更新成本函数和Theta参数时的简单线性回归误差

Question

I wrote a piece code to make a simple linear regression model using Python. However, I am having trouble getting the correct cost function, and most importantly the correct theta parameters. The model is implemented from scratch and not using Scikit learn module. I have used Andrew NG's notes from his ML Coursera course to create the model. The correct values of theta are [[-3.630291] [1.166362]].

Would be really grateful if someone could offer their expertise, and point out what I'm doing wrong.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

#Load The Dataset
dataset = pd.read_csv("Population vs Profit.txt",names=["Population" , 
"Profit"])
print (dataset.head())

col = len(dataset.columns)
x = dataset.iloc[:,:col-1].values
y = dataset.iloc[:,col-1].values

#Visualizing The Dataset
plt.scatter(x, y, color="red", marker="x", label="Profit")
plt.title("Population vs Profit")
plt.xlabel("Population")
plt.ylabel("Profit")
plt.legend()
plt.show()

#Preprocessing Data
dataset.insert(0,"x0",1)
col = len(dataset.columns)
x = dataset.iloc[:,:col-1].values
b = np.zeros(col-1)
m = len(y)
costlist = []
alpha = 0.001
iteration = 10000

#Defining Functions
def hypothesis(x,b,y):
    h = x.dot(b.T) - y
    return h

def cost(x,b,y,m):
    j = np.sum(hypothesis(x,b,y)**2)
    j = j/(2*m)
    return j

print (cost(x,b,y,m))

def gradient_descent(x,b,y,m,alpha):
    for i in range (iteration):
        h = hypothesis(x,b,y)
        product = np.sum(h.dot(x))
        b = b - ((alpha/m)*product)
        costlist.append(cost(x,b,y,m))

    return b,cost(x,b,y,m)

b , mincost = gradient_descent(x,b,y,m,alpha)
print (b , mincost)
print (cost(x,b,y,m))
plt.plot(b,color="green")
plt.show()

The dataset I'm using is the following text.

6.1101,17.592
5.5277,9.1302
8.5186,13.662
7.0032,11.854
5.8598,6.8233
8.3829,11.886
7.4764,4.3483
8.5781,12
6.4862,6.5987
5.0546,3.8166
5.7107,3.2522
14.164,15.505
5.734,3.1551
8.4084,7.2258
5.6407,0.71618
5.3794,3.5129
6.3654,5.3048
5.1301,0.56077
6.4296,3.6518
7.0708,5.3893
6.1891,3.1386
20.27,21.767
5.4901,4.263
6.3261,5.1875
5.5649,3.0825
18.945,22.638
12.828,13.501
10.957,7.0467
13.176,14.692
22.203,24.147
5.2524,-1.22
6.5894,5.9966
9.2482,12.134
5.8918,1.8495
8.2111,6.5426
7.9334,4.5623
8.0959,4.1164
5.6063,3.3928
12.836,10.117
6.3534,5.4974
5.4069,0.55657
6.8825,3.9115
11.708,5.3854
5.7737,2.4406
7.8247,6.7318
7.0931,1.0463
5.0702,5.1337
5.8014,1.844
11.7,8.0043
5.5416,1.0179
7.5402,6.7504
5.3077,1.8396
7.4239,4.2885
7.6031,4.9981
6.3328,1.4233
6.3589,-1.4211
6.2742,2.4756
5.6397,4.6042
9.3102,3.9624
9.4536,5.4141
8.8254,5.1694
5.1793,-0.74279
21.279,17.929
14.908,12.054
18.959,17.054
7.2182,4.8852
8.2951,5.7442
10.236,7.7754
5.4994,1.0173
20.341,20.992
10.136,6.6799
7.3345,4.0259
6.0062,1.2784
7.2259,3.3411
5.0269,-2.6807
6.5479,0.29678
7.5386,3.8845
5.0365,5.7014
10.274,6.7526
5.1077,2.0576
5.7292,0.47953
5.1884,0.20421
6.3557,0.67861
9.7687,7.5435
6.5159,5.3436
8.5172,4.2415
9.1802,6.7981
6.002,0.92695
5.5204,0.152
5.0594,2.8214
5.7077,1.8451
7.6366,4.2959
5.8707,7.2029
5.3054,1.9869
8.2934,0.14454
13.394,9.0551
5.4369,0.61705

Answer 1

One issue is with your "product". It is currently a number when it should be a vector. I was able to get the values [-3.24044334 1.12719788] by rerwitting your for-loop as follows:

def gradient_descent(x,b,y,m,alpha):
    for i in range (iteration):
        h = hypothesis(x,b,y)
        #product = np.sum(h.dot(x))
        xvalue = x[:,1]
        product = h.dot(xvalue)
        hsum = np.sum(h)
        b = b - ((alpha/m)* np.array([hsum , product]) )
        costlist.append(cost(x,b,y,m))

    return b,cost(x,b,y,m)

There's possibly another issue besides this as it doesn't converge to your answer. You should make sure you are using the same alpha also.