目錄

• 【1】解析法實現(xiàn)一元線性回歸
• • python列表實現(xiàn)
  • 利用Numpy實現(xiàn)
  • 利用TensorFlow實現(xiàn)
  • 數(shù)據(jù)和模型可視化
• 【2】解析法實現(xiàn)多元線性回歸
• • 利用Numpy實現(xiàn)
  • • 需要用到的NumPy數(shù)組運算函數(shù)
  • 數(shù)據(jù)和模型可視化
  • • 繪制空間點集：
    • 繪制空間平面圖：
    • 繪制線框圖并且與散點圖對比：

【1】解析法實現(xiàn)一元線性回歸

公式：

代碼：

python列表實現(xiàn)

#加載樣本數(shù)據(jù) x=[137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21] y=[145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30] meanX=sum(x)/len(x) meanY=sum(y)/len(y) sumXY=0.0 sumX=0.0 for i in range(len(x)):sumXY+=(x[i]-meanX)*(y[i]-meanY)sumX+=(x[i]-meanX)*(x[i]-meanX) w=sumXY/sumX b=meanY-w*meanX print("w=",w) print("b=",b) x_test=[128.15,45.00,141.43,106.27,99.00,53.84,85.36,70.00] print("面積\t估計房價") for i in range(len(x_test)):print(x_test[i],"\t",round(w*x_test[i]+b,2))

結(jié)果：

利用Numpy實現(xiàn)

import numpy as np x=np.array([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) y=np.array([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) meanX=np.mean(x) meanY=np.mean(y) sumXY=np.sum((x-meanX)*(y-meanY)) sumX=np.sum((x-meanX)*(x-meanX)) w=sumXY/sumX b=meanY-w*meanX print("w=",w) print("b=",b) x_test=np.array([128.15,45.00,141.43,106.27,99.00,53.84,85.36,70.00]) y_pred = w*x_test+b print("面積\t估計房價") for i in range(y_pred.size):print(x_test[i],"\t",np.round(y_pred[i],2))

利用TensorFlow實現(xiàn)

import tensorflow as tf x=tf.constant([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) y=tf.constant([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) meanX=tf.reduce_mean(x) meanY=tf.reduce_mean(y) sumXY=tf.reduce_sum((x-meanX)*(y-meanY)) sumX=tf.reduce_sum((x-meanX)*(x-meanX)) w=sumXY/sumX b=meanY-w*meanX print("w=",w) print("b=",b) x_test=tf.constant([128.15,45.00,141.43,106.27,99.00,53.84,85.36,70.00]) y_pred = w*x_test+b print(y_pred)

數(shù)據(jù)和模型可視化

#解析法實現(xiàn)一元線性回歸 # #Realization of one variable linear regression by analytic method #導(dǎo)入庫 import tensorflow as tf import numpy as np import matplotlib.pyplot as plt #設(shè)置字體 plt.rcParams['font.sans-serif'] =['SimHei'] #加載樣本數(shù)據(jù) x=tf.constant([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) y=tf.constant([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) #學(xué)習(xí)模型-計算w,b meanX=tf.reduce_mean(x) meanY=tf.reduce_mean(y) sumXY=tf.reduce_sum((x-meanX)*(y-meanY)) sumX=tf.reduce_sum((x-meanX)*(x-meanX)) w=sumXY/sumX b=meanY-w*meanX print("權(quán)值w=",w.numpy()) print("偏置b=",b.numpy()) print("線性模型：y=",w.numpy(),"*x+",b.numpy()) #預(yù)測房價 x_test=np.array([128.15,45.00,141.43,106.27,99.00,53.84,85.36,70.00]) y_pred = (w*x_test+b).numpy()#將張量類型轉(zhuǎn)化為numpy數(shù)組類型 print("面積\t估計房價") n=len(x_test) for i in range(n):print(x_test[i],"\t",round(y_pred[i],2)) #模型和數(shù)據(jù)可視化 plt.figure() #繪制散點圖 #張量和數(shù)組都可以作為散點函數(shù)的輸入提供點坐標(biāo) plt.scatter(x,y,color="red",label="銷售記錄") plt.scatter(x_test,y_pred,color="blue",label="預(yù)測房價") plt.plot(x_test,y_pred,color="green",label="擬合直線",linewidth=2) #設(shè)置坐標(biāo)軸的標(biāo)簽文字和字號 plt.xlabel("面積（平方米）",fontsize=14) plt.xlabel("價格（萬元）",fontsize=14) #設(shè)置坐標(biāo)軸的范圍 plt.xlim((40,150)) plt.ylim((40,150)) #設(shè)置標(biāo)題文字和字號 plt.suptitle("商品房銷售價格評估系統(tǒng)v1.0",fontsize=20) #在左上方顯示圖例 plt.legend(loc="upper left") #顯示整個繪圖 plt.show()

