nnarmax2.m

类神经网路─MATLAB的应用(范例程式)

function [W1,W2,PI_vector,iteration,lambda]=nnarmax2(NetDef,NN,W1,W2,trparms,skip,Y,U)
%  NNARMAX2
%  --------
%          [W1,W2,critvec,iteration,lambda]=...
%                                   nnarmax2(NetDef,NN,W1,W2,trparms,skip,Y,U)
%          Determines a nonlinear ARMAX model
%                          y(t)=f(y(t-1),...,u(t-k),...,e(t-1),...)
%          of a dynamic system by training a two layer neural network with
%          the Marquardt method. The function can handle multi-input
%          systems (MISO). 
%
%  INPUTS:
%  U       : Input signal (= control signal) (left out in the nnarma case)
%            dim(U) = [(inputs) * (# of data)]
%  Y       : Output signal. dim(Y) = [1 * # of data]
%  NN      : NN=[na nb nc nk].
%            na = # of past outputs used for determining the prediction
%            nb = # of past inputs
%            nc = # of past residuals (= order of C)
%            nk = time delay (usually 1)
%            For multi-input systems nb and nk contain as many columns as
%            there are inputs.
%  W1,W2   : Input-to-hidden layer and hidden-to-output layer weights.
%            If they are passed as [] they are initialized automatically
%  trparms : Contains parameters associated with the training (see MARQ)
%            if trparms=[] it is reset to trparms = [500 0 1 0]
%  skip    : Don't use the first 'skip' samples for training in order to
%            reduce the influence from the transient occuring because of the
%            unknown initial prediction error and gradient. If skip=[]
%            it is reset to skip=0.
%
%           For time series (NNARMA models) NN=[na nc] only.
% 
%
%  NB NB NB! 
%  --------- 
%  See the function MARQ for an explanation of the remaining input arguments
%  as well as of the returned variables.
%
%                                                                                
%  Programmed by : Magnus Norgaard, IAU/IMM
%  LastEditDate  : July 16, 1996

%----------------------------------------------------------------------------------
%--------------             NETWORK INITIALIZATIONS                   -------------
%----------------------------------------------------------------------------------
if isempty(skip),
  skip=0;
end
skip=skip+1;
Ndat     = length(Y);                   % # of data
na = NN(1);
if length(NN)==2                        % nnarma model
  nc     = NN(2);
  nb     = 0;
  nk     = 0;
  nu     = 0;
else                                    % nnarmax model
  [nu,Ndat]= size(U); 
  nb     = NN(2:1+nu);
  nc     = NN(2+nu);
  nk     = NN(2+nu+1:2+2*nu);
end
nmax     = max([na,nb+nk-1,nc]);        % 'Oldest' signal used as input to the model
N        = Ndat - nmax;                 % Size of training set
N2       = N-skip+1;
nab      = na+sum(nb);                  % na+nb
nabc     = nab+nc;                      % na+nb+nc
hidden   = length(NetDef(1,:));         % Number of hidden neurons
inputs   = nabc;                        % Number of inputs to the network
outputs  = 1;                           % Only one output 
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
y1       = zeros(hidden,N);             % Hidden layer outputs
y1       = [y1;ones(1,N)];
y2       = zeros(outputs,N);            % Network output
E        = zeros(outputs,N);            % Initialize prediction error vector
E_new    = zeros(outputs,N);            % Initialize prediction error vector
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
parameters1= hidden*(inputs+1);         % # of input-to-hidden weights
parameters2= outputs*(hidden+1);        % # of hidden-to-output weights
parameters=parameters1 + parameters2;   % Total # of weights
ones_h   = ones(hidden+1,1);            % A vector of ones
ones_i   = ones(inputs+1,1);            % Another vector of ones
if isempty(W1) | isempty(W2),           % Initialize weights if nescessary
  if nb==0,
    [W1,W2]=nnarx(NetDef,na,[],[],[100 trparms(2:length(trparms))],Y);
  else
    [W1,W2]=nnarx(NetDef,[na nb nk],[],[],[100 trparms(2:length(trparms))],Y,U);
  end
  W1=[W1(:,1:nab),0.05*rand(hidden,nc)-0.025, W1(:,nab+1)];                          
end
                                        % 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
dy2de    = zeros(nc,N);                 % Der. of outputs wrt. the past residuals
index4   = nab+1:nabc;                  % And a fourth
PSI_red  = zeros(reduced,N);            % Deriv. of output w.r.t. each weight
RHO      = zeros(parameters,N);         % Partial -"-  -"-

