模式識別-線性擬合實(shí)驗(yàn)報告-黃志強(qiáng)

1、線性擬合-實(shí)驗(yàn)報告內(nèi)蒙古大學(xué)計(jì)算機(jī)學(xué)院31209003黃志強(qiáng)實(shí)驗(yàn)方法：1最小二乘法2梯度下降法二.公式推導(dǎo)1最小二乘用線性函數(shù)h a(x)=a0+a1*x來擬合y=f(x)；構(gòu)造代價函數(shù)J(a)：代價函數(shù)分別對a0和al求偏導(dǎo)，連個偏導(dǎo)數(shù)都等于0成為兩個方程，兩個方程聯(lián)合求解得到a0和al；2梯度下降構(gòu)造代價函數(shù)J(a)，J(a)對a0，al分別求偏導(dǎo)得到梯度， J(a)/a0= n*a0+a1*sumx-sumy; J(a)/a1 = ai* sumx*sumx+a0*sumx-sumx*sumy;tidu_a0=n*a0+a1*sumx-sumy;tidu_a1=a1*sumxx+a0*s

2、umx-sumxy;設(shè)置步長為1，迭代m次delta_r=sqrt(tidu_a0*tidu_a0+tidu_a1*tidu_a1);a0=a0-l*(tidu_a0/tidu_r);a1=a1-l*(tidu_a1/tidu_r);每次迭代顯示得到的直線和mse，并修訂學(xué)習(xí)率%顯示直線x2=-0.1,1.1;y2=x2.*a1+a0;plot(x2,y2color,1-i/m,1-i/m,1-i/m);%顯示錯誤error=0;for j=1:nerror=error+(y(j)-(a1*x(j)+a0)*(y(j)-(a1*x(j)+a0);endmse=error/n;l=mse;mse

3、三.matlab代碼1最小二乘法代碼：%in是一個100行2列的矩陣，兩列分別為x和y。用一條直線y=x*a+b擬合x和y的關(guān)系； %用最小二乘法計(jì)算a和b。x=in(1:100,1);y=in(1:100,2);sumx=0;sumy=0;sumxx=0;sumyy=0;sumxy=0;for i=1:1:100sumx=sumx+x(i);sumy=sumy+y(i);sumxx=sumxx+x(i)*x(i);sumyy=sumyy+y(i)*y(i);sumxy=sumxy+x(i)*y(i);endplot(in(:,1),in(:,2),r)；用紅色的點(diǎn)畫出100個樣本點(diǎn)hold

4、on; %保留當(dāng)前繪圖，不被下次繪圖遮蓋n=100;b,a=solve(n*a0+a1*sumx=sumy,a0*sumx+a1*sumxx=sumxy,a0,a1);%解二元一次方程組，未知數(shù)為a0，a1，結(jié)果返回給b和a。a=eval(a);%eval(str)，把str當(dāng)做一條語句執(zhí)行b=eval(b);x2=0,1;%知道解析式y(tǒng)=a*x+b，畫直線的方法y2=x2.*a+b;%因?yàn)閤2是一個向量，所以用x2.表示plot(x2,y2);%制動化一條以x2為x,以y2為y的直線mse=0;error=0;for i=1:nerror=error+(y(i)-(a*x(i)+b)*(y(

5、i)-(a*x(i)+b);endmse=error/n;mse2梯度下降法代碼：x=in(1:100,1);y=in(1:100,2);sumx=0;sumy=0;sumxx=0;sumyy=0;sumxy=0;for i=1:1:100sumx=sumx+x(i);sumy=sumy+y(i);sumxx=sumxx+x(i)*x(i);sumyy=sumyy+y(i)*y(i);sumxy=sumxy+x(i)*y(i);endplot(in(:,1),in(:,2),r.);hold on;a0=2;a1=1;l=0.5;n=100;m=5 0;for i=0:1:mtidu_a0=n

6、*a0+a1*sumx-sumy;tidu_a1=a1*sumxx+a0*sumx-sumxy;tidu_r=sqrt(tidu_a0*tidu_a0+tidu_a1*tidu_a1);a0=a0-l*(tidu_a0/tidu_r);a1=a1-l*(tidu_a1/tidu_r);x2=-0.1,1.1;y2=x2.*a1+a0;plot(x2,y2 color,1-i/m,1-i/m,1-i/m);error=0;for j=1:nerror=error+(y(j)-(a1*x(j)+a0)*(y(j)-(a1*x(j)+a0);endmse=error/n;l=mse;mseend四.

7、運(yùn)行結(jié)果1最小二乘法結(jié)果2梯度下降法結(jié)果誤差1最小二乘法A1=3.679365985769617A0=-1.030876273676726均方誤差mse=0.04292梯度下降法（起點(diǎn)為a0=2, a1=1迭代次數(shù)為50次）A1=3.67860477725630A0=-1.00565713447357均方誤差mse =0.0436 附數(shù)據(jù)in.txt0.9005 1.91130.4480 0.92180.2689 -0.46540.5538 1.46670.1788 -0.23930.8597 1.70480.2320 -0.21350.1681 -0.25490.0267 -1.09280.

8、3224 0.29850.5552 0.79310.8245 2.01720.8042 2.22730.0244 -0.88880.3715 0.56870.4919 0.77950.4661 0.53480.0417 -0.79690.6170 1.24030.5780 1.51130.2988 -0.11200.4357 0.57820.1366 -0.84070.2997 0.38070.7614 1.89590.0353 -0.63990.2695 -0.10720.9963 2.72330.4469 0.86040.1528 -0.54720.8862 2.33980.0314 -1

9、.21900.1160 -0.68320.2509 -0.14950.7597 1.61760.8983 1.95520.2234 -0.16960.6733 1.48590.8188 2.10080.9489 2.65170.8743 2.00690.3937 0.45570.9370 2.44270.4369 0.80250.1625 -0.26760.3098 -0.06410.6811 1.10380.9341 2.24060.9474 2.65010.5991 1.16170.9489 2.41700.4040 0.30190.0410 -1.02710.2938 0.12610.0

10、319 -0.78420.8645 2.24680.4325 0.58290.0928 -0.47670.1378 -0.58010.2420 -0.16170.2230 -0.42450.8677 2.19760.7642 1.74470.3447 0.01780.3848 0.48110.5949 1.20160.5351 1.33880.3336 0.28380.8547 2.21270.2656 -0.10610.9339 2.18400.3898 0.15150.6831 1.54170.2750 0.27060.0280 -0.87500.9406 2.61790.5340 0.82420.6712 1.49270.6075 1.14170.7509 1.56650.9813 2.72670.7277 1.58300.8573 1.47560.9918 3.00380.7595 1.69700.1460 -0.43690.3263 0.06280.0288