一、單變量房價預測

采用一元線性回歸實現單變量房價預測。通過房屋面積與房價建立線性關系，通過梯度下降進行訓練，擬合權重和偏置參數，使用訓練到的參數進行房價預測。

1、房屋面積與房價數據

32.50234527	31.70700585
53.42680403	68.77759598
61.53035803	62.5623823
47.47563963	71.54663223
59.81320787	87.23092513
55.14218841	78.21151827
52.21179669	79.64197305
39.29956669	59.17148932
48.10504169	75.3312423
52.55001444	71.30087989
45.41973014	55.16567715
54.35163488	82.47884676
44.1640495	62.00892325
58.16847072	75.39287043
56.72720806	81.43619216
48.95588857	60.72360244
44.68719623	82.89250373
60.29732685	97.37989686
45.61864377	48.84715332
38.81681754	56.87721319

2、面積與房價分布散點圖

3、實現步驟。

訓練部分：

第一步，數據預處理，采用歸一化。

第二步，建立線性關系，y=w*x+b，y為房價，x為面積，w權重，b偏置。

第三步，通過偏導數計算梯度。w_gradient=SUM(2*x*((w*x+b)-y) / N)，b_gradient=SUM(2*((w*x+b)-y) / N)，N為訓練數據個數。初始化權重和偏置，一般初始化為0，通過偏導數計算權重和偏置的梯度，通過梯度和學習率更新權重和偏置。

第四步，計算誤差，采用均方差計算全局誤差。

實現代碼：

import numpy as np

#計算誤差

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))

#計算梯度

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，循環計算權重和偏置

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]

#導入數據，開始計算

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()

測試部分：

#偏置、權重和輸入值

def predict(b, w, x):

????return w*x + b

if __name__ == '__main__':

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

二、多變量房價預測

在面積基礎上增加一維房間數，將單變量預測改為多變量預測。

2104	3	399900
1600	3	329900
2400	3	369000
1416	2	232000
3000	4	539900
1985	4	299900
1534	3	314900
1427	3	198999
1380	3	212000
1494	3	242500
1940	4	239999
2000	3	347000
1890	3	329999
4478	5	699900
1268	3	259900
2300	4	449900
1320	2	299900
1236	3	199900
2609	4	499998
3031	4	599000

采用多元線性回歸實現。根據偏導數的計算公式，w_gradient=SUM(2*x*((w*x+b)-y) / N)，假設x恒為1，該公式即變換為偏置參數的計算公式。只需要對訓練數據加一列值1，即x=1，該列數據對應偏置參數，即可將偏置的計算歸入到權重計算方式中，實現一元線性回歸和多元線性回歸的統一。

實現代碼：

import numpy as np#計算誤差 def compute_error(w, points):total_error = 0for 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) ** 2return total_error / float(len(points))#計算梯度 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))#對整個數據集進行一次迭代，計算梯度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.#根據當前參數預測值for j in range(0, len(w_current)):y_predict += w_current[j] * x[j]#根據梯度下降計算梯度for j in range(0, len(w_current)):w_gradient[j] += 2 * x[j] * (y_predict - y) / N#根據梯度更新權重for i in range(0, len(w_current)):new_w[i] = w_current[i] - (learn_rate * w_gradient[i])return new_w#梯度下降，循環計算權重和偏置 def gradient_descent_runner(points, starting_w, learn_rate, iteration_num):w = starting_wfor i in range(iteration_num):w = step_gradient(w, np.array(points), learn_rate)return w#導入數據，開始計算 def run():points = np.genfromtxt("data0.csv", delimiter=",")learn_rate = 0.0001initial_w = np.zeros(2)iteration_num = 2000initial_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()

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