Table of contents
3. Code - multiple linear regression
3.3 Multiple linear regression model
3.3.1 Multiple Linear Regression-OLS
3.3.2 Relative error of prediction value of multiple linear regression model
3.3.4 Predicted value and actual value distribution chart
4.2 Response surface rendering code
4.3 Call matching_3D to draw the response surface
1. Write in front
Polynomial regression combined with the response surface analysis method can use the response surface diagram to intuitively reflect the complex three-dimensional relationship, so as to clearly present the technical method of the relationship between two independent variables and one dependent variable.
2. Data
In addition to 25 sets of data, a set of benchmark data under natural conditions is also added:
| No. | Temperature | pH | Fe2+ | Cu2+ | Fe3+ | Y |
| 0 | 30 | 6.5 | 0 | 0 | 0 | 7.55 |
3. Code - multiple linear regression
3.1 Import library
from mpl_toolkits.mplot3d.axes3d import Axes3D
from matplotlib import cm
from pylab import *
from numpy import *
import matplotlib.pyplot as plt
import pandas as pd
import math
import numpy as np
import copy
plt.rcParams['axes.unicode_minus']=False #用于解决不能显示负号的问题
mpl.rcParams['font.sans-serif'] = ['SimHei']
3.2 Import data
#26
xArr = [
[1,30,6.5,0,0,0],
[1,30,1.5,0,0,0],
[1,30,2,1,3,1],
[1,30,2.25,3,5,3],
[1,30,2.27,5,8,5],
[1,30,2.41,8,10,8],
[1,35,1.5,1,5,5],
[1,35,2,3,8,8],
[1,35,2.5,5,10,0],
[1,35,3,8,0,1],
[1,35,2.61,0,3,3],
[1,40,1.5,3,10,1],
[1,40,2,5,0,3],
[1,40,2.5,8,3,5],
[1,40,2.28,0,5,8],
[1,40,3.23,1,8,0],
[1,45,1.5,5,3,8],
[1,45,2,8,5,0],
[1,45,2.5,0,8,1],
[1,45,2.44,1,10,3],
[1,45,2.26,3,0,5],
[1,50,1.5,8,8,3],
[1,50,2,0,10,5],
[1,50,2.17,1,0,8],
[1,50,3,3,3,0],
[1,50,2.82,5,5,1]
]
#26
yArr = [
7.55,
7.14,
7.2,
7.05,
6.82,
6.51,
6.73,
6.69,
6.46,
6.75,
6.70,
6.55,
6.3,
6.21,
6.18,
5.97,
5.95,
5.9,
5.5,
5.72,
5.6,
5.62,
5.29,
5.57,
5.30,
5.21
]
# print(len(xArr),len(yArr))3.3 Multiple linear regression model
3.3.1 Multiple Linear Regression-OLS
Here is something unrelated to the topic, like this kind of data with many X and corresponding Y, you can consider performing multiple linear regression
The regression code is as follows: (least square method)
#最小二乘法 OLS
def standRegres(xArr,yArr):
xMat = mat(xArr)
yMat = mat(yArr).T
xTx = xMat.T*xMat
if linalg.det(xTx) == 0.0:
print("This matrix is singular, cannot do inverse")
return
ws = xTx.I * (xMat.T*yMat)
# print(ws)
return ws3.3.2 Relative error of prediction value of multiple linear regression model
The relative error of each set of data was verified by multiple linear regression equation:
mySum = 0
sse = 0
yPerList = []
#采用全部数据进行训练
ws = standRegres(xArr,yArr) #ws即为方程系数
print(ws)
for index,x in enumerate(xArr):
yPer = float(x*ws) #yPer即为预测值
yPerList.append(yPer)
mySum += abs(yPer-yArr[index])*100
sse = (yPer-yArr[index])**2
error = abs(yPer-yArr[index])/yArr[index]*100 #相对误差
# print(yArr[index],round(yPer,2),str(round(error,2))+"%")
plt.plot(index,error,"o")
