[数据结构与算法]python使用线性回归实现房价预测

一、单变量房价预测

采用一元线性回归实现单变量房价预测。通过房屋面积与房价建立线性关系，通过梯度下降进行训练，拟合权重和偏置参数，使用训练到的参数进行房价预测。

1、房屋面积与房价数据

2、面积与房价分布散点图

3、实现步骤。

训练部分：

第一步，数据预处理，采用归一化。

第二步，建立线性关系，y=w*x+b，y为房价，x为面积，w权重，b偏置。

第三步，通过偏导数计算梯度。w_gradient=SUM(2*x*((w*x+b)-y) / N)，b_gradient=SUM(2*((w*x+b)-y) / N)，N为训练数据个数。初始化权重和偏置，一般初始化为0，通过偏导数计算权重和偏置的梯度，通过梯度和学习率更新权重和偏置。

第四步，计算误差，采用均方差计算全局误差。

实现代码：

import numpy as np

#计算误差

def compute_error(w, b, points):

????total_error = 0

????for i in range(0, len(points)):

????????x = points[i, 0]

????????y = points[i, 1]

????????total_error += (y - (w * x + b)) ** 2

????return total_error / float(len(points))

#计算梯度

def step_gradient(w_current, b_current, points, learn_Rate):

????b_gradient = 0

????w_gradient = 0

????N = float(len(points))

????for i in range(0, len(points)):

????????x = points[i, 0]

????????y = points[i, 1]

????????b_gradient += (2/N) * ((w_current * x + b_current) - y)

????????w_gradient += (2/N) * ((w_current * x + b_current) - y) * x

????new_b = b_current - (learn_Rate * b_gradient)

????new_w = w_current - (learn_Rate * w_gradient)

????return [new_b, new_w]

#梯度下降，循环计算权重和偏置

def gradient_descent_runner(points, starting_w, starting_b, learn_rate, iteration_num):

????b = starting_b

????w = starting_w

????for i in range(iteration_num):

????????b, w = step_gradient(w, b, np.array(points), learn_rate)

????return [b, w]

#导入数据，开始计算

def run():

????points = np.genfromtxt("data0.csv", delimiter=",")

????learn_rate = 0.0001

????initial_b = 0

????initial_w = 0

????iteration_num = 2000

????print("start gradient b = {0}, w = {1}, error = {2}"

??????????.format(initial_b, initial_w, compute_error(initial_w, initial_b, points)))

????[b, w] = gradient_descent_runner(points, initial_w, initial_b, learn_rate, iteration_num)

????print("after iteration b = {0}, w = {1}, error = {2}"

??????????.format(b, w, compute_error(w, b, points)))

if __name__ == '__main__':

????run()

测试部分：

#偏置、权重和输入值

def predict(b, w, x):

????return w*x + b

if __name__ == '__main__':

????print(predict(0.5523011954231988, 1.2331875596916462, 90))

二、多变量房价预测

在面积基础上增加一维房间数，将单变量预测改为多变量预测。

采用多元线性回归实现。根据偏导数的计算公式，w_gradient=SUM(2*x*((w*x+b)-y) / N)，假设x恒为1，该公式即变换为偏置参数的计算公式。只需要对训练数据加一列值1，即x=1，该列数据对应偏置参数，即可将偏置的计算归入到权重计算方式中，实现一元线性回归和多元线性回归的统一。

实现代码：

import numpy as np

#计算误差

def compute_error(w, points):

????total_error = 0

????for i in range(0, len(points)):

????????x = np.zeros(len(w))

????????for j in range(0, len(w)):

????????????x[j] = points[i, j]

????????y = points[i, len(w)]

????????y_predict = 0.

????????for j in range(0, len(w)):

????????????y_predict += w[j] * x[j]

????total_error += (y - y_predict) ** 2

????return total_error / float(len(points))

#计算梯度

def step_gradient(w_current, points, learn_rate):

????w_gradient = np.zeros(len(w_current))

????new_w = np.zeros(len(w_current))

????N = float(len(points))

????for i in range(0, len(points)):

????????x = np.zeros(len(w_current))

????????for j in range(0, len(w_current)):

????????????x[j] = points[i, j]

????????y = points[i, len(w_current)]

????????y_predict = 0.

????????for j in range(0, len(w_current)):

????????????y_predict += w_current[j] * x[j]

????????for j in range(0, len(w_current)):

????????????w_gradient[j] = 2 * x[j] * (y_predict - y) / N

????????for j in range(0, len(w_current)):

????????????new_w[j] = w_current[j] - (learn_rate * w_gradient[j])

????return new_w

#梯度下降，循环计算权重和偏置

def gradient_descent_runner(points, starting_w, learn_rate, iteration_num):

????w = starting_w

????for i in range(iteration_num):

????????w = step_gradient(w, np.array(points), learn_rate)

????return w

#导入数据，开始计算

def run():

????points = np.genfromtxt("data0.csv", delimiter=",")

????learn_rate = 0.0001

????initial_w = np.zeros(3)

????iteration_num = 20000

????initial_error = compute_error(initial_w, points)

????print(initial_error)

????w = gradient_descent_runner(points, initial_w, learn_rate, iteration_num)

????error = compute_error(w, points)

????print(error)

if __name__ == '__main__':

????run()