marq.m

来自「基于MATLAB的神经网络非线性系统辨识软件包.」· M 代码 · 共 308 行

function [W1,W2,PI_vector,iteration,lambda]=marq(NetDef,W1,W2,PHI,Y,trparms)
%  MARQ
%  ----
%              Train a two layer neural network with the Levenberg-Marquardt
%              method.
%
%          If desired, it is possible to use regularization by
%          weight decay. Also pruned (ie. not fully connected) networks can
%          be trained.
%
%          Given a set of corresponding input-output pairs and an initial
%          network,
%          [W1,W2,critvec,iteration,lambda]=marq(NetDef,W1,W2,PHI,Y,trparms)
%          trains the network with the Levenberg-Marquardt method.
% 
%          The activation functions can be either linear or tanh. The
%          network architecture is defined by the matrix 'NetDef' which
%          has two rows. The first row specifies the hidden layer and the
%          second row specifies the output layer.
% 
%          E.g.:    NetDef = ['LHHHH' 
%                             'LL---']
%          (L = Linear, H = tanh)
%
%          A weight is pruned by setting it to zero.
%
%          The Marquardt method is described in:
%          K. Madsen: 'Optimering' (Haefte 38), IMM, DTU, 1991
%  
%          Notice that the bias is included as the last column in the weight
%          matrices.
%
% 
%  INPUT:
%  NetDef : Network definition .
%  W1     : Input-to-hidden layer weights. The matrix dimension is
%           [(# of hidden units)-by-(inputs + 1)] (the 1 is due to the bias).
%           Use [] for a random initialization.
%  W2     : hidden-to-output layer weights. Dimension is
%           [(outputs)  *  (# of hidden units + 1)].
%           Use [] for a random initialization.
%  PHI    : Input vector. dim(PHI) = [(inputs)  *  (# of data)].
%  Y      : Output data. dim(Y) = [(outputs)  * (# of data)].
%  trparms: Data structure with parameters associated with the
%           training algorithm (optional). Use the function SETTRAIN if
%           you do not want to use the default values.
% 
%  OUTPUT:
%  W1, W2   : Weight matrices after training.
%  critvec:   Vector containing the criterion evaluated at each iteration
%  iteration: # of iterations
%  lambda   : The final value of lambda. Relevant only if retraining is desired

%  Written by : Magnus Norgaard, IAU/IMM, Tecnical University of Denmark
%  LastEditDate  : Jan 15, 2000

%----------------------------------------------------------------------------------
%--------------             NETWORK INITIALIZATIONS                   -------------
%----------------------------------------------------------------------------------
[outputs,N] = size(Y);                  % # of outputs and # of data
[inputs,N] = size(PHI);                 % # of hidden units
L_hidden = find(NetDef(1,:)=='L')';     % Location of linear hidden neurons
H_hidden = find(NetDef(1,:)=='H')';     % Location of tanh hidden neurons
L_output = find(NetDef(2,:)=='L')';     % Location of linear output neurons
H_output = find(NetDef(2,:)=='H')';     % Location of tanh output neurons
hidden = length(L_hidden)+length(H_hidden);
if isempty(W1) | isempty(W2),           % Initialize weights if nescessary
  W1 = rand(hidden,inputs+1)-0.5;
  W2 = rand(outputs,hidden+1)-0.5;
end
if (size(W1,2)~=inputs+1 | size(W1,1)~=hidden |... % Check dimensions
      size(W2,2)~=hidden+1 | size(W2,1)~=outputs)
   error('Dimension mismatch in weights, data, or NetDef.');
end
y1       = [zeros(hidden,N);ones(1,N)]; % Hidden layer outputs
y2       = zeros(outputs,N);            % Network output
index = outputs*(hidden+1) + 1 + [0:hidden-1]*(inputs+1); % A useful vector!
index2 = (0:N-1)*outputs;               % Yet another useful vector
iteration = 1;                          % Counter variable
dw       = 1;                           % Flag telling that the weights are new
PHI      = [PHI;ones(1,N)];             % Augment PHI with a row containg ones
parameters1= hidden*(inputs+1);         % # of input-to-hidden weights
parameters2= outputs*(hidden+1);        % # of hidden-to-output weights
parameters = parameters1 + parameters2; % Total # of weights
PSI      = zeros(parameters,outputs*N); % Deriv. of each output w.r.t. each weight
ones_h   = ones(hidden+1,1);            % A vector of ones
ones_i   = ones(inputs+1,1);            % Another vector of ones
                                        % Parameter vector containing all weights
theta = [reshape(W2',parameters2,1) ; reshape(W1',parameters1,1)];
theta_index = find(theta);              % Index to weights<>0
theta_red = theta(theta_index);         % Reduced parameter vector
reduced  = length(theta_index);         % The # of parameters in theta_red
index3   = 1:(reduced+1):(reduced^2);   % A third useful vector
lambda_old = 0;
if nargin<6 | isempty(trparms) % Default training parameters
  trparms = settrain;
  lambda  = trparms.lambda;
  D       = trparms.D;
else                                    % User specified values
  if ~isstruct(trparms),
     error('''trparms'' must be a structure variable.');
  end
  if ~isfield(trparms,'infolevel')
     trparms = settrain(trparms,'infolevel','default');
  end
  if ~isfield(trparms,'maxiter')
     trparms = settrain(trparms,'maxiter','default');
  end
  if ~isfield(trparms,'critmin')
     trparms = settrain(trparms,'critmin','default');
  end
  if ~isfield(trparms,'critterm')
     trparms = settrain(trparms,'critterm','default');
  end
  if ~isfield(trparms,'gradterm')
     trparms = settrain(trparms,'gradterm','default');
  end
  if ~isfield(trparms,'paramterm')
     trparms = settrain(trparms,'paramterm','default');
  end
  if ~isfield(trparms,'lambda')
     trparms = settrain(trparms,'lambda','default');
  end
  lambda    = trparms.lambda;
  if ~isfield(trparms,'D')
     trparms = settrain(trparms,'D','default');
     D = trparms.D;
  else
    if length(trparms.D)==1,              % Scalar weight decay parameter
      D = trparms.D(ones(1,reduced));      
    elseif length(trparms.D)==2,          % Two weight decay parameters
      D = trparms.D([ones(1,parameters2) 2*ones(1,parameters1)])';
      D = D(theta_index);
    elseif length(trparms.D)>2,           % Individual weight decay
      D = trparms.D(:);
    end
  end
end
D = D(:);
critdif  = trparms.critterm+1;            % Initialize stopping variables
gradmax  = trparms.gradterm+1;
paramdif = trparms.paramterm+1;
PI_vector = zeros(trparms.maxiter,1);     % Vector for storing criterion values