plt.title(" ",fontsize=13) #图片上方留白
plt.rc('font',family='Arial') #设置字体
plt.rcParams['xtick.direction'] = 'in' #刻度线朝内
plt.rcParams['ytick.direction'] = 'in'
plt.tick_params(labelsize=18) #刻度大小
plt.xlabel("Test No.",fontsize=18)
plt.ylabel("Relative Error/%",fontsize=18)
plt.savefig("线性回归模型各拟合值相对误差",dpi=500,bbox_inches = 'tight') #dpi-清晰度
plt.show()
print("SSE=",sse,"平均相对误差=",round(mySum/sum(yArr),2))
print(corrcoef(yPerList,yArr)[0][1])Effect:
3.3.3 Residual plot
Relevant code:
#残差图
mySum = 0
sse = 0
yPerList = []
#采用全部数据进行训练
ws = standRegres(xArr,yArr)
# print(ws)
for index,x in enumerate(xArr):
yPer = float(x*ws)
residua = yArr[index] - yPer
plt.plot(index,residua,"bo")
x = np.linspace(0,25,100)
plt.plot(x,np.zeros(len(x)),"r")
plt.title(" ",fontsize=13)
plt.rcParams['xtick.direction'] = 'in'
plt.rcParams['ytick.direction'] = 'in'
plt.rc('font',family='Arial')
plt.tick_params(labelsize=18)
plt.xlabel("Test No.",fontsize=18)
plt.ylabel("Residua",fontsize=18)
plt.ylim((-0.5, 0.5))
plt.savefig("残差图.jpg",dpi=500,bbox_inches = 'tight')
plt.show()
Effect:
3.3.4 Predicted value and actual value distribution chart
Relevant code:
#分布图
mySum = 0
sse = 0
yPerList = []
#采用全部数据进行训练
ws = standRegres(xArr,yArr)
# print(ws)
for index,x in enumerate(xArr):
yPer = float(x*ws)
yPerList.append(yPer)
plt.plot(list(range(len(xArr))),yPerList,"bo",label="Fitted value")
plt.plot(list(range(len(xArr))),yArr,"r*",label="Measurement value")
plt.title(" ",fontsize=13)
plt.rc('font',family='Arial')
plt.rcParams['xtick.direction'] = 'in'
plt.rcParams['ytick.direction'] = 'in'
plt.tick_params(labelsize=18)
plt.legend(framealpha=0,loc=(0.45, 0.55),fontsize=15)
plt.xlabel("Test No.",fontsize=18)
plt.ylabel("Oxygen solubility/mg·${
{L^-}^1}$",fontsize=18)
plt.savefig("分布图.jpg",dpi=500,bbox_inches = 'tight')
plt.show()
Effect:
4. Code-response surface
4.1 Gaussian elimination
Relevant code:
#最小二乘法曲面拟合
def fun(x):
round(x, 2)
if x >= 0:
return '+'+str(x)
else:
return str(x)
def get_res(X, Y, Z, n):
# 求方程系数
sigma_x = 0
for i in X: sigma_x += i
sigma_y = 0
for i in Y: sigma_y += i
sigma_z = 0
for i in Z: sigma_z += i
sigma_x2 = 0
for i in X: sigma_x2 += i * i
sigma_y2 = 0
for i in Y: sigma_y2 += i * i
sigma_x3 = 0
for i in X: sigma_x3 += i * i * i
sigma_y3 = 0
for i in Y: sigma_y3 += i * i * i
sigma_x4 = 0
for i in X: sigma_x4 += i * i * i * i
sigma_y4 = 0
for i in Y: sigma_y4 += i * i * i * i
sigma_x_y = 0
for i in range(n):
sigma_x_y += X[i] * Y[i]
# print(sigma_xy)
sigma_x_y2 = 0
for i in range(n): sigma_x_y2 += X[i] * Y[i] * Y[i]
sigma_x_y3 = 0
for i in range(n): sigma_x_y3 += X[i] * Y[i] * Y[i] * Y[i]
sigma_x2_y = 0
for i in range(n): sigma_x2_y += X[i] * X[i] * Y[i]
sigma_x2_y2 = 0
for i in range(n): sigma_x2_y2 += X[i] * X[i] * Y[i] * Y[i]
sigma_x3_y = 0
for i in range(n): sigma_x3_y += X[i] * X[i] * X[i] * Y[i]
sigma_z_x2 = 0
for i in range(n): sigma_z_x2 += Z[i] * X[i] * X[i]
sigma_z_y2 = 0
for i in range(n): sigma_z_y2 += Z[i] * Y[i] * Y[i]