結(jié)果：

【2】解析法實現(xiàn)多元線性回歸

利用Numpy實現(xiàn)

需要用到的NumPy數(shù)組運算函數(shù)

功能函數(shù)

數(shù)組堆疊	np.stack( )
改變數(shù)組形狀	np.reshape( )
矩陣相乘	np.matmul( )
矩陣轉(zhuǎn)置	np.transpose( )
矩陣求逆	np.linalg.inv( )

公式：

代碼：

#解析法實現(xiàn)多元線性回歸 #Realization of multiple linear regression by analytic method #導(dǎo)入庫 import numpy as np #=======================【1】加載樣本數(shù)據(jù)=============================================== x1=np.array([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) x2=np.array([3,2,2,3,1,2,3,2,2,3,1,1,1,1,2,2]) y=np.array([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) #=======================【2】數(shù)據(jù)處理=============================================== x0=np.ones(len(x1)) #堆疊屬性數(shù)組，構(gòu)造屬性矩陣 #從(16,)到(16,3),因為新出現(xiàn)的軸是第二個軸所以axis為1 X=np.stack((x0,x1,x2),axis=1) print(X) #得到形狀為一列的數(shù)組 Y=np.array(y).reshape(-1,1) print(Y) #=======================【3】求解模型參數(shù)=============================================== Xt=np.transpose(X) #計算X' XtX_1=np.linalg.inv(np.matmul(Xt,X)) #計算(X'X)-1 XtX_1_Xt=np.matmul(XtX_1,Xt) #計算(X'X)-1X' W=np.matmul(XtX_1_Xt,Y) #W=((X'X)-1)X'Y print(W) W=W.reshape(-1) print(W) print("多元線性回歸方程") print("Y=",W[1],"*x1+",W[2],"*x2+",W[0]) #=======================【4】預(yù)測房價=============================================== print("請輸入房屋面積和房間數(shù)，預(yù)測房屋銷售價格") x1_test=float(input("商品房面積:")) x2_test=int(input("房間數(shù):")) y_pred=W[1]*x1_test+W[2]*x2_test+W[0] print("預(yù)測價格：",round(y_pred,2),"萬元")

效果：

數(shù)據(jù)和模型可視化

繪制空間點集：

#解析法實現(xiàn)多元線性回歸 #Realization of multiple linear regression by analytic method #導(dǎo)入庫與模塊 import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D #=======================【1】加載樣本數(shù)據(jù)=============================================== x1=np.array([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) x2=np.array([3,2,2,3,1,2,3,2,2,3,1,1,1,1,2,2]) y=np.array([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) #=======================【2】數(shù)據(jù)處理=============================================== x0=np.ones(len(x1)) #堆疊屬性數(shù)組，構(gòu)造屬性矩陣 #從(16,)到(16,3),因為新出現(xiàn)的軸是第二個軸所以axis為1 X=np.stack((x0,x1,x2),axis=1) print(X) #得到形狀為一列的數(shù)組 Y=np.array(y).reshape(-1,1) print(Y) #=======================【3】求解模型參數(shù)=============================================== Xt=np.transpose(X) #計算X' XtX_1=np.linalg.inv(np.matmul(Xt,X)) #計算(X'X)-1 XtX_1_Xt=np.matmul(XtX_1,Xt) #計算(X'X)-1X' W=np.matmul(XtX_1_Xt,Y) #W=((X'X)-1)X'Y print(W) W=W.reshape(-1) print(W) print("多元線性回歸方程") print("Y=",W[1],"*x1+",W[2],"*x2+",W[0]) #模型和數(shù)據(jù)可視化 fig=plt.figure(figsize=(8,6)) #創(chuàng)建3D繪圖對象 ax3d=Axes3D(fig) #用來改變視角的函數(shù) #elev:視角的水平高度 水平旋轉(zhuǎn)的角度 #ax3d.view_init(elev=0,azim=-90) #繪制散點圖 #張量和數(shù)組都可以作為散點函數(shù)的輸入提供點坐標(biāo) ax3d.scatter(x1,x2,y,color="b",marker="*") ax3d.set_xlabel('Area',color='r',fontsize=16) ax3d.set_ylabel('Room',color='r',fontsize=16) ax3d.set_zlabel('Price',color='r',fontsize=16) #設(shè)置y軸坐標(biāo)值刻度 ax3d.set_yticks([1,2,3]) ax3d.set_zlim3d(30,160) plt.show()

ax3d.view_init(elev=0,azim=-90)

ax3d.view_init(elev=0,azim=0)

