注：這裡的練習鑑於當時理解不完全，可能會有些錯誤，關於神經網路的實踐可以參考我的這篇博文
這裡的程式碼只是簡單的練習，不涉及程式碼優化，也不涉及神經網路優化，所以我用了最能體現原理的方式來寫的程式碼。
啟用函式用的是h = 1/(1+exp(-y))，其中y=sum([X Y].*w)。
代價函式用的是E = 1/2*(t-h)^2，其中t為目標值，t為1代表是該類，t為0代表不是該類。
權值更新採用BP演算法。
網路1形式如下，沒有隱含層，1個偏置量，輸入直接連線輸出：
分類結果：
程式碼如下：
clear all; close all; clc; n=5; randn('seed',1); mu1=[0 0]; S1=[0.5 0; 0 0.5]; P1=mvnrnd(mu1,S1,n); mu2=[0 6]; S2=[0.5 0; 0 0.5]; P2=mvnrnd(mu2,S2,n); mu3=[6 3]; S3=[0.5 0; 0 0.5]; P3=mvnrnd(mu3,S3,n); P=[P1;P2;P3]; meanP=mean(P); P=[P(:,1)-meanP(1) P(:,2)-meanP(2)]; sigma = 5; X=P(:,1); Y=P(:,2); B=rand(3*n,1); w1 = rand(3*n,1); w2 = rand(3*n,1); w3 = rand(3*n,1); w4 = rand(3*n,1); w5 = rand(3*n,1); w6 = rand(3*n,1); for i=1:3*n i while 1 y1 = X(i)*w1(i) + Y(i)*w4(i) + B(i); y2 = X(i)*w2(i) + Y(i)*w5(i) + B(i); y3 = X(i)*w3(i) + Y(i)*w6(i) + B(i); h1 = 1/(1+exp(-y1)); h2 = 1/(1+exp(-y2)); h3 = 1/(1+exp(-y3)); e1 = 1/2*(1 - h1)^2; e2 = 1/2*(1 - h2)^2; e3 = 1/2*(1 - h3)^2; if i<=n && e1<=0.0000001 break; elseif i>n && i<=2*n && e2<0.0000001 break; elseif i>2*n && e3<0.0000001 break; end if i<=n w1(i) = w1(i)-sigma*(h1-1)*h1*(1-h1)*X(i); w2(i) = w2(i)-sigma*(h2-0)*h2*(1-h2)*X(i); w3(i) = w3(i)-sigma*(h3-0)*h3*(1-h3)*X(i); w4(i) = w4(i)-sigma*(h1-1)*h1*(1-h1)*Y(i); w5(i) = w5(i)-sigma*(h2-0)*h2*(1-h2)*Y(i); w6(i) = w6(i)-sigma*(h3-0)*h3*(1-h3)*Y(i); B(i) =B(i)- sigma*((h1-1)*h1*(1-h1)+(h2-0)*h2*(1-h2)+(h3-0)*h3*(1-h3)); elseif i>n && i<=2*n w1(i) = w1(i)-sigma*(h1-0)*h1*(1-h1)*X(i); w2(i) = w2(i)-sigma*(h2-1)*h2*(1-h2)*X(i); w3(i) = w3(i)-sigma*(h3-0)*h3*(1-h3)*X(i); w4(i) = w4(i)-sigma*(h1-0)*h1*(1-h1)*Y(i); w5(i) = w5(i)-sigma*(h2-1)*h2*(1-h2)*Y(i); w6(i) = w6(i)-sigma*(h3-0)*h3*(1-h3)*Y(i); B(i) =B(i)- sigma*((h1-0)*h1*(1-h1)+(h2-1)*h2*(1-h2)+(h3-0)*h3*(1-h3)); else w1(i) = w1(i)-sigma*(h1-0)*h1*(1-h1)*X(i); w2(i) = w2(i)-sigma*(h2-0)*h2*(1-h2)*X(i); w3(i) = w3(i)-sigma*(h3-1)*h3*(1-h3)*X(i); w4(i) = w4(i)-sigma*(h1-0)*h1*(1-h1)*Y(i); w5(i) = w5(i)-sigma*(h2-0)*h2*(1-h2)*Y(i); w6(i) = w6(i)-sigma*(h3-1)*h3*(1-h3)*Y(i); B(i) =B(i)- sigma*((h1-0)*h1*(1-h1)+(h2-0)*h2*(1-h2)+(h3-1)*h3*(1-h3)); end end end plot(P(:,1),P(:,2),'o'); hold on; flag = 0; M=[]; for x=-8:0.3:8 for y=-8:0.3:8 H=[]; for i=1:3*n y1 = x*w1(i)+y*w4(i) +B(i); y2 = x*w2(i)+y*w5(i) +B(i); y3 = x*w3(i)+y*w6(i) +B(i); h1=1/(1+exp(-y1)); h2=1/(1+exp(-y2)); h3=1/(1+exp(-y3)); H=[H;h1 h2 h3]; end % H1 = mean(H(1:n,1)); % H2 = mean(H(n:2*n,2)); % H3 = mean(H(2*n:3*n,3)); meanH = mean(H); H1 = meanH(1); H2 = meanH(2); H3= meanH(3); if H1>H2 && H1>H3 plot(x,y,'g.') elseif H2 > H1 && H2 > H3 plot(x,y,'r.') elseif H3 > H1 && H3 > H2 plot(x,y,'b.') end end end
網路2形式如下，有1個隱含層，2個偏置量：
分類結果：
程式碼如下：
clear all; close all; clc; n=5; randn('seed',1); mu1=[0 0]; S1=[0.5 0; 0 0.5]; P1=mvnrnd(mu1,S1,n); mu2=[0 6]; S2=[0.5 0; 0 0.5]; P2=mvnrnd(mu2,S2,n); mu3=[6 3]; S3=[0.5 0; 0 0.5]; P3=mvnrnd(mu3,S3,n); P=[P1;P2;P3]; meanP=mean(P); P=[P(:,1)-meanP(1) P(:,2)-meanP(2)]; sigma = 5; X=P(:,1); Y=P(:,2); B1=rand(3*n,1); B2=rand(3*n,1); w1 = rand(3*n,1); w2 = rand(3*n,1); w3 = rand(3*n,1); w4 = rand(3*n,1); w5 = rand(3*n,1); for i=1:3*n i while 1 y0 = X(i)*w1(i) + Y(i)*w2(i) + B1(i); h0 = 1/(1+exp(-y0)); y1 = h0*w3(i) + B2(i); y2 = h0*w4(i) + B2(i); y3 = h0*w5(i) + B2(i); h1 = 1/(1+exp(-y1)); h2 = 1/(1+exp(-y2)); h3 = 1/(1+exp(-y3)); e1 = 1/2*(1 - h1)^2; e2 = 1/2*(1 - h2)^2; e3 = 1/2*(1 - h3)^2; if i<=n && e1<=0.0000001 break; elseif i>n && i<=2*n && e2<0.0000001 break; elseif i>2*n && e3<0.0000001 break; end %e1 if i<=n w1(i) = w1(i)- sigma*((h1-1)*h1*(1-h1)*w3(i)*h0*(1-h0)*X(i) + (h2-0)*h2*(1-h2)*w4(i)*h0*(1-h0)*X(i) + (h3-0)*h3*(1-h3)*w5(i)*h0*(1-h0)*X(i)); w2(i) = w2(i)- sigma*((h1-1)*h1*(1-h1)*w3(i)*h0*(1-h0)*Y(i) + (h2-0)*h2*(1-h2)*w4(i)*h0*(1-h0)*Y(i) + (h3-0)*h3*(1-h3)*w5(i)*h0*(1-h0)*Y(i)); B1(i) = B1(i)- sigma*((h1-1)*h1*(1-h1)*w3(i)*h0*(1-h0) + (h2-0)*h2*(1-h2)*w4(i)*h0*(1-h0) + (h3-0)*h3*(1-h3)*w5(i)*h0*(1-h0)); w3(i) = w3(i)-sigma*(h1-1)*h1*(1-h1)*h0; w4(i) = w4(i)-sigma*(h2-0)*h2*(1-h2)*h0; w5(i) = w5(i)-sigma*(h3-0)*h3*(1-h3)*h0; B2(i) =B2(i)- sigma*((h1-1)*h1*(1-h1)+(h2-0)*h2*(1-h2)+(h3-0)*h3*(1-h3)); elseif i>n && i<=2*n w1(i) = w1(i)-sigma*((h1-0)*h1*(1-h1)*w3(i)*h0*(1-h0)*X(i) + (h2-1)*h2*(1-h2)*w4(i)*h0*(1-h0)*X(i) + (h3-0)*h3*(1-h3)*w5(i)*h0*(1-h0)*X(i)); w2(i) = w2(i)-sigma*((h1-0)*h1*(1-h1)*w3(i)*h0*(1-h0)*Y(i) + (h2-1)*h2*(1-h2)*w4(i)*h0*(1-h0)*Y(i) + (h3-0)*h3*(1-h3)*w5(i)*h0*(1-h0)*Y(i)); B1(i) =B1(i)- sigma*((h1-0)*h1*(1-h1)*w3(i)*h0*(1-h0) + (h2-1)*h2*(1-h2)*w4(i)*h0*(1-h0) + (h3-0)*h3*(1-h3)*w5(i)*h0*(1-h0)); w3(i) = w3(i)-sigma*(h1-0)*h1*(1-h1)*h0; w4(i) = w4(i)-sigma*(h2-1)*h2*(1-h2)*h0; w5(i) = w5(i)-sigma*(h3-0)*h3*(1-h3)*h0; B2(i) =B2(i)- sigma*((h1-0)*h1*(1-h1)+(h2-1)*h2*(1-h2)+(h3-0)*h3*(1-h3)); else w1(i) = w1(i)-sigma*((h1-0)*h1*(1-h1)*w3(i)*h0*(1-h0)*X(i) + (h2-0)*h2*(1-h2)*w4(i)*h0*(1-h0)*X(i) + (h3-1)*h3*(1-h3)*w5(i)*h0*(1-h0)*X(i)); w2(i) = w2(i)-sigma*((h1-0)*h1*(1-h1)*w3(i)*h0*(1-h0)*Y(i) + (h2-0)*h2*(1-h2)*w4(i)*h0*(1-h0)*Y(i) + (h3-1)*h3*(1-h3)*w5(i)*h0*(1-h0)*Y(i)); B1(i) =B1(i)- sigma*((h1-0)*h1*(1-h1)*w3(i)*h0*(1-h0) + (h2-0)*h2*(1-h2)*w4(i)*h0*(1-h0) + (h3-1)*h3*(1-h3)*w5(i)*h0*(1-h0)); w3(i) = w3(i)-sigma*(h1-0)*h1*(1-h1)*h0; w4(i) = w4(i)-sigma*(h2-0)*h2*(1-h2)*h0; w5(i) = w5(i)-sigma*(h3-1)*h3*(1-h3)*h0; B2(i) =B2(i)- sigma*((h1-0)*h1*(1-h1)+(h2-0)*h2*(1-h2)+(h3-1)*h3*(1-h3)); end end end plot(P(:,1),P(:,2),'o'); hold on; flag = 0; M=[]; for x=-8:0.3:8 for y=-8:0.3:8 H=[]; for i=1:3*n y0 = x*w1(i)+y*w2(i) +B1(i); h0=1/(1+exp(-y0)); y1 = h0*w3(i) + B2(i); y2 = h0*w4(i) + B2(i); y3 = h0*w5(i) + B2(i); h1 =1/(1+exp(-y1)); h2 =1/(1+exp(-y2)); h3 =1/(1+exp(-y3)); H=[H;h1 h2 h3]; end meanH = mean(H); H1 = meanH(1); H2 = meanH(2); H3= meanH(3); if H1>H2 && H1>H3 plot(x,y,'g.') elseif H2 > H1 && H2 > H3 plot(x,y,'r.') elseif H3 > H1 && H3 > H2 plot(x,y,'b.') end end end
網路3形式如下，有2個隱含層，2個偏置量：
分類結果：
程式碼如下：