%----------------------------------------------------------------------------------
%--------------                   TRAIN NETWORK                       -------------
%----------------------------------------------------------------------------------
clc;
c=fix(clock);
fprintf('Network training started at %2i.%2i.%2i\n\n',c(4),c(5),c(6));


% >>>>>>>>>>>>>>>>>>>>>  COMPUTE NETWORK OUTPUT  y2(theta)   <<<<<<<<<<<<<<<<<<<<<<
h1 = W1*PHI;  
y1(H_hidden,:) = pmntanh(h1(H_hidden,:));
y1(L_hidden,:) = h1(L_hidden,:);    

h2 = W2*y1;
y2(H_output,:) = pmntanh(h2(H_output,:));
y2(L_output,:) = h2(L_output,:);

E        = Y - y2;                      % Training error
E_vector = E(:);                        % Reshape E into a long vector
SSE      = E_vector'*E_vector;          % Sum of squared errors (SSE)
PI       = (SSE+theta_red'*(D.*theta_red))/(2*N); % Performance index

% Iterate until stopping criterion is satisfied
while (iteration<=trparms.maxiter & PI>trparms.critmin & lambda<1e7 & ...
       (critdif>trparms.critterm | gradmax>trparms.gradterm | ...
       paramdif>trparms.paramterm))
    
    if dw==1,
   % >>>>>>>>>>>>>>>>>>>>>>>>>   COMPUTE THE PSI MATRIX   <<<<<<<<<<<<<<<<<<<<<<<<<
   % (The derivative of each network output (y2) with respect to each weight)

    % ==========   Elements corresponding to the linear output units   ============
    for i = L_output'
      index1 = (i-1) * (hidden + 1) + 1;

      % -- The part of PSI corresponding to hidden-to-output layer weights --
      PSI(index1:index1+hidden,index2+i) = y1;
      % ---------------------------------------------------------------------
 
      % -- The part of PSI corresponding to input-to-hidden layer weights ---
      for j = L_hidden',
        PSI(index(j):index(j)+inputs,index2+i) = W2(i,j)*PHI;
      end

      for j = H_hidden',
        tmp = W2(i,j)*(1-y1(j,:).*y1(j,:)); 
        PSI(index(j):index(j)+inputs,index2+i) = tmp(ones_i,:).*PHI;
      end 
      % ---------------------------------------------------------------------    
    end

    % ============  Elements corresponding to the tanh output units   =============
    for i = H_output',
      index1 = (i-1) * (hidden + 1) + 1;

      % -- The part of PSI corresponding to hidden-to-output layer weights --
      tmp = 1 - y2(i,:).*y2(i,:);
      PSI(index1:index1+hidden,index2+i) = y1.*tmp(ones_h,:);
      % ---------------------------------------------------------------------
     
