python 做回归

1 一元线性回归

线性回归是一种简单的模型，但受到广泛应用，比如预测商品价格，成本评估等，都可以用一元线性模型。y = f(x) 叫做一元函数，回归意思就是根据已知数据复原某些值，线性回归（regression）就是用线性的模型做回归复原。

基本思想：已知一批(x, y)来复原另外未知的值，例如（1,1），（2,2），（3,3），那么（4，？），大家很容易知道（4,4），这就是一元线性回归求解问题

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.font_manager import FontProperties
from sklearn.linear_model import LinearRegression

x = [[1], [2],[3], [4], [5], [6]]
y = [[1], [2.1], [2.9], [4.2], [5.1], [5.8]]
model = LinearRegression()
model.fit(x, y)

predicted = model.predict([13])[0]
x2 = [[0], [2.5],[5.3],[7.6],[9.1]]
y2 = model.predict(x2)
print model.score(x2, y2)
print predicted

fonr = FontProperties()
plt.figure()
plt.title("linearRegression")
plt.xlabel('x')
plt.ylabel('y')
plt.axis([0, 10, 0, 10])
plt.grid(True)
plt.pause(0.4)
plt.plot(x , y, 'k.')
plt.plot(x2 , y2, 'g.')
plt.show()

print np.var(x2, ddof = 0)
print model.score(x2, y2)

2 多元线性回归

假设方程为：,求解系数，因为不一定是方阵，所以不能直接求，所以我们对方程两边同乘以,则，由于是方阵，所以两边同时乘以的逆，就可以得到：。

设计二元一次方程：y=1+2x₁+3x₂

取样本为(1,1,1),(1,1,2),(1,2,1)，计算得y=(6,9,8)

注意：这里面常数项1相当于1*x₀，只不过这里的x₀永远取1

X = [[1,1,1],[1,1,2],[1,2,1]]

y = [[6],[9],[8]]

from numpy.linalg import inv
from numpy import dot, transpose
from numpy.linalg import lstsq

x = [[1,1,1], [1,1,2], [1,2,1]]
y = [[6], [9], [8]]
print dot(inv(dot(transpose(x),x)),dot(transpose(x), y))
##===最小二乘法=====================
print lstsq(x, y)[0]

from sklearn.linear_model import LinearRegression

x = [[1,1,1], [1,1,2], [1,2,1]]
y = [[6], [9], [8]]

model = LinearRegression()
model.fit(x, y)
x2 = [[1,3,5]]
y2 = model.predict(x2)
print y2

3 .采用多项式回归

import sys
reload(sys)
sys.setdefaultencoding('utf-8')

import matplotlib.pyplot as plt
import numpy as np
from sklearn.linear_model import LinearRegression#导入线性回归模型
from sklearn.preprocessing import PolynomialFeatures# 导入多项式回归模型

plt.figure()
plt.title('single variable')
plt.xlabel('x')
plt.ylabel('y')
plt.axis([30, 400, 100, 400])
plt.grid(True)

x = [[50], [100], [150], [200], [250], [300]]
y = [[150], [200], [250], [280], [310], [330]]
x_test = [[250], [300]]
y_test = [[310], [330]]
plt.plot(x, y, 'g.',markersize =20)

model = LinearRegression()
x2 = [[30], [400]]
model.fit(x, y)
y2 = model.predict(x2)
plt.plot(x2, y2,label = '$y = ax + c$')
plt.legend()

xx = np.linspace(30, 400, 100)
quadratic_featurizer = PolynomialFeatures(degree = 2)#实例化一个二次多项式
x_train_quadratic = quadratic_featurizer.fit_transform(x)#用二次多项式多样本x做变换
xx_quadratic = quadratic_featurizer.transform(xx.reshape(xx.shape[0], 1))# 把训练好X值的多项式特征实例应用到一系列点上,形成矩阵

regressor_quadratic = LinearRegression()
regressor_quadratic.fit(x_train_quadratic, y)
plt.plot(xx, regressor_quadratic.predict(xx_quadratic),label="$y = ax^2 + bx + c$")
plt.legend()

cubic_featurizer = PolynomialFeatures(degree = 3)
x_train_cubic = cubic_featurizer.fit_transform(x)
xx_cubie = cubic_featurizer.transform(xx.reshape(xx.shape[0], 1))

regressor_cubic = LinearRegression()
regressor_cubic.fit(x_train_cubic, y)
plt.plot(xx, regressor_cubic.predict(xx_cubie),label="$y = a_1x^3 + a_2x^2 + a_3x +c $")
plt.legend()

print '一元线性回归 r-squared', model.score(x_test, y_test)
x_test_quadratic = quadratic_featurizer.transform(x_test)
print '二次性回归 r-squared', regressor_quadratic.score(x_test_quadratic, y_test)
x_test_cubic = cubic_featurizer.transform(x_test)
print '三次线性回归 r-squared', regressor_cubic.score(x_test_cubic, y_test)
plt.show()