Newsgroups: comp.ai.neural-nets
From: John@tpsujmfh.demon.co.uk (John R Guppy)
Path: cantaloupe.srv.cs.cmu.edu!nntp.club.cc.cmu.edu!godot.cc.duq.edu!news.duke.edu!MathWorks.Com!europa.eng.gtefsd.com!howland.reston.ans.net!pipex!demon!tpsujmfh.demon.co.uk!John
Subject: Re: Help with limit determination ???
References: <1704FA2F7S86.RVANRAAM@bcsc02.gov.bc.ca>
Organization: Myorganisation
Reply-To: John@tpsujmfh.demon.co.uk
X-Newsreader: Demon Internet Simple News v1.29
Lines: 61
Date: Sun, 16 Oct 1994 12:05:59 +0000
Message-ID: <782309159snz@tpsujmfh.demon.co.uk>
Sender: usenet@demon.co.uk

In article <1704FA2F7S86.RVANRAAM@bcsc02.gov.bc.ca>
           RVANRAAM@bcsc02.gov.bc.ca  writes:

> In my Neural network self study program I came up with the
> following problem:
>  
> ( A ) ------> ( R )
>          |--> (   )
> ( B ) ---|
>  
>  
> 1) A(i) = random number
> 2) B(i) = random number
> 3) R(i) = A(i)/constant1+ B(i)/constant2
>  
> 4) U(i+1) = -U(i) + (A(i)-A(i-1))x(R(i)-R(i-1))
> 5) V(i+1) = -V(i) + (B(i)-B(i-1))x(R(i)-R(i-1))
>  
> 6) SUMU = SUMU + U(i) i=1,2,3,4,...
> 7) SUMV = SUMV + V(i) i=1,2,3,4,...
>  
> 8) limit i-> Big  SUMU/SUMV = ?
>  
> What is the best way to solve these kinds of problems.
> 4 and 5 come from a neural network learning model (Hebbian
> learning).

I apologise in advance if I have not understood the question.

I think you can proceed as follows:

Assuming that the random numbers have constant statistics
Let the expected value of A(i) be E(A) ie the mean of A.
Let the expected value of A(i)^2 be E(A^2) ie the mean of A squared.
Similarly E(B) and E(B^2).  These terms can be calculated from the
characteristics of the source of the random numbers.

Now calculate the expected value of SUMU/SUMV

  E(SUMU/SUMV)  =  E(SUMU)/E(SUMV)

 (assuming that there is a limit as i-> Big)


From 4) and 5) above calculate E(U) and E(V) 
If A(i), A(i-1), B(i) and B(i-1) are independent, no correlation etc

  	E(U) = (E(A^2) - E(A)^2)/constant1
	E(V) = (E(B^2) - E(B)^2)/constant2
	
Giving the limit that you want 

E(SUMU/SUMV) = constant2/constant1*(E(A^2) - E(A)^2)/(E(B^2)-E(B)^")
    ---------------------------------------------------------------

If the A and B random variables have some auto or cross correlation
then the method becomes a little more complicated but still ok I think.

Regards
-- 
John@tpsujmfh.demon.co.uk
