七月算法机器学习5 回归分析与工程应用 小案例
2024-05-15 04:44:50
目录
逻辑回归
逻辑斯特回归假设
损失函数
向量化的损失函数(矩阵形式)
最小化损失函数(梯度下降)
预测部分
画决策边界
加正则化项的逻辑斯特回归
data
逻辑回归
# %load ../../standard_import.txt
import pandas as pd
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as pltfrom scipy.optimize import minimizefrom sklearn.preprocessing import PolynomialFeaturespd.set_option('display.notebook_repr_html', False)
pd.set_option('display.max_columns', None)
pd.set_option('display.max_rows', 150)
pd.set_option('display.max_seq_items', None)#%config InlineBackend.figure_formats = {'pdf',}
%matplotlib inlineimport seaborn as sns
sns.set_context('notebook')
sns.set_style('white')def loaddata(file, delimeter):data = np.loadtxt(file, delimiter=delimeter)print('Dimensions: ',data.shape)print(data[1:6,:])return(data)def plotData(data, label_x, label_y, label_pos, label_neg, axes=None):# 获得正负样本的下标(即哪些是正样本，哪些是负样本)neg = data[:,2] == 0pos = data[:,2] == 1if axes == None:axes = plt.gca()axes.scatter(data[pos][:,0], data[pos][:,1], marker='+', c='k', s=60, linewidth=2, label=label_pos)axes.scatter(data[neg][:,0], data[neg][:,1], c='y', s=60, label=label_neg)axes.set_xlabel(label_x)axes.set_ylabel(label_y)axes.legend(frameon= True, fancybox = True);
X = np.c_[np.ones((data.shape[0],1)), data[:,0:2]]
y = np.c_[data[:,2]]np.r_中的r是row（行）的缩写，是按行叠加两个矩阵的意思，也可以说是按列连接两个矩阵，就是把两矩阵上下相加，要求列数相等，类似于pandas中的concat()。
np.c_中的c是column（列）的缩写，是按列叠加两个矩阵的意思，也可以说是按行连接两个矩阵，就是把两矩阵左右相加，要求行数相等，类似于pandas中的merge()。
逻辑斯特回归假设
#定义sigmoid函数
def sigmoid(z):return(1 / (1 + np.exp(-z)))其实scipy包里有一个函数可以完成一样的功能:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.expit.html#scipy.special.expit
损失函数
向量化的损失函数(矩阵形式)
#定义损失函数
def costFunction(theta, X, y):m = y.sizeh = sigmoid(X.dot(theta))J = -1*(1/m)*(np.log(h).T.dot(y)+np.log(1-h).T.dot(1-y))if np.isnan(J[0]):return(np.inf)return(J[0])
#求解梯度
def gradient(theta, X, y):m = y.sizeh = sigmoid(X.dot(theta.reshape(-1,1)))grad =(1/m)*X.T.dot(h-y)return(grad.flatten())initial_theta = np.zeros(X.shape[1])
cost = costFunction(initial_theta, X, y)
grad = gradient(initial_theta, X, y)
print('Cost: \n', cost)
print('Grad: \n', grad)
最小化损失函数(梯度下降)
# 这里偷懒了，直接调用scipy里面的最小化损失函数的minimize函数
res = minimize(costFunction, initial_theta, args=(X,y), method=None, jac=gradient, options={'maxiter':400})
res
预测部分
def predict(theta, X, threshold=0.5):p = sigmoid(X.dot(theta.T)) >= thresholdreturn(p.astype('int'))
画决策边界
plt.scatter(45, 85, s=60, c='r', marker='v', label='(45, 85)')
plotData(data, 'Exam 1 score', 'Exam 2 score', 'Pass', 'Failed')
x1_min, x1_max = X[:,1].min(), X[:,1].max(),
x2_min, x2_max = X[:,2].min(), X[:,2].max(),
xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))
h = sigmoid(np.c_[np.ones((xx1.ravel().shape[0],1)), xx1.ravel(), xx2.ravel()].dot(res.x))
h = h.reshape(xx1.shape)
plt.contour(xx1, xx2, h, [0.5], linewidths=1, colors='b');
加正则化项的逻辑斯特回归
def costFunctionReg(theta, reg, *args):m = y.sizeh = sigmoid(XX.dot(theta))J = -1*(1/m)*(np.log(h).T.dot(y)+np.log(1-h).T.dot(1-y)) + (reg/(2*m))*np.sum(np.square(theta[1:]))if np.isnan(J[0]):return(np.inf)return(J[0])
def gradientReg(theta, reg, *args):m = y.sizeh = sigmoid(XX.dot(theta.reshape(-1,1)))grad = (1/m)*XX.T.dot(h-y) + (reg/m)*np.r_[[[0]],theta[1:].reshape(-1,1)]return(grad.flatten())fig, axes = plt.subplots(1,3, sharey = True, figsize=(17,5))# 决策边界，咱们分别来看看正则化系数lambda太大太小分别会出现什么情况
# Lambda = 0 : 就是没有正则化，这样的话，就过拟合咯
# Lambda = 1 : 这才是正确的打开方式
# Lambda = 100 : 卧槽，正则化项太激进，导致基本就没拟合出决策边界for i, C in enumerate([0, 1, 100]):# 最优化 costFunctionRegres2 = minimize(costFunctionReg, initial_theta, args=(C, XX, y), method=None, jac=gradientReg, options={'maxiter':3000})# 准确率accuracy = 100*sum(predict(res2.x, XX) == y.ravel())/y.size # 对X,y的散列绘图plotData(data2, 'Microchip Test 1', 'Microchip Test 2', 'y = 1', 'y = 0', axes.flatten()[i])# 画出决策边界x1_min, x1_max = X[:,0].min(), X[:,0].max(),x2_min, x2_max = X[:,1].min(), X[:,1].max(),xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))h = sigmoid(poly.fit_transform(np.c_[xx1.ravel(), xx2.ravel()]).dot(res2.x))h = h.reshape(xx1.shape)axes.flatten()[i].contour(xx1, xx2, h, [0.5], linewidths=1, colors='g'); axes.flatten()[i].set_title('Train accuracy {}% with Lambda = {}'.format(np.round(accuracy, decimals=2), C))
data
data1.txt
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data2.txt
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