牛顿法、梯度下降法、最小二乘法的原理以及利用它们解决实际问题的python编程
一、牛顿法原理
1、产生背景
2、牛顿迭代公式
二、梯度下降法原理
根据计算梯度时所用数据量不同，可以分为三种基本方法：批量梯度下降法（Batch Gradient Descent, BGD）、小批量梯度下降法（Mini-batch Gradient Descent, MBGD）以及随机梯度下降法（Stochastic Gradient Descent, SGD）。
梯度下降法的一般求解框架
三、最小二乘法原理
最小二乘法（又称最小平方法）是一种数学优化技术。它通过最小化误差的平方和寻找数据的最佳函数匹配。利用最小二乘法可以简便地求得未知的数据，并使得这些求得的数据与实际数据之间误差的平方和为最小。最小二乘法还可用于曲线拟合。其他一些优化问题也可通过最小化能量或最大化熵用最小二乘法来表达。
四、运用梯度下降法原理解决实际问题的python编程举例
1、问题如下
2、导入所需要的包
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import math
from mpl_toolkits.mplot3d import Axes3D
import warnings
3、画出函数图像
def f2(x1,x2):
return x1 ** 2 + 2 * x2 ** 2 -4*x1- 2 * x1*x2
X1 = np.arange(-4,4,0.2)
X2 = np.arange(-4,4,0.2)
X1, X2 = np.meshgrid(X1, X2)
Y = np.array(list(map(lambda t : f2(t[0],t[1]),zip(X1.flatten(),X2.flatten()))))
Y.shape = X1.shape
%matplotlib inline
fig = plt.figure(facecolor='w')
ax = Axes3D(fig)
ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
ax.set_title(u'$ x1 ** 2 + 2 * x2 ** 2 -4*x1- 2 * x1*x2 $')
plt.show()
4、求极小点和极值点
mpl.rcParams['font.sans-serif'] = [u'SimHei']
mpl.rcParams['axes.unicode_minus'] = False
%matplotlib inline
def f2(x, y):
return x ** 2 + 2 * y ** 2 -4*x- 2 * x*y
def hx1(x, y):
return 2*x-4-2*y
def hx2(x, y):
return 4*y-2*x
x1 = 4
x2 = 4
alpha = 0.001
GD_X1 = [x1]
GD_X2 = [x2]
GD_Y = [f2(x1,x2)]
y_change = f2(x1,x2)
iter_num = 0
while(iter_num < 10000) :
tmp_x1 = x1 - alpha * hx1(x1,x2)
tmp_x2 = x2 - alpha * hx2(x1,x2)
tmp_y = f2(tmp_x1,tmp_x2)
f_change = np.absolute(tmp_y - f2(x1,x2))
x1 = tmp_x1
x2 = tmp_x2
GD_X1.append(x1)
GD_X2.append(x2)
GD_Y.append(tmp_y)
iter_num += 1
print(u"最终结果为:(%.5f, %.5f, %.5f)" % (x1, x2, f2(x1,x2)))
print(u"迭代过程中X的取值，迭代次数:%d" % iter_num)
print(GD_X1)
fig = plt.figure(facecolor='w',figsize=(20,18))
ax = Axes3D(fig)
ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
ax.plot(GD_X1,GD_X2,GD_Y,'ko-')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_title(u'函数;\n学习率:%.3f; 最终解:(%.3f, %.3f, %.3f);迭代次数:%d' % (alpha, x1, x2, f2(x1,x2), iter_num))
plt.show()
最终结果为:(4.00043, 2.00027, -8.00000)
迭代过程中X的取值，迭代次数:10000
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4.000547986569924, 4.000547567945435, 4.000547149640747, 4.000546731655615, 4.000546313989796, 4.000545896643045, 4.000545479615118, 4.000545062905773, 4.000544646514765, 4.000544230441851, 4.000543814686789, 4.000543399249335, 4.000542984129248, 4.000542569326283, 4.0005421548402005, 4.000541740670757, 4.000541326817711, 4.00054091328082, 4.000540500059843, 4.00054008715454, 4.000539674564667, 4.000539262289985, 4.000538850330253, 4.00053843868523, 4.0005380273546765, 4.0005376163383515, 4.0005372056360144, 4.0005367952474264, 4.000536385172348, 4.000535975410538, 4.0005355659617585, 4.00053515682577, 4.000534748002334, 4.000534339491211, 4.000533931292163, 4.000533523404951, 4.000533115829337, 4.000532708565084, 4.000532301611952, 4.000531894969705, 4.000531488638105, 4.000531082616915, 4.000530676905897, 4.000530271504815, 4.000529866413432, 4.000529461631511, 4.000529057158816, 4.0005286529951105, 4.000528249140158, 4.000527845593725, 4.000527442355573, 4.000527039425467, 4.000526636803173, 4.000526234488454, 4.0005258324810775, 4.000525430780807, 4.000525029387408, 4.000524628300646, 4.000524227520287, 4.000523827046097, 4.000523426877842, 4.000523027015288, 4.000522627458203, 4.000522228206352, 4.000521829259502, 4.00052143061742, 4.000521032279874, 4.00052063424663, 4.000520236517457, 4.000519839092122, 4.000519441970392, 4.000519045152037, 4.000518648636825, 4.000518252424523, 4.0005178565149, 4.000517460907726, 4.000517065602768, 4.000516670599796, 