Re: [boost] Automated error analysis with correlation

From: Steven Watanabe (watanabesj_at_[hidden])
Date: 2008-04-14 13:13:29

AMDG

Hervé Brönnimann wrote:
> Steven: You seem to be following a partial differential approach,  
> where you record the partial derivative of an expression wrt to each  
> of the variables that compose it.  Your particular approach is using  
> a sparse representation of the vector of derivatives wrt to all the  
> variables in your system.  Wrt the evaluation of the uncertainty (sqrt 
> (0.2*0.2 + 0.2*0.2) = 0.28 in your example), this just corresponds to  
> a norm for the vector, here the Euclidean norm, with a twist that  
> you're not dividing by the total number of variables (not even by the  
> number of variables that are actually involved in your expression).   
> (Note that with a finite number of variables, all norms are  
> equivalent anyway.) It's not an especially new idea.  If you go to  
> second derivatives, you can do the same with the Jacobian matrix of  
> your expression.

It should in theory be possible to generalize this to an arbitrary number
of derivatives, specified as a template parameter, right?

>   The discussion then becomes sparse vs. dense  
> representation.  Your idea seems to boil down to using a sparse  
> representation of this otherwise well known aproach.  Am I missing  
> something?
>   

So, I've just reinvented yet another wheel...
Ok.  What are the problems with this approach.
For me the most important thing is that I can hide
the  messy calculations behind a nice API.  The sparse
representation makes this easy.

> BTW,  error propagation can also be handled by boost.interval.  Note,  
> I am not talking about tracking dependencies (it's the same kind of  
> thing that makes square(x) have a smaller interval enclosure than x *  
> x).  Although in general, for small errors, there isn't much  
> difference and we've used interval analysis exactly for that purpose,  
> assuming all the variables are always independent (you get an  
> guaranteed enclosure either way, just not as tight as it could be if  
> you had know the dependencies).
>   

I happen to need dependency tracking.

In Christ,
Steven Watanabe