if isempty(trparms),                    % Default training parameters
  max_iter  = 500;
  stop_crit = 0;
  lambda    = 1;
  D         = 0;
else                                    % User specified values
  max_iter  = trparms(1);
  stop_crit = trparms(2);
  lambda    = trparms(3);
  if length(trparms)==4,                % Scalar weight decay parameter
    D = trparms(4*ones(1,reduced))';      
  elseif length(trparms)==5,            % Two weight decay parameters
    D = trparms([4*ones(1,parameters2+nc) 5*ones(1,parameters1)])';
    D = D(theta_index);
  elseif length(trparms)>5,             % Individual weight decay
    D = trparms(4:length(trparms))';
  end
end
PI_vector = zeros(max_iter,1);          % A vector containing the accumulated SSE


% >>>>>>>>>>>>>>>>>>>>  CONSTRUCT THE REGRESSION MATRIX PHI   <<<<<<<<<<<<<<<<<<<<<
PHI = zeros(nabc,N);
jj  = nmax+1:Ndat;
for k = 1:na, PHI(k,:)    = Y(jj-k); end
index5 = na;
for kk = 1:nu,
  for k = 1:nb(kk), PHI(k+index5,:) = U(kk,jj-k-nk(kk)+1); end
  index5 = index5 + nb(kk);
end
PHI_aug = [PHI;ones(1,N)];              % Augment PHI with a row containg ones
Y       = Y(nmax+1:Ndat);               % Extract the 'target' part of Y


%----------------------------------------------------------------------------------
%--------------                   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)   <<<<<<<<<<<<<<<<<<<<<<
for t=1:N,
  h1 = W1*PHI_aug(:,t);  
  y1(H_hidden,t) = pmntanh(h1(H_hidden));
  y1(L_hidden,t) = h1(L_hidden);    

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

  E(:,t) = Y(:,t) - y2(:,t);            % Prediction error
  for d=1:min(nc,N-t),
    PHI_aug(nab+d,t+d) = E(:,t);
  end
end

SSE      = E(skip:N)*E(skip:N)';        % Sum of squared errors (SSE)
PI       = (SSE+theta_red'*(D.*theta_red))/(2*N2); % Performance index


while iteration<=max_iter    
if dw==1,
% >>>>>>>>>>>>>>>>>>>>>>>>>>>   COMPUTE THE RHO MATRIX   <<<<<<<<<<<<<<<<<<<<<<<<<<
% Partial derivative of output (y2) with respect to each weight and neglecting
% that the model inputs (the residuals) depends on the weights

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

    % -- The part of RHO corresponding to hidden-to-output layer weights --
    RHO(index1:index1+hidden,index2+i) = y1;
    % ---------------------------------------------------------------------
 
    % -- The part of RHO corresponding to input-to-hidden layer weights ---
    for j = L_hidden',
      RHO(index(j):index(j)+inputs,index2+i) = W2(i,j)*PHI_aug;
    end
     
    for j = H_hidden',
      tmp = W2(i,j)*(1-y1(j,:).*y1(j,:)); 
      RHO(index(j):index(j)+inputs,index2+i) = tmp(ones_i,:).*PHI_aug;
    end 
    % ---------------------------------------------------------------------    
  end

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

    % -- The part of RHO corresponding to hidden-to-output layer weights --
    tmp = 1 - y2(i,:).*y2(i,:);
    RHO(index1:index1+hidden,index2+i) = y1.*tmp(ones_h,:);
    % ---------------------------------------------------------------------
         
    % -- The part of RHO corresponding to input-to-hidden layer weights ---
    for j = L_hidden',
      tmp = W2(i,j)*(1-y2(i,:).*y2(i,:));
      RHO(index(j):index(j)+inputs,index2+i) = tmp(ones_i,:).* PHI_aug;
    end
      
    for j = H_hidden',
      tmp  = W2(i,j)*(1-y1(j,:).*y1(j,:));
      tmp2 = (1-y2(i,:).*y2(i,:));
      RHO(index(j):index(j)+inputs,index2+i) = tmp(ones_i,:)...
                                                .*tmp2(ones_i,:).* PHI_aug;
      end
      % ---------------------------------------------------------------------
    end 
    RHO_red = RHO(theta_index(1:reduced),:);


