MATLAB: NARXNET closed-loop vs open-loop

Deep Learning Toolboxnarxnetneural network

I have three questions regarding the difference between a closed-loop and an open-loop narxnet, and it's behavior.
First, a little about the problem I'm trying to solve. I have an 2xN dimensional matrix of observation (X), from which I'm trying to predict an output Y, 1xN. Now, a narxnet takes as input both X and Y. The Matlab documentation says that an open-loop narxnet finds a function 'f' where y(t) = f( y(t-1), y(t-2), x(t-1), x(t-2) ), for a delay of 2. However, the results that I get are much more accurate than I expect them to be. This suggests that the narxnet uses the actual y(t) as input as well. When I convert the open-loop to a closed-loop, and retrain it, I get much more reasonable results, not good, but reasonable.
1.a) What is the actual input to an open-loop narxnet, and the closed-loop. When trying to predict y(t), are the inputs [ y(t), y(t-1), y(t-2), x(t-1), x(t-2) ] or [ y(t-1), y(t-2), x(t-1), x(t-2) ], for both?
1.b) For my problem I need the inputs to be [ y(t-1), y(t-2), x(t), x(t-1), x(t-2) ], 0 to 2 delays on the X, and 1 to 2 delays on the Y. How can I do that?
2. For comparison, I've trained my open-loop, and tested it, then convert the open-loop to a closed-loop using
    netc = closeloop(net);
then trained the closed-loop from scratch and tested it. I've read on MATLAB Answers that I should start training the closed-loop with the weights from the open-loop net. How can I used the weights of one net as the initial weights for training another?
3. I understand that the narxnet uses a combination of NNs and difference equations. I problem I run into when using a closed-loop narxnet is that my predictions begin to oscillate and explode exponential. I know that this problem occurs in a difference equation, such as y(t)=a*y(t-1)+b, when 'a' is greater than 1. What, in a closed-loop narxnet could cause this similar issue?
I didn't include any code because my questions need high-level understanding and explanations, which I am lacking.
A huge thank you in advance,
– Yevgeniy

Best Answer

 % NARXNET closed-loop vs open-loop
 % Asked by Yevgeniy Arabadzhi 
 % 8:38 PM Thursday, November 3, 2016 
 %  
% I have three questions regarding the difference between 
% a closed-loop and an open-loop narxnet, and it's behavior.
% 
% First, a little about the problem I'm trying to solve. I 
% have an 2xN dimensional matrix of observation (X), from 
% which I'm trying to predict an output Y, 1xN. Now, a 
% narxnet takes as input both X and Y. The Matlab 
% documentation says that an open-loop narxnet finds a 
% function 'f' where y(t) = f( y(t-1), y(t-2), x(t-1), 
% x(t-2) ), for a delay of 2.
What you are overlooking is the role of the target; In fact, you never mention the target!
The target is used to design and operate the OL net. To emphasize this point I prefer to use the following notation:
 Uppercase for cells
 Lowercase for doubles and integers
 X,x for inputs
 T,t for targets
 Y,y for outputs
 i,j for timestep indices
% However, the results that I % get are much more accurate than I expect them to be. % This suggests that the narxnet uses the actual y(t) % as input as well.
ABSOLUTELY NOT! You just happened to have an easy problem (which you did not identify!). There are a variety of sample datasets in the help and doc documentations. Use the commands
 help nndatasets
 doc nndatsets
% When I convert the open-loop to a closed-loop, and retrain it, % I get much more reasonable results, not good, but reasonable.
Typically, when you convert from OL to CL and compare the results over the length of the target, you will get a spectrum of results: from excellent to lousy. Again, this will be demonstated if you try some of the sample datasets.
I have posted tutorials on narxnet design. I gather from this post that you are not familiar with them.
% 1.a) What is the actual input to an open-loop narxnet, % and the closed-loop. When trying to predict y(t), are the % inputs [ y(t), y(t-1), y(t-2), x(t-1), x(t-2) ] or % [ y(t-1), y(t-2), x(t-1), x(t-2) ], for both?
 ID = 1:2, FD = 1:2, H = 10    % DEFAULTs
 OL: y(i) =  f(x(i-1),x(i-2),t(i-1),t(i-2)); i = 3:N
 CL: y(i) =  f(x(i-1),x(i-2),y(i-1),y(i-2)); i = 3:N
 ID = 0:2, FD = 0:2, H = 10    
 OL: y(i) =  f(x(i),x(i-1),x(i-2),t(i),t(i-1),t(i-2)); i=3:N
 CL: y(i) =  f(x(i),x(i-1),x(i-2),y(i-1),y(i-2)); i = 3:N
NOTE THAT y(i) FEEDBACK IS NOT ALLOWED FOR CL !
% 1.b) For my problem I need the inputs to be [ y(t-1), % y(t-2), x(t), x(t-1), x(t-2) ], 0 to 2 delays on the X, % and 1 to 2 delays on the Y. How can I do that?
See above
% 2. For comparison, I've trained my open-loop, and tested % it, then convert the open-loop to a closed-loop using % netc = closeloop(net);
Next, compare the OL and CL on the OL design data.
% then trained the closed-loop from scratch and tested it. % I've read on MATLAB Answers that I should start training % the closed-loop with the weights from the open-loop net. % How can I used the weights of one net as the initial % weights for training another?
netc = closeloop(neto); netc = train(netc,X,Xoi,Aoi);
% 3. I understand that the narxnet uses a combination of % NNs and difference equations. I problem I run into when % using a closed-loop narxnet is that my predictions begin % to oscillate and explode exponential. I know that this % problem occurs in a difference equation, such as % y(t)=a*y(t-1)+b, when 'a' is greater than 1. What, in a % closed-loop narxnet could cause this similar issue?
Since the output containing errors is fed back to the input, the cause is, basically, similar.
% I didn't include any code because my questions need % high-level understanding and explanations, which I am % lacking. % % A huge thank you in advance, % % Yevgeniy
No, thank you. I'm sure more people than you realize will be thankful for this post.
Hope this helps
Greg
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