繪制空間平面圖：

#解析法實現(xiàn)多元線性回歸 #Realization of multiple linear regression by analytic method #導(dǎo)入庫與模塊 import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D #=======================【1】加載樣本數(shù)據(jù)=============================================== x1=np.array([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) x2=np.array([3,2,2,3,1,2,3,2,2,3,1,1,1,1,2,2]) y=np.array([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) #=======================【2】數(shù)據(jù)處理=============================================== x0=np.ones(len(x1)) #堆疊屬性數(shù)組，構(gòu)造屬性矩陣 #從(16,)到(16,3),因為新出現(xiàn)的軸是第二個軸所以axis為1 X=np.stack((x0,x1,x2),axis=1) print(X) #得到形狀為一列的數(shù)組 Y=np.array(y).reshape(-1,1) print(Y) #=======================【3】求解模型參數(shù)=============================================== Xt=np.transpose(X) #計算X' XtX_1=np.linalg.inv(np.matmul(Xt,X)) #計算(X'X)-1 XtX_1_Xt=np.matmul(XtX_1,Xt) #計算(X'X)-1X' W=np.matmul(XtX_1_Xt,Y) #W=((X'X)-1)X'Y print(W) W=W.reshape(-1) print(W) print("多元線性回歸方程") print("Y=",W[1],"*x1+",W[2],"*x2+",W[0]) #模型和數(shù)據(jù)可視化 X1,X2=np.meshgrid(x1,x2) #平面方程 Y_PRED=W[0]+W[1]*X1+W[2]*X2 #創(chuàng)建3D繪圖對象 fig=plt.figure() ax3d=Axes3D(fig) #繪制空間平面圖 ax3d.plot_surface(X1,X2,Y_PRED,cmap="coolwarm") ax3d.set_xlabel('Area',color='r',fontsize=14) ax3d.set_ylabel('Room',color='r',fontsize=14) ax3d.set_zlabel('Price',color='r',fontsize=14) ax3d.set_yticks([1,2,3]) plt.show()

繪制線框圖并且與散點圖對比：

#解析法實現(xiàn)多元線性回歸 #Realization of multiple linear regression by analytic method #導(dǎo)入庫與模塊 import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D #=======================【1】加載樣本數(shù)據(jù)=============================================== x1=np.array([137.97,104.50,100.00,124.32,79.20,99.00,124.00,114.00,106.69,138.05,53.75,46.91,68.00,63.02,81.26,86.21]) x2=np.array([3,2,2,3,1,2,3,2,2,3,1,1,1,1,2,2]) y=np.array([145.00,110.00,93.00,116.00,65.32,104.00,118.00,91.00,62.00,133.00,51.00,45.00,78.50,69.65,75.69,95.30]) #=======================【2】數(shù)據(jù)處理=============================================== x0=np.ones(len(x1)) #堆疊屬性數(shù)組，構(gòu)造屬性矩陣 #從(16,)到(16,3),因為新出現(xiàn)的軸是第二個軸所以axis為1 X=np.stack((x0,x1,x2),axis=1) print(X) #得到形狀為一列的數(shù)組 Y=np.array(y).reshape(-1,1) print(Y) #=======================【3】求解模型參數(shù)=============================================== Xt=np.transpose(X) #計算X' XtX_1=np.linalg.inv(np.matmul(Xt,X)) #計算(X'X)-1 XtX_1_Xt=np.matmul(XtX_1,Xt) #計算(X'X)-1X' W=np.matmul(XtX_1_Xt,Y) #W=((X'X)-1)X'Y print(W) W=W.reshape(-1) print(W) print("多元線性回歸方程") print("Y=",W[1],"*x1+",W[2],"*x2+",W[0]) y_pred=W[0]+W[1]*x1+W[2]*x2 #設(shè)置字體 plt.rcParams['font.sans-serif'] =['SimHei'] #模型和數(shù)據(jù)可視化 X1,X2=np.meshgrid(x1,x2) #平面方程 Y_PRED=W[0]+W[1]*X1+W[2]*X2 #創(chuàng)建3D繪圖對象 fig=plt.figure() ax3d=Axes3D(fig) #繪制散點圖與線框圖 ax3d.scatter(x1,x2,y,color='b',marker="*",label="銷售記錄") ax3d.scatter(x1,x2,y_pred,color='r',label="預(yù)測房價") ax3d.plot_wireframe(X1,X2,Y_PRED,color='c',linewidth=0.5,label="擬合平面")ax3d.set_xlabel('Area',color='r',fontsize=14) ax3d.set_ylabel('Room',color='r',fontsize=14) ax3d.set_zlabel('Price',color='r',fontsize=14) ax3d.set_yticks([1,2,3]) plt.suptitle("商品房銷售回歸模型",fontsize=20) plt.legend(loc="upper left") plt.show()

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