4.00051627589858, 4.000515881498888, 4.000515487400492, 4.000515093603159, 4.000514700106661, 4.000514306910768, 4.000513914015249, 4.000513521419876, 4.000513129124419, 4.000512737128649, 4.0005123454323375, 4.000511954035255, 4.000511562937173, 4.0005111721378634, 4.000510781637098, 4.000510391434649, 4.000510001530288, 4.000509611923787, 4.0005092226149195, 4.000508833603457, 4.0005084448891735, 4.000508056471841, 4.000507668351234, 4.000507280527123, 4.000506892999284, 4.00050650576749, 4.000506118831514, 4.000505732191131, 4.000505345846116, 4.000504959796242, 4.000504574041283, 4.000504188581015, 4.000503803415213, 4.0005034185436505, 4.000503033966105, 4.00050264968235, 4.000502265692162, 4.000501881995316, 4.000501498591588, 4.000501115480755, 4.000500732662592, 4.000500350136876, 4.000499967903384, 4.000499585961892, 4.000499204312177, 4.000498822954017, 4.000498441887189, 4.00049806111147, 4.000497680626638, 4.00049730043247, 4.000496920528745, 4.000496540915241, 4.000496161591735, 4.000495782558007, 4.000495403813835, 4.0004950253589975, 4.000494647193274, 4.000494269316443, 4.000493891728284, 4.000493514428577, 4.000493137417101, 4.000492760693637, 4.0004923842579645, 4.0004920081098625, 4.000491632249112, 4.0004912566754935, 4.000490881388788, 4.0004905063887755, 4.000490131675238, 4.000489757247956, 4.000489383106711, 4.000489009251284, 4.0004886356814575, 4.000488262397013, 4.000487889397733, 4.000487516683399, 4.000487144253793, 4.000486772108697, 4.000486400247896, 4.000486028671171, 4.000485657378305, 4.000485286369082, 4.000484915643284, 4.000484545200696, 4.000484175041101, 4.000483805164283, 4.000483435570025, 4.000483066258113, 4.000482697228329, 4.00048232848046, 4.000481960014288, 4.0004815918296, 4.0004812239261796, 4.000480856303812, 4.000480488962284, 4.000480121901379, 4.000479755120884, 4.000479388620585, 4.000479022400267, 4.000478656459716, 4.000478290798719, 4.000477925417061, 4.000477560314531, 4.000477195490914, 4.000476830945997, 4.000476466679569, 4.000476102691414, 4.000475738981322, 4.00047537554908, 4.000475012394475, 4.000474649517296, 4.0004742869173295, 4.000473924594366, 4.000473562548192, 4.000473200778597, 4.000472839285369, 4.000472478068297, 4.000472117127171, 4.0004717564617795, 4.000471396071911, 4.000471035957357, 4.000470676117906, 4.000470316553347, 4.000469957263471, 4.000469598248068, 4.000469239506929, 4.000468881039843, 4.000468522846602, 4.000468164926996, 4.000467807280817, 4.0004674499078545, 4.000467092807901, 4.000466735980748, 4.000466379426186, 4.000466023144008, 4.000465667134005, 4.00046531139597, 4.000464955929694, 4.00046460073497, 4.0004642458115915, 4.00046389115935, 4.000463536778038, 4.00046318266745, 4.000462828827378, 4.000462475257616, 4.000462121957957, 4.000461768928195, 4.0004614161681245, 4.0004610636775375, 4.000460711456229, 4.000460359503995, 4.000460007820628, 4.000459656405924, 4.0004593052596755, 4.0004589543816795, 4.0004586037717305, 4.000458253429624, 4.000457903355155, 4.000457553548119, 4.0004572040083115, 4.000456854735528, 4.0004565057295665, 4.0004561569902215, 4.000455808517289, 4.000455460310567, 4.000455112369851, 4.000454764694938, 4.000454417285625, 4.000454070141709, 4.0004537232629875, 4.000453376649258, 4.000453030300317, 4.000452684215964, 4.000452338395996, 4.00045199284021, 4.000451647548405, 4.00045130252038, 4.000450957755933, 4.000450613254863, 4.000450269016968, 4.000449925042047, 4.0004495813298995, 4.000449237880325, 4.000448894693123, 4.000448551768092, 4.000448209105032, 4.000447866703745, 4.000447524564028, 