线性回归
回归问题 输入变量和输出都是连续的问题
一维线性问题
模型选择
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y = wx + b
y=wx+b 损失函数 （表示预测值和数据真实值的偏差）
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L(y,y_i) = \sum_{i=1}^n (y - y_i)^2
L(y,yi)=i=1∑n(y−yi)2 损失函数使用的是均值误差（定义为图中 预测值(黄线) 和 真实值(蓝点)的距离的平方）。还可以为,曼哈顿距离，马氏距离，总偏差平方和,回归平方和，残差平方和等 求解 求解斜率w和截距b。既可以得到一维线性方程。
损失函数最小值则表示训练数据整体和真实数据的距离最小。此时的预测值越接近真实值。
对b 偏导数
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\frac{\partial f}{\partial b} = -2(\sum_{i=1}^n y_i -nb - w\sum_{i=1}^n x_i)
∂b∂f=−2(i=1∑nyi−nb−wi=1∑nxi) 对w 偏导数
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\frac{\partial f}{\partial w} = -2(\sum_{i=1}^n y_i * x_i -\sum_{i=0}^n x_i - w\sum_{i=1}^n x_i)
∂w∂f=−2(i=1∑nyi∗xi−i=0∑nxi−wi=1∑nxi) 令
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\frac{\partial f}{\partial w}=0
∂w∂f=0 以及
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\frac{\partial f}{\partial b}=0
∂b∂f=0
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w = \frac{n\varSigma y_i * x_i -\varSigma x_i*\varSigma y_i}{n\varSigma x_i^2 -(\varSigma x_i)^2}
w=nΣxi2−(Σxi)2nΣyi∗xi−Σxi∗Σyi
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b = \frac{\varSigma x_i^2 *\varSigma y_i -\varSigma x_i*\varSigma y_i x_i}{n\varSigma x_i^2 -(\varSigma x_i)^2}
b=nΣxi2−(Σxi)2Σxi2∗Σyi−Σxi∗Σyixi ==>
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w = \frac{\overline{X} * \overline{Y} - \overline{(XY)}}{\overline{X}^2 - \overline{X^2}}
w=X2−X2X∗Y−(XY)
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b = \overline{Y} - w\overline{X}
b=Y−wX 编程验证1：手动实现 import numpy as np
import matplotlib.pyplot as plt
#测试数据
train_x = np.array([2, 6, 8, 8, 12, 16, 20, 20, 22, 26])
train_y = np.array([58, 105, 88, 118, 117, 137, 157, 169, 149, 202])
#测试数据
test_x = np.array([3, 7, 17]).reshape(-1, 1)
test_y = np.array([70, 100, 155]).reshape(-1, 1)
class OneLineaRegression:
def __init__(self, train_x, train_y):
self.train_x = train_x
self.train_y = train_y
self.train = np.hstack(
(train_x.reshape(-1, 1), train_y.reshape(-1, 1)))
self.w = 0
self.b = 0
def calcuate_loss(self):
#计算线性参数 w ,b
# x 均值
x_avg = np.mean(self.train_x)
# y 均值
y_avg = np.mean(self.train_y)
# x*y的均值
xy_avg = np.mean(self.train_x * self.train_y)
# 平方的均值
xx_avg = np.mean(self.train_x ** 2)
# 均值的平方
x_avg_x = x_avg * x_avg
self.w = (x_avg * y_avg - xy_avg) / (x_avg_x - xx_avg)
self.b = y_avg - self.w * x_avg
def predict(self, x):
return self.w * x + self.b
def show_line(self, pre_value):
line_x = np.arange(0, 30)
line_y = self.w * line_x + self.b
plt.scatter(self.train_x, self.train_y)
plt.plot(line_x, line_y,label='y = %.2fx + %.2f' % (self.w, self.b))
plt.xlabel('x')
plt.ylabel('y')
plt.ylim(0, 250)
# 预测值
test_pre_y = self.predict(test_x)
plt.scatter(test_x, test_pre_y,c='r',label='predict')
plt.scatter(test_x, test_y,c='orange',label='test')
plt.legend()
