基于标准反向传播算法的改进BP神经网络算法（Matlab代码实现）
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目录
1 概述
2 运行结果
3 参考文献
4 Matlab代码实现
文献来源，然后复现之：
1 概述
SBP算法已成为用于训练多层感知器的标准算法，如图1所示。它是一种广义最小均方 （LMS） 算法，它最小化等于实际输出和所需输出之间误差平方和的标准。这个标准是：
详细文章下载链接：
https://ieeexplore.ieee.org/document/914537
2 运行结果
W1W1 =-0.1900 -0.7425 -2.9507-0.3955 0.2059 -0.8937-0.4751 0.5315 -1.2644-2.5390 -1.8319 3.2741-1.0816 -0.6534 0.7954-1.4622 -0.0331 -0.4283-0.3125 0.0840 -0.8470-0.6496 -0.2922 -0.6272b1b1 =9.42431.54310.18602.46300.10970.53901.6335-0.4229W2W2 =-1.6282 0.5796 0.2008 1.0366 0.9238 -0.3099 0.6122 -0.0681b2b2 =0.3461Mean Error Square at Iter = 2000eSq =0.0016eSq_v =0.0022eSq_t =0.0017Trained时间已过 1540.739160 秒。No. of Iterations = 2001Final Mean Squared Error at Iter = 2001eSq =0.0016>>
部分代码：
%****Load the Input File******
load ./nnm_train.txt
redData = nnm_train(:,2);
nir1Data = [nnm_train(:,3) ./ redData]';
nir2Data = [nnm_train(:,4) ./ redData]';
nir3Data = [nnm_train(:,5) ./ redData]';
pg = [ nir1Data; nir2Data; nir3Data];
targetData = nnm_train(:,7) ;
%*******Validate Data*******
load ./nnm_validate.txt
redData_v = nnm_validate(:,2);
nir1Data_v = [nnm_validate(:,3) ./ redData_v]';
nir2Data_v = [nnm_validate(:,4) ./ redData_v]';
nir3Data_v = [nnm_validate(:,5) ./ redData_v]';
targetData_v = nnm_validate(:,7) ;
pValidate = [nir1Data_v; nir2Data_v; nir3Data_v];
%*******Test Data*******
load ./nnm_test.txt
redData_t = nnm_test(:,2);
nir1Data_t = [nnm_test(:,3) ./ redData_t]';
nir2Data_t = [nnm_test(:,4) ./ redData_t]';
nir3Data_t = [nnm_test(:,5) ./ redData_t]';
targetData_t = nnm_test(:,7) ;
pTest = [nir1Data_t; nir2Data_t; nir3Data_t];
%---Plot the Original Function----
pa = 1 : length(targetData_t);
actLine = 0:0.1:0.8;
subplot(2,1,2), plot (actLine, actLine); legend('Actual');%scatter(targetData_t, targetData_t,'^b');
hold on
%-----Randomized First Layer Weights & Bias-------
fprintf( 'Initial Weights and Biases');
%****3-8-1******
W1 = [ -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand; -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand;...
-0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand]'; %Uniform distribution [-0.5 0.5]
b1 = [ -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand]';
W2 = [ -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand];
%-----Randomized Second Layer Bias------
b2 = [ -0.5+rand ];
if (lambda == 0) % Save the Weights and Bias on SBP
W1_initial = W1;
b1_initial = b1;
W2_initial = W2;
b2_initial = b2;
else % Reuse the Weights and Bias on MBP
W1 = W1_initial;
b1 = b1_initial;
W2 = W2_initial;
b2 = b2_initial;
end
%-----RandPermutation of Input Training Set-------
j = randperm(length(targetData));
j_v = randperm(length(targetData_v));
j_t = randperm(length(targetData_t));
%--Set Max. Iterations---
maxIter = 2000;
tic
for train = 1 : maxIter +1
eSq = 0; eSq_v = 0; eSq_t = 0;
% **** Mean Square Error ****
%if ( train <= maxIter )
for p = 1 : length(targetData)
n1 = W1*pg(:,p)+ b1 ;
a1 = logsig(n1);
a2 = poslin( W2 * a1 + b2 );
e = targetData(p) - a2 ;
eSq = eSq + (e^2);
end
eSq = eSq/length(targetData);
%*******Validate Error**********
for p = 1 : length(targetData_v)
n1 = W1*pValidate(:,p)+ b1 ;
a1 = logsig(n1);
a2 = poslin( W2 * a1 + b2 );
e = targetData_v(p) - a2 ;
eSq_v = eSq_v + (e^2);
end
eSq_v = eSq_v/length(targetData_v);
%********Use Validate Error for Early Stopping********
if ( train > 200 )
earlyStopCount = earlyStopCount + 1;
% fprintf('EarlyStop = %d', earlyStopCount);
if (earlyStopCount == 50)