4.000447182685683, 4.000446841068509, 4.000446499712308, 4.00044615861688, 4.000445817782025, 4.0004454772075455, 4.000445136893242, 4.000444796838915, 4.000444457044366, 4.000444117509397, 4.00044377823381, 4.000443439217406, 4.000443100459988, 4.000442761961357, 4.000442423721316, 4.000442085739668, 4.000441748016215, 4.000441410550759, 4.000441073343104, 4.000440736393053, 4.000440399700409, 4.000440063264975, 4.000439727086555, 4.000439391164952, 4.000439055499971, 4.000438720091415, 4.0004383849390885, 4.000438050042796, 4.000437715402341, 4.000437381017528, 4.000437046888163, 4.00043671301405, 4.000436379394994, 4.0004360460308, 4.0004357129212735, 4.000435380066221, 4.000435047465446, 4.0004347151187565, 4.000434383025957, 4.000434051186853, 4.0004337196012525, 4.00043338826896, 4.000433057189784, 4.000432726363528, 4.0004323957900025, 4.000432065469012, 4.0004317354003645, 4.000431405583867, 4.000431076019327, 4.000430746706551, 4.000430417645349, 4.000430088835527, 4.0004297602768935, 4.0004294319692555, 4.000429103912423]
从上面求出的结果可以得到极小点为（4,2），极小值为-8
5、用Excel求上个函数的极小点和极小值
实验结果如下图：
依旧可以清晰地看出该函数的极小点为（4,2），极值点为-8
五、使用梯度下降法和最小二乘法求解多元函数python编程举例
1、问题如下
用梯度下降法和最小二乘法根据以下图片的数据进行多元线性回归并求解相关系数
将数据写入Excel中
2、使用梯度下降法求解多元函数
代码如下：
import matplotlib.pyplot as plt
import numpy as np
from numpy import array
import pandas as pd
df=pd.read_csv("C:/Users/LOL/Desktop/店铺多元回归.csv")
%matplotlib notebook
df=df.values
x_data=df[:,:2]
y_data=df[:,2]
lr=0.00001
theta0=0
theta1=0
theta2=0
epochs=10000
def compute_error(theta0,theta1,theta2,x_data,y_data):
totalerror=0
for i in range(0,len(x_data)):
totalerror=totalerror+(y_data[i]-(theta1*x_data[i,0]+theta2*x_data[i,1]+theta0))**2
return totalerror/float(len(x_data))/2
def gradient_descent_runner(x_data,y_data,theta0,theta1,theta2,lr,epochs):
m=len(x_data)
for i in range(epochs):
theta0_grad=0
theta1_grad=0
theta2_grad=0
for j in range(0,m):
theta0_grad-=(1/m)*(-(theta1*x_data[j,0]+theta2*x_data[j,1]+theta2)+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])
theta0=theta0-lr*theta0_grad
theta1=theta1-lr*theta1_grad
theta2=theta2-lr*theta2_grad
return theta0,theta1,theta2
theta0,theta1,theta2=gradient_descent_runner(x_data,y_data,theta0,theta1,theta2,lr,epochs)
print('迭代次数：{0} 学习率：{1}之后 a0={2},a1={3},a2={4},代价函数为{5}'.format(epochs,lr,theta0,theta1,theta2,compute_error(theta0,theta1,theta2,x_data,y_data)))
print("多元线性回归方程为:y=",theta1,"X1+",theta2,"X2+",theta0)
迭代次数：10000 学习率：1e-05之后 a0=5.3774162274868,a1=45.0533119768975,a2=-0.19626929358281256,代价函数为366.7314528822914
多元线性回归方程为:y= 45.0533119768975 X1+ -0.19626929358281256 X2+ 5.3774162274868
3、用最小二乘法求解
代码如下：
import numpy as np
import pandas as pd
X=[]
Y=[]
B=[]
Q_e=0
Q_E=0
def get_data(file_name):
data=pd.read_csv(file_name,header=0)
data=np.array(data)
Y=data[:,data.shape[1]-1]
X=data[:,0:data.shape[1]-1]
print(X.shape)
return X,Y
return X,Y
X,Y=get_data('C:/Users/LOL/Desktop/店铺多元回归.csv')
X=np.mat(np.c_[np.ones(X.shape[0]),X])
Y=np.mat(Y)
B=np.linalg.inv(X.T*X)*(X.T)*(Y.T)
print("第一项为常数项，其他为回归系数",B)
print("输入店铺面积，距离最近的车站距离，预测营业额:",np.mat([1,10,80])*B ,"万日元")
Y_mean=np.mean(Y)
for i in range(Y.size):
Q_e+=pow(np.array((Y.T)[i]-X[i]*B),2)
Q_E+=pow(np.array(X[i]*B)-Y_mean,2)
R2=Q_E/(Q_e+Q_E)
print("R2的值：",R2)
(10, 2)
第一项为常数项，其他为回归系数 [[65.32391639]
[41.51347826]
[-0.34088269]]
输入店铺面积，距离最近的车站距离，预测营业额: [[453.1880841]] 万日元
R2的值： [[0.94523585]]
4、使用Excel求解
如图
由上面三种方法求到的结果对比，可以清晰地看出用最小二乘法和Excel求出的结果一致，而用梯度下降法求解有一定的误差
六、总结
最小二乘法：能通过最小化误差的平方和寻找数据的最佳函数匹配，但是使用有一定局限性，在回归过程中，回归的关联式不可能全部通过每个回归数据点。
梯度下降法：是通过梯度方向和步长，直接求解目标函数的最小值时的参数，越接近最优值时，步长应该不断减小，否则会在最优值附近来回震荡。
七、参考文献
