線性回歸和梯度下降代碼demo

程序所用文件：https://files.cnblogs.com/files/henuliulei/%E5%9B%9E%E5%BD%92%E5%88%86%E7%B1%BB%E6%95%B0%E6%8D%AE.zip

線性回歸

 

 

 

 

 

 

 

 

 

 

 

 

 

決定系數越接近一那么預測效果越好

 對於多元線性回歸和一元線性回歸推導理論是一致的，只不過參數是多個參數而已

 

 

 

 

 

 

 

 

梯度下降

 

 

梯度下降法存在局部最小值

 

 

 

太小迭代次數多，太大將無法迭代到最優質

梯度下降發容易到達局部最小值

 

 

 

 凸函數使用局部下降法一定可以到全部最小值，所以不存在局部最小值才可以

下面兩個demo是一元函數的擬合

 1使用梯度下降法的數學公式進行的機器學習代碼

 

import numpy as np
from matplotlib import  pyplot as plt
#讀取數據
data = np.genfromtxt('data.csv',delimiter=',')
x_data = data[:, 0]
y_data = data[:, 1]
#plt.scatter(x_data, y_data)
#plt.show()
lr = 0.0001
k = 0
b = 0
epochs = 500
def compute_loss(x_data, y_data, b, k):#計算損失函數
    m = float(len(x_data))
    sum = 0
    for i in range(0, len(x_data)):
        sum += (y_data[i] - (k*x_data[i] + b))**2
    return sum/(2*m)
def gradient(x_data, y_data, k, b, lr, epochs):#進行梯度下降
    m = float(len(x_data))

    for i in range(0,epochs):
        k_gradient = 0
        b_gradiet = 0
        for j in range(0,len(x_data)):
            k_gradient += (1/m)*((x_data[j] * k + b) - y_data[j])
            b_gradiet += (1/m)*((x_data[j] * k + b) - y_data[j]) * x_data[j]
        k -= lr * k_gradient
        b -= lr * b_gradiet


        if i % 50 == 0:
            print(i)
            plt.plot(x_data, y_data, 'b.')
            plt.plot(x_data, k*x_data + b, 'r')
            plt.show()

    return k, b

k,b = gradient(x_data, y_data, 0, 0, lr, epochs)
plt.plot(x_data, k * x_data + b, 'r')
plt.plot(x_data, y_data, 'b.')
print('loss =:',compute_loss(x_data, y_data, b, k),'b =:',b,'k =:',k)
plt.show()

2 使用Python的sklearn庫

import numpy as np
from matplotlib import  pyplot as plt
from sklearn.linear_model import LinearRegression
#讀取數據
data = np.genfromtxt('data.csv',delimiter=',')
x_data = data[:, 0]
y_data = data[:, 1]
plt.scatter(x_data, y_data)
plt.show()
x_data = data[:, 0, np.newaxis]#使一位數據編程二維數據
y_data = data[:, 1, np.newaxis]
model =LinearRegression()
model.fit(x_data, y_data)#傳進的參數必須是二維的
plt.plot(x_data, y_data, 'b.')
plt.plot(x_data, model.predict(x_data), 'r')#畫出預測的線條
plt.show()

 

 3使用梯度下降法完成多元線性回歸（以二元為例）

import numpy as np
from numpy import genfromtxt
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D #用來畫3D圖的包
# 讀入數據
data = genfromtxt(r"Delivery.csv",delimiter=',')
print(data)
# 切分數據
x_data = data[:,:-1]
y_data = data[:,-1]
print(x_data)
print(y_data)
# 學習率learning rate
lr = 0.0001
# 參數
theta0 = 0
theta1 = 0
theta2 = 0
# 最大迭代次數
epochs = 1000

# 最小二乘法
def compute_error(theta0, theta1, theta2, x_data, y_data):
    totalError = 0
    for i in range(0, len(x_data)):
        totalError += (y_data[i] - (theta1 * x_data[i,0] + theta2*x_data[i,1] + theta0)) ** 2
    return totalError / float(len(x_data))