% >>>>>>>>>>>>>>>>>>>>>>>>>>>   COMPUTE THE PSI MATRIX   <<<<<<<<<<<<<<<<<<<<<<<<<<
    % ---------- Find derivative of output wrt. the past residuals ----------
    for t=1:N,
      dy2dy1 = W2(:,1:hidden);
      for j = H_output',
        dy2dy1(j,:) = W2(j,1:hidden)*(1-y2(j,t).*y2(j,t));
      end
      
      % Matrix with partial derivatives of the output from each hidden neurons with
      % respect to each input:
      dy1de = W1(:,index4);
      for j = H_hidden',
        dy1de(j,:) = W1(j,index4)*(1-y1(j,t).*y1(j,t));
      end

      % Matrix with partial derivative of each output with respect to each input
      dy2de(:,t)= (dy2dy1 * dy1de)';
    end


    % ---------- Determine PSI by "filtering" ----------
    for t=1:N,
      PSI_red(:,t)=RHO_red(:,t);
      for t1=1:min(nc,t-1),
        PSI_red(:,t)  = PSI_red(:,t)-dy2de(t1,t)*PSI_red(:,t-t1);
      end
    end
 
   
% >>>>>>>>>>>>>>>>>>>>>>>>>>>        COMPUTE h_k        <<<<<<<<<<<<<<<<<<<<<<<<<<<
    % -- Gradient --
    G = PSI_red(:,skip:N)*E(skip:N)'-D.*theta_red;
    
    % -- Mean square error part of Hessian  --
    R = PSI_red(:,skip:N)*PSI_red(:,skip:N)';
    dw = 0;
  end
  
  % -- Hessian  --
  H = R;
  H(index3) = H(index3)'+lambda+D;                  % 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)   <<<<<<<<<<<<<<<<<<<<
for t=1:N,
  h1 = W1_new*PHI_aug(:,t);  
  y1(H_hidden,t) = pmntanh(h1(H_hidden));
  y1(L_hidden,t) = h1(L_hidden);    

  h2 = W2_new*y1(:,t);
  y2(H_output,t) = pmntanh(h2(H_output,:));
  y2(L_output,t) = h2(L_output,:);

  E_new(:,t) = Y(:,t) - y2(:,t);          % Prediction error
  for d=1:min(nc,N-t),
    PHI_aug(nab+d,t+d) = E_new(:,t);
  end
end

SSE_new  = E_new(skip:N)*E_new(skip:N)';% Sum of squared errors (SSE)
PI_new   = (SSE_new + theta_red_new'*(D.*theta_red_new))/(2*N2); % PI
    

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

  % Decrease lambda if SSE has fallen 'sufficiently'
  if 2*N2*(PI - PI_new) > (0.75*L),
    lambda = lambda/2;
  
  % Increase lambda if SSE has grown 'sufficiently'
  elseif 2*N2*(PI-PI_new) <= (0.25*L),
    lambda = 2*lambda;
  end


% >>>>>>>>>>>>>>>>>>>>       UPDATES FOR NEXT ITERATION        <<<<<<<<<<<<<<<<<<<<
  % Update only if criterion has decreased
  if PI_new < PI,                      
    W1 = W1_new;
    W2 = W2_new;
    theta_red = theta_red_new;
    E = E_new;
    PI = PI_new;
    dw = 1;
    iteration = iteration + 1;
    PI_vector(iteration-1) = PI;                             % Collect PI in vector
    fprintf('iteration # %i   PI = %4.3e\r',iteration-1,PI); % Print on-line inform
  end

  % Check if stop condition is fulfilled
  if (PI < stop_crit) | (lambda>1e7), break, end             
end
%----------------------------------------------------------------------------------
%--------------              END OF NETWORK TRAINING                  -------------
%----------------------------------------------------------------------------------
PI_vector = PI_vector(1:iteration-1);
c=fix(clock);
fprintf('\n\nNetwork training ended at %2i.%2i.%2i\n',c(4),c(5),c(6));