      % -- The part of PSI corresponding to input-to-hidden layer weights ---
      for j = L_hidden',
        tmp = W2(i,j)*(1-y2(i,:).*y2(i,:));
        PSI(index(j):index(j)+inputs,index2+i) = tmp(ones_i,:).*PHI;
      end
      
      for j = H_hidden',
        tmp  = W2(i,j)*(1-y1(j,:).*y1(j,:));
        tmp2 = (1-y2(i,:).*y2(i,:));
        PSI(index(j):index(j)+inputs,index2+i) = tmp(ones_i,:)...
                                                  .*tmp2(ones_i,:).*PHI;
      end
      % ---------------------------------------------------------------------
    end
    PSI_red = PSI(theta_index,:);
    
    % -- Gradient --
    G = PSI_red*E_vector-D.*theta_red;

    % -- Means square error part Hessian  --
    H = PSI_red*PSI_red';
    H(index3) = H(index3)'+D;                       % Add diagonal matrix  
    dw = 0;
  end
  
   
% >>>>>>>>>>>>>>>>>>>>>>>>>>>        COMPUTE h_k        <<<<<<<<<<<<<<<<<<<<<<<<<<<
  % -- Hessian  --
  H(index3) = H(index3)'+(lambda-lambda_old);       % Add diagonal matrix

  % -- Search direction --
  h = H\G;                                          % Solve for search direction

  % -- Compute 'apriori' iterate --
  theta_red_new = theta_red + h;                    % Update parameter vector
  theta(theta_index) = theta_red_new;

  % -- Put the parameters back into the weight matrices --
  W1_new = reshape(theta(parameters2+1:parameters),inputs+1,hidden)';
  W2_new = reshape(theta(1:parameters2),hidden+1,outputs)';

    
% >>>>>>>>>>>>>>>>>>>>   COMPUTE NETWORK OUTPUT  y2(theta+h)   <<<<<<<<<<<<<<<<<<<<
  h1 = W1_new*PHI;  
  y1(H_hidden,:) = pmntanh(h1(H_hidden,:));
  y1(L_hidden,:) = h1(L_hidden,:);
    
  h2 = W2_new*y1;
  y2(H_output,:) = pmntanh(h2(H_output,:));
  y2(L_output,:) = h2(L_output,:);

  E_new        = Y - y2;                 % Training error
  E_new_vector = E_new(:);               % Reshape E into a long vector
  SSE_new  = E_new_vector'*E_new_vector; % Sum of squared errors (SSE)
  PI_new   = (SSE_new + theta_red_new'*(D.*theta_red_new))/(2*N); % PI


% >>>>>>>>>>>>>>>>>>>>>>>>>>>       UPDATE  lambda     <<<<<<<<<<<<<<<<<<<<<<<<<<<<
  L = h'*G + h'*(h.*(D+lambda));
  lambda_old = lambda;

  % Decrease lambda if SSE has fallen 'sufficiently'
  if 2*N*(PI - PI_new) > (0.75*L),
    lambda = lambda/2;
  
  % Increase lambda if SSE has grown 'sufficiently'
  elseif 2*N*(PI-PI_new) <= (0.25*L),
    lambda = 2*lambda;
  end


% >>>>>>>>>>>>>>>>>>>>       UPDATES FOR NEXT ITERATION        <<<<<<<<<<<<<<<<<<<<
  % Update only if criterion has decreased
  if PI_new < PI,
    critdif  = PI-PI_new;                           % Criterion difference
    gradmax  = max(abs(G))/N;                       % Maximum gradient
    paramdif = max(abs(theta_red_new - theta_red)); % Maximum parameter dif.
    W1 = W1_new;
    W2 = W2_new;
    theta_red = theta_red_new;
    E_vector = E_new_vector;
    PI = PI_new;
    dw = 1;
    lambda_old = 0;
    iteration = iteration + 1;
    PI_vector(iteration-1) = PI;                             % Collect PI in vector
    switch(trparms.infolevel)                                % Print on-line inform
       case 1
          fprintf('# %i   W=%4.3e  critdif=%3.2e  maxgrad=%3.2e  paramdif=%3.2e\n',...
                                                  iteration-1,PI,critdif,gradmax,paramdif);
       otherwise
          fprintf('iteration # %i   W = %4.3e\r',iteration-1,PI);
    end
  end
end
%----------------------------------------------------------------------------------
%--------------              END OF NETWORK TRAINING                  -------------
%----------------------------------------------------------------------------------
iteration = iteration-1;
PI_vector = PI_vector(1:iteration);
c=fix(clock);
fprintf('\n\nNetwork training ended at %2i.%2i.%2i\n',c(4),c(5),c(6));