sigma_z_x_y = 0
for i in range(n): sigma_z_x_y += Z[i] * X[i] * Y[i]
sigma_z_x = 0
for i in range(n): sigma_z_x += Z[i] * X[i]
sigma_z_y = 0
for i in range(n): sigma_z_y += Z[i] * Y[i]
# print("-----------------------")
# 给出对应方程的矩阵形式
a = np.array([[sigma_x4, sigma_x3_y, sigma_x2_y2, sigma_x3, sigma_x2_y, sigma_x2],
[sigma_x3_y, sigma_x2_y2, sigma_x_y3, sigma_x2_y, sigma_x_y2, sigma_x_y],
[sigma_x2_y2, sigma_x_y3, sigma_y4, sigma_x_y2, sigma_y3, sigma_y2],
[sigma_x3, sigma_x2_y, sigma_x_y2, sigma_x2, sigma_x_y, sigma_x],
[sigma_x2_y, sigma_x_y2, sigma_y3, sigma_x_y, sigma_y2, sigma_y],
[sigma_x2, sigma_x_y, sigma_y2, sigma_x, sigma_y, n]])
b = np.array([sigma_z_x2, sigma_z_x_y, sigma_z_y2, sigma_z_x, sigma_z_y, sigma_z])
# 高斯消元解线性方程
res = np.linalg.solve(a, b)
return res
labelName = ["Oxygen solubility/mg·${
{L^-}^1}$",
"T/$^\circ$C",
"pH",
"c(${
{Fe^2}^+}$)/g·${
{L^-}^1}$",
"c(${
{Cu^2}^+}$)/g·${
{L^-}^1}$",
"c(${
{Fe^3}^+}$)/g·${
{L^-}^1}$",]
print(labelName)4.2 Response surface rendering code
The core code of drawing is here, if you are interested, do your own research.
The coefficient of the variable res is the coefficient generated by the Gaussian elimination algorithm in Section 4.1
def matching_3D(X, Y, Z,xLabelIndex,yLabelIndex,name,arg1=37,arg2=-72):
n = len(X)
res = get_res(X, Y, Z, n)
# 输出方程形式
print("z=%.6s*x^2%.6s*xy%.6s*y^2%.6s*x%.6s*y%.6s" % (
fun(res[0]), fun(res[1]), fun(res[2]), fun(res[3]), fun(res[4]), fun(res[5])))
# 画曲面图和离散点
fig = plt.figure() # 建立一个空间
ax = fig.add_subplot(111, projection='3d') # 3D坐标
xgrid = np.linspace(min(X),max(X),100)
ygrid = np.linspace(min(Y),max(Y),100)
x,y = np.meshgrid(xgrid,ygrid)
# 给出方程
z = res[0] * x * x + res[1] * x * y + res[2] * y * y + res[3] * x + res[4] * y + res[5]
# 画出曲面
sp = ax.plot_surface(x, y, z, rstride=3, cstride=3, cmap=cm.jet)
ax.contourf(x,y,z,zdir='z',offset=5,cmap = plt.get_cmap('rainbow'))
# 画出点
ax.scatter(X, Y, Z, c='r',label="实测点",alpha=0)
plt.rc('font',family='Arial')
plt.xlabel(labelName[xLabelIndex])
plt.xticks(rotation=30,fontsize=9)
plt.ylabel(labelName[yLabelIndex])
ax.set_zlabel(labelName[0])
# show_text(ax)
# ax.legend()
fig.colorbar(sp)
ax.view_init(elev=arg1, azim=arg2)
plt.rcParams['xtick.direction'] = 'in'
plt.rcParams['ytick.direction'] = 'in'
plt.savefig(name,dpi=500,bbox_inches = 'tight')
fig.show()4.3 Call matching_3D to draw the response surface
code:
Compare with the parameters of matching_3D in Section 4.2 :
X and Y are lists of independent variables - X-axis and Y-axis respectively
yArr is the dependent variable-Z axis coordinate
1 and 4 are the names corresponding to the labelName list in section 4.1, which are used to automatically generate the coordinate names of the response
T-cu2+ is the name of the saved picture file
37, -72 is used to adjust the picture angle
%matplotlib notebook
X = []
Y = []
for x,y in zip(xArrMat[:,1],xArrMat[:,4]):
X.append(float(x))
Y.append(float(y))
matching_3D(X,Y,yArr,1,4,"T-cu2+曲面图",37,-72)Effect:
