minimize(fun = J_func, x0 = theta, args = (X,Y), method = 'TNC', jac = gradient)

D:\ProgramData\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: divide by zero encountered in log

D:\ProgramData\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: invalid value encountered in multiply

result_2

fun: 0.09302529598313683

jac: array([-4.49764720e-03, 1.75005873e-01, -8.49928704e-01, 4.58961567e+01,

-9.03255063e+01, -2.62541929e+01])

message: 'Converged (|f_n-f_(n-1)| ~= 0)'

nfev: 80

nit: 5

status: 1

success: True

x: array([-1.86643292e-02, -3.75077940e-01, -3.09429907e-01, 1.06962108e-03,

3.50023874e-04, 1.03079001e-02])

画二元多项式的方法，用a,b造一个网格，用c表示a和b的函数值(即c=f(a,b)c=f(a,b)c=f(a,b)c=f(a,b)或 c=f(x1,x2)c=f(x1,x2)c=f(x1,x2)c=f(x1,x2))，相当于用坐标轴的连续点做出x1,x2对应的c的等高线

a = np.arange(20,110,1)

b = np.arange(20,110,1)

xs, ys = np.meshgrid(a,b)

c = result[0] + result[1]*xs + result[2]*ys + result[3]*(xs**2) + result[4]*(ys**2) + result[5]*(xs*ys)

#散点图做法，有两种点，所以把每类点的数据分开，分别同一张图上画

positive = data1[data1['y'] == 1]

negative = data1[data1['y'] == 0]

positive_x1 = positive.loc[:,'x1']

negative_x2 = negative.loc[:,'x2']

positive_x2 = positive.loc[:,'x2']

negative_x1 = negative.loc[:,'x1']

plt.figure(figsize= (15,15))

plt.contour(xs,ys,c,0, label = 'decision boundary') #画出c的等高线，等于0的那条

plt.scatter(x = positive_x1, y = positive_x2, marker = 'x', color = 'r', label = 'positive')

plt.scatter(x = negative_x1, y = negative_x2, label = 'negative')

plt.legend()

曲线分类的效果比线性分类直观上拟合效果更佳

原文链接:https://blog.csdn.net/qq_17541489/article/details/105198647