def gradient_descent_runner(x_data, y_data, theta0, theta1, theta2, lr, epochs):
    # 計算總數據量
    m = float(len(x_data))
    # 循環epochs次
    for i in range(epochs):
        theta0_grad = 0
        theta1_grad = 0
        theta2_grad = 0
        # 計算梯度的總和再求平均
        for j in range(0, len(x_data)):
            theta0_grad += (1/m) * ((theta1 * x_data[j,0] + theta2*x_data[j,1] + theta0) - y_data[j])
            theta1_grad += (1/m) * x_data[j,0] * ((theta1 * x_data[j,0] + theta2*x_data[j,1] + theta0) - y_data[j])
            theta2_grad += (1/m) * x_data[j,1] * ((theta1 * x_data[j,0] + theta2*x_data[j,1] + theta0) - y_data[j])
        # 更新b和k
        theta0 = theta0 - (lr*theta0_grad)
        theta1 = theta1 - (lr*theta1_grad)
        theta2 = theta2 - (lr*theta2_grad)
    return theta0, theta1, theta2
print("Starting theta0 = {0}, theta1 = {1}, theta2 = {2}, error = {3}".
      format(theta0, theta1, theta2, compute_error(theta0, theta1, theta2, x_data, y_data)))
print("Running...")
theta0, theta1, theta2 = gradient_descent_runner(x_data, y_data, theta0, theta1, theta2, lr, epochs)
print("After {0} iterations theta0 = {1}, theta1 = {2}, theta2 = {3}, error = {4}".
      format(epochs, theta0, theta1, theta2, compute_error(theta0, theta1, theta2, x_data, y_data)))
ax = Axes3D(plt.figure())#和下面的代碼功能一樣
#ax = plt.figure().add_subplot(111, projection='3d')#plt.figure().add_subplot和plt.subplot的作用是一致的
ax.scatter(x_data[:, 0], x_data[:, 1], y_data, c='r', marker='o', s=100)  # 點為紅色三角形
x0 = x_data[:, 0]
x1 = x_data[:, 1]
# 生成網格矩陣
x0, x1 = np.meshgrid(x0, x1)#生成一個網格矩陣，矩陣的每個點的第一個軸的取值來自於x0范圍內，第二個坐標軸的取值來自於x1范圍內
z = theta0 + x0 * theta1 + x1 * theta2
# 畫3D圖
ax.plot_surface(x0, x1, z)
# 設置坐標軸
ax.set_xlabel('Miles')
ax.set_ylabel('Num of Deliveries')
ax.set_zlabel('Time')

# 顯示圖像
plt.show()

 

 4：使用Python的sklearn庫完成多元線性回歸

import numpy as np
from numpy import genfromtxt
from sklearn import linear_model
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# 讀入數據
data = genfromtxt(r"Delivery.csv",delimiter=',')
print(data)
# 切分數據
x_data = data[:,:-1]
y_data = data[:,-1]
print(x_data)
print(y_data)
# 創建模型
model = linear_model.LinearRegression()
model.fit(x_data, y_data)
# 系數
print("coefficients:",model.coef_)

# 截距
print("intercept:",model.intercept_)

# 測試
x_test = [[102,4]]
predict = model.predict(x_test)
print("predict:",predict)
ax = plt.figure().add_subplot(111, projection='3d')
ax.scatter(x_data[:, 0], x_data[:, 1], y_data, c='r', marker='o', s=100)  # 點為紅色三角形
x0 = x_data[:, 0]
x1 = x_data[:, 1]
# 生成網格矩陣
x0, x1 = np.meshgrid(x0, x1)
z = model.intercept_ + x0*model.coef_[0] + x1*model.coef_[1]
# 畫3D圖
ax.plot_surface(x0, x1, z)#參數是二維的，而model.prodict(x_data)是一維的。
# 設置坐標軸
ax.set_xlabel('Miles')
ax.set_ylabel('Num of Deliveries')
ax.set_zlabel('Time')

# 顯示圖像
plt.show()

5 多項式回歸擬合

 

 

 

import numpy as np
import matplotlib.pyplot as plt
from sklearn.preprocessing import PolynomialFeatures#多項式
from sklearn.linear_model import LinearRegression

# 載入數據
data = np.genfromtxt("job.csv", delimiter=",")
x_data = data[1:,1]
y_data = data[1:,2]
plt.scatter(x_data,y_data)
plt.show()
x_data
x_data = x_data[:,np.newaxis]
y_data = y_data[:,np.newaxis]
x_data
# 創建並擬合模型
model = LinearRegression()
model.fit(x_data, y_data)
# 畫圖
plt.plot(x_data, y_data, 'b.')
plt.plot(x_data, model.predict(x_data), 'r')
plt.show()
# 定義多項式回歸,degree的值可以調節多項式的特征
poly_reg  = PolynomialFeatures(degree=5)
# 特征處理
x_poly = poly_reg.fit_transform(x_data)
# 定義回歸模型
lin_reg = LinearRegression()
# 訓練模型
lin_reg.fit(x_poly, y_data)
# 畫圖
plt.plot(x_data, y_data, 'b.')
plt.plot(x_data, lin_reg.predict(poly_reg.fit_transform(x_data)), c='r')
plt.title('Truth or Bluff (Polynomial Regression)')
plt.xlabel('Position level')
plt.ylabel('Salary')
plt.show()
# 畫圖
plt.plot(x_data, y_data, 'b.')
x_test = np.linspace(1,10,100)
x_test = x_test[:,np.newaxis]
plt.plot(x_test, lin_reg.predict(poly_reg.fit_transform(x_test)), c='r')
plt.title('Truth or Bluff (Polynomial Regression)')
plt.xlabel('Position level')
plt.ylabel('Salary')
plt.show()