plt.legend()
plt.show()
# 测试
model = OneLineaRegression(train_x, train_y)
# 训练数据
model.calcuate_loss()
# 显示一维显示方程参数
print("一维线性回归：",'斜率：',model.w,' 截距：',model.b)
# 一维线性回归： 斜率： 5 截距： 60
# 预测并显示
model.show_line()
编程实现2:sklearn 机器学习包 from sklearn.linear_model import LinearRegression
import matplotlib.pyplot as plt
# 和方式1 一样的训练数据
train_x = np.array([2, 6, 8, 8, 12, 16, 20, 20, 22, 26]).reshape((-1, 1))
train_y = np.array([58, 105, 88, 118, 117, 137, 157,
169, 149, 202]).reshape((-1, 1))
# sklearn 提供的线性回归函数
model = LinearRegression()
#训练数据
model.fit(train_x, train_y)
# 画图
def show_line(w, b):
line_x = np.arange(0, 30)
line_y = w * line_x + b
pre_value = w * pre_x + b
plt.plot(line_x, line_y)
plt.scatter(train_x, train_y)
plt.xlabel('预测输入量',fontproperties="SimSun")
plt.ylabel('预测值',fontproperties="SimSun")
plt.ylim(0, 250)
plt.xlim(0, 30)
test_pre_y = model.predict(test_x)
plt.scatter(test_x, test_pre_y,c='r',label='predict2')
plt.scatter(test_x, test_y,c='orange',label='test2')
plt.show()
print('线性回归函数参数:', '斜率:', model.coef_[0, 0], ',截距:', model.intercept_[0])
#线性回归函数参数: 斜率: 5.0 ,截距: 60.0
show_line(model.coef_[0, 0], model.intercept_[0] )
一元多项式预测函数
数据并非一元线性方程，数据走向适合二次项，三次项多项式模型选择
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y=\beta_0 +\beta_1 * x+ \beta_2 * x^2+ \beta_3 * x^3+ \beta_4 * x^4+ \beta_5 * x^5 + b
y=β0+β1∗x+β2∗x2+β3∗x3+β4∗x4+β5∗x5+b例如该数据如果用一元线性方程不能切合所有数据，而更适合二项式函数一元多项式# 模拟一元多项式回归
from sklearn.preprocessing import StandardScaler, MaxAbsScaler
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['font.family'] = ['SimHei']
# 生成模拟数据 y = 10.5 - 2.4 *x + 1.4 * x^2 + 30
impurity_x = np.arange(0, 20, 0.5).reshape(-1, 1)
impurity_y = 10.5 - 2.4 * impurity_x + 1.4 * impurity_x ** 2 + 30 + np.random.uniform(-30, 30, impurity_x.shape)
train_x = impurity_x[:, :]
train_y = impurity_y[:, :]
# 一元线性回归
def showLine1():
plt.subplot(121)
plt.scatter(train_x, train_y, label='训练数据')
# 一元线性回归
model = LinearRegression()
model.fit(train_x, train_y)
line_x = np.arange(0, 20, 1).reshape(-1, 1)
line_y = model.predict(line_x)
plt.plot(line_x, line_y, label='预测函数:一元线性回归', c='r')
plt.xlabel('预测输入量', fontproperties="SimSun")
plt.ylabel('预测值', fontproperties="SimSun")
plt.legend()
#一元多项式回归
def showLine2():
plt.subplot(122)
# 一元多项式回归
# 生成二次项特征值
quad_model = PolynomialFeatures(degree=2)
train_x_quad = quad_model.fit_transform(train_x)
model2 = LinearRegression()
model2.fit(train_x_quad, train_y)
test_x = np.arange(0, 20, 1).reshape(-1, 1)
quad_test_x = quad_model.fit_transform(test_x)
test_y = model2.predict(quad_test_x)
plt.scatter(train_x, train_y, label='训练数据')
plt.plot(test_x, test_y, label='预测函数:一元多项式回归', c='orange')
plt.xlabel('预测输入量', fontproperties="SimSun")
plt.ylabel('预测值', fontproperties="SimSun")
plt.legend()
plt.figure(figsize=(10, 5))
showLine1()
showLine2()
plt.show()
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