clear all; close all; clc; n=5; randn('seed',1); mu1=[0 0]; S1=[0.5 0; 0 0.5]; P1=mvnrnd(mu1,S1,n); mu2=[0 6]; S2=[0.5 0; 0 0.5]; P2=mvnrnd(mu2,S2,n); mu3=[6 3]; S3=[0.5 0; 0 0.5]; P3=mvnrnd(mu3,S3,n); P=[P1;P2;P3]; meanP=mean(P); P=[P(:,1)-meanP(1) P(:,2)-meanP(2)]; sigma = 20; X=P(:,1); Y=P(:,2); B1=rand(3*n,1); B2=rand(3*n,1); w1 = rand(3*n,1); w2 = rand(3*n,1); w3 = rand(3*n,1); w4 = rand(3*n,1); w5 = rand(3*n,1); w6 = rand(3*n,1); w7 = rand(3*n,1); w8 = rand(3*n,1); w9 = rand(3*n,1); w10 = rand(3*n,1); for i=1:3*n i while 1 y1 = X(i)*w1(i) + Y(i)*w3(i) + B1(i); y2 = X(i)*w2(i) + Y(i)*w4(i) + B1(i); h1 = 1/(1+exp(-y1)); h2 = 1/(1+exp(-y2)); dh1 = h1*(1-h1); dh2 = h2*(1-h2); y3 = h1*w5(i) + h2*w8(i)+ B2(i); y4 = h1*w6(i) + h2*w9(i)+ B2(i); y5 = h1*w7(i) + h2*w10(i)+ B2(i); h3 = 1/(1+exp(-y3)); h4 = 1/(1+exp(-y4)); h5 = 1/(1+exp(-y5)); dh3 = h3*(1-h3); dh4 = h4*(1-h4); dh5 = h5*(1-h5); e1 = 1/2*(1 - h3)^2; e2 = 1/2*(1 - h4)^2; e3 = 1/2*(1 - h5)^2; if i<=n && e1<=0.0000001 break; elseif i>n && i<=2*n && e2<0.0000001 break; elseif i>2*n && e3<0.0000001 break; end %e1 if i<=n w1(i) = w1(i) -sigma * ((h3-1)*dh3*w5(i)+(h4-0)*dh4*w6(i)+(h5-0)*dh5*w7(i)) * dh1*X(i); w2(i) = w2(i) -sigma * ((h3-1)*dh3*w8(i)+(h4-0)*dh4*w9(i)+(h5-0)*dh5*w10(i)) * dh2*X(i); w3(i) = w3(i) -sigma * ((h3-1)*dh3*w5(i)+(h4-0)*dh4*w6(i)+(h5-0)*dh5*w7(i)) * dh1*Y(i); w4(i) = w4(i) -sigma * ((h3-1)*dh3*w8(i)+(h4-0)*dh4*w9(i)+(h5-0)*dh5*w10(i)) * dh2*Y(i); B1(i) = B1(i)- sigma*(((h3-1)*dh3*w5(i)+(h4-0)*dh4*w6(i)+(h5-0)*dh5*w7(i))*dh1+((h3-1)*dh3*w8(i)+(h4-0)*dh4*w9(i)+(h5-0)*dh5*w10(i))*dh2); w5(i) = w5(i)-sigma*(h3-1)*dh3*h1; w6(i) = w6(i)-sigma*(h4-0)*dh4*h1; w7(i) = w7(i)-sigma*(h5-0)*dh5*h1; w8(i) = w8(i)-sigma*(h3-1)*dh3*h2; w9(i) = w9(i)-sigma*(h4-0)*dh4*h2; w10(i) = w10(i)-sigma*(h5-0)*dh5*h2; B2(i) =B2(i)- sigma*((h3-1)*dh3+(h4-0)*dh4+(h5-0)*dh5); elseif i>n && i<=2*n w1(i) = w1(i) -sigma * ((h3-0)*dh3*w5(i)+(h4-1)*dh4*w6(i)+(h5-0)*dh5*w7(i)) * dh1*X(i); w2(i) = w2(i) -sigma * ((h3-0)*dh3*w8(i)+(h4-1)*dh4*w9(i)+(h5-0)*dh5*w10(i)) * dh2*X(i); w3(i) = w3(i) -sigma * ((h3-0)*dh3*w5(i)+(h4-1)*dh4*w6(i)+(h5-0)*dh5*w7(i)) * dh1*Y(i); w4(i) = w4(i) -sigma * ((h3-0)*dh3*w8(i)+(h4-1)*dh4*w9(i)+(h5-0)*dh5*w10(i)) * dh2*Y(i); B1(i) = B1(i)- sigma*(((h3-0)*dh3*w5(i)+(h4-1)*dh4*w6(i)+(h5-0)*dh5*w7(i))*dh1+((h3-0)*dh3*w8(i)+(h4-1)*dh4*w9(i)+(h5-0)*dh5*w10(i))*dh2); w5(i) = w5(i)-sigma*(h3-0)*dh3*h1; w6(i) = w6(i)-sigma*(h4-1)*dh4*h1; w7(i) = w7(i)-sigma*(h5-0)*dh5*h1; w8(i) = w8(i)-sigma*(h3-0)*dh3*h2; w9(i) = w9(i)-sigma*(h4-1)*dh4*h2; w10(i) = w10(i)-sigma*(h5-0)*dh5*h2; B2(i) =B2(i)- sigma*((h3-0)*dh3+(h4-1)*dh4+(h5-0)*dh5); else w1(i) = w1(i) -sigma * ((h3-0)*dh3*w5(i)+(h4-0)*dh4*w6(i)+(h5-1)*dh5*w7(i)) * dh1*X(i); w2(i) = w2(i) -sigma * ((h3-0)*dh3*w8(i)+(h4-0)*dh4*w9(i)+(h5-1)*dh5*w10(i)) * dh2*X(i); w3(i) = w3(i) -sigma * ((h3-0)*dh3*w5(i)+(h4-0)*dh4*w6(i)+(h5-1)*dh5*w7(i)) * dh1*Y(i); w4(i) = w4(i) -sigma * ((h3-0)*dh3*w8(i)+(h4-0)*dh4*w9(i)+(h5-1)*dh5*w10(i)) * dh2*Y(i); B1(i) = B1(i)- sigma*(((h3-0)*dh3*w5(i)+(h4-0)*dh4*w6(i)+(h5-1)*dh5*w7(i))*dh1+((h3-0)*dh3*w8(i)+(h4-0)*dh4*w9(i)+(h5-1)*dh5*w10(i))*dh2); w5(i) = w5(i)-sigma*(h3-0)*dh3*h1; w6(i) = w6(i)-sigma*(h4-0)*dh4*h1; w7(i) = w7(i)-sigma*(h5-1)*dh5*h1; w8(i) = w8(i)-sigma*(h3-0)*dh3*h2; w9(i) = w9(i)-sigma*(h4-0)*dh4*h2; w10(i) = w10(i)-sigma*(h5-1)*dh5*h2; B2(i) =B2(i)- sigma*((h3-0)*dh3+(h4-0)*dh4+(h5-1)*dh5); end end end plot(P(:,1),P(:,2),'o'); hold on; flag = 0; M=[]; for x=-8:0.3:8 for y=-8:0.3:8 % x=-1; % y=2; H=[]; for i=1:3*n y1 = x*w1(i) + y*w3(i) + B1(i); y2 = x*w2(i) + y*w4(i) + B1(i); h1 = 1/(1+exp(-y1)); h2 = 1/(1+exp(-y2)); dh1 = h1*(1-h1); dh2 = h2*(1-h2); y3 = h1*w5(i) + h2*w8(i)+ B2(i); y4 = h1*w6(i) + h2*w9(i)+ B2(i); y5 = h1*w7(i) + h2*w10(i)+ B2(i); h3 = 1/(1+exp(-y3)); h4 = 1/(1+exp(-y4)); h5 = 1/(1+exp(-y5)); H=[H;h3 h4 h5]; end % H1 = mean(H(1:n,1)); % H2 = mean(H(n+1:2*n,2)); % H3 = mean(H(2*n+1:3*n,3)); meanH = mean(H); H1 = meanH(1); H2 = meanH(2); H3= meanH(3); M=[M;H1 H2 H3 x y]; if H1>H2 && H1>H3 plot(x,y,'g.') elseif H2 > H1 && H2 > H3 plot(x,y,'r.') elseif H3 > H1 && H3 > H2 plot(x,y,'b.') end end end
後面我計劃對網路分別使用softmax，權重初始化，正則化，ReLu啟用函式，交叉熵代價函式與卷積的形式進行優化。