if ( (prev_eSq_v - eSq_v) < 0 )
W2 = W2_25;
b2 = b2_25;
W1 = W1_25;
b1 = b1_25;
break;
end
prev_eSq_v = eSq_v; % Store previous validation error
earlyStopCount = 0; % Reset Early Stopping
%----Save the weights and biases-------
disp('Saving'); eSq
W2_25 = W2;
b2_25 = b2;
W1_25 = W1;
b1_25 = b1;
end
else
% ----Initialize the Weights----
if ( train == 200 )
W2_25 = W2;
b2_25 = b2;
W1_25 = W1;
b1_25 = b1;
end
prev_eSq_v = eSq_v; % Store previous validation error
end
%*******Test Error**********
for p = 1 : length(targetData_t)
n1 = W1*pTest(:,p)+ b1 ;
a1 = logsig(n1);
a2 = poslin( W2 * a1 + b2 );
e = targetData_t(p) - a2 ;
eSq_t = eSq_t + (e^2);
end
eSq_t = eSq_t/length(targetData_t);
if (train == 1 || mod (train, 100) == 0 )
fprintf( 'Weights and Biases at Iter = %d\n',train);
fprintf('W1');
(W1)
fprintf('b1')
b1
fprintf('W2')
W2
fprintf('b2')
b2
fprintf( 'Mean Error Square at Iter = %d',train);
eSq
eSq_v
eSq_t
end
subplot(2,1,1),
xlabel('No. of Iterations');
ylabel('Mean Square Error');
title('Convergence Characteristics ');
loglog(train, eSq, '*r'); hold on
loglog(train, eSq_v, '*g'); hold on
loglog(train, eSq_t, '*c'); hold on
legend('Training Error', 'Validation Error', 'Testing Error');
%************Train Data**********************
% Update only when the error is decreasing
% if ( earlyStopCount == 0 )
for p = 1 : length(targetData)
%----Output of the 1st Layer-----------
n1 = W1*pg(:,j(p))+ b1 ;
a1 = logsig(n1) ;
%-----Output of the 2nd Layer----------
n2 = W2 * a1 + b2;
a2 = (poslin( n2 ));
%a2 = (logsig( n2 ));
%-----Error-----
t = targetData(j(p));
e = t - a2;
%******CALCULATE THE SENSITIVITIES************
%-----Derivative of logsig function----
%f1 = dlogsig(n1,a1)
% f1 = [(1-a1(1))*a1(1) 0; 0 (1-a1(2))*a1(2)] ;
f1 = diag((1-a1).*a1);
%-----Derivative of purelin function---
f2 = 1;
%f2 = diag((1-a2).*a2);
%------Last Layer (2nd) Sensitivity----
S2 = -2 * f2 * e;
S2mbp = ((t)-n2);
%------First Layer Sensitivity---------
S1 = f1 *(W2' * S2);
S1mbp = f1 * (W2' * S2mbp);
%******UPDATE THE WEIGHTS**********************
%-----Second Layer Weights & Bias------
W2 = W2 - (alpha * S2*(a1)') - (alpha * lambda * S2mbp *(a1)');
b2 = b2 - alpha * S2 - (alpha * lambda * S2mbp);
%-----First Layer Weights & Bias-------
W1 = W1 - alpha * S1*(pg(:,j(p)))' - (alpha * lambda * S1mbp *(pg(:,j(p)))');
b1 = b1 - alpha * S1 - (alpha * lambda * S1mbp );
% end
end
%end
% End of 21 Input Training Sets
% ********** Function Apporx. *****************
if (train == 1 || mod (train, 100) == 0 || train == maxIter )
disp('Trained');
subplot(2,1,2),
xlabel('Actual Fraction of Weeds in 3 sq feet of grass area');
ylabel('Estimated Fraction of Weeds in 3 sq feet of grass area');
title('Correlation of Estimated Value with respect to the Actual Function using Standard Backpropagation');
legend('Estimated');
for p = 1 : length(targetData_t)
n1 = W1*pTest(:,p)+ b1 ; % Test Data
a1 = logsig(n1) ;
a2(p) = (poslin( W2 * a1 + b2 ));
end %end for
%scatter(targetData_t, a2); hold on;
end
end
toc
%-------End of Iterations------------
%***Plot of Final Function******
subplot(2,1,2),
for p = 1 : length(targetData_t)
n1 = W1*pTest(:,p)+ b1 ; % Test Data
a1 = logsig(n1) ;
a2(p) = (poslin( W2 * a1 + b2 ));
end %end for
3 参考文献
[1]S. Abid, F. Fnaiech and M. Najim, "A fast feedforward training algorithm using a modified form of the standard backpropagation algorithm," in IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 424-430, March 2001, doi: 10.1109/72.914537.