[sdpd] Re: UPPW-5 solution - UPPW-6 problem

[Alan Coelho <alan.coelho...@attglobal.net>]

hello Robin and list members

Your mail Robin was most appreciated and informative on my part. A 
Herculean effort at both defending the term "exhaustive search" and 
keeping up to date on the multitude of indexing algorithms.

>An exhaustive search is one that definitively reports all the possible
>solutions within its search domain, so that one never has to search 
 >that particular domain again.

The statement is technically correct and the phrase that gets you out of 
jail is the "within its search domain". I am wrong as I must have missed 
the qualifying phrases surrounding the term "exhaustive".

I will for the sake of argument speak for the uninitiated as 
"exhaustive" maybe misinterpreted where a method that claims to be 
exhaustive should not have too many restrictions placed on it.

Lets take for example the successive dichotomy method. If it were 
exhaustive then it should not miss high de Wolff values. But of course 
it would if the delta-2Th (or delta-d if you prefer) does not encompass 
the true solution. Thus it is exhaustive if and only if the errors in 
the peak positions are known before hand - which need I say is 
impossible to ascertain on difficult problems with a lot of peak overlap.

Lets take "Index permutation" which I am unfamiliar with but I worry 
about the phrase:
>  Full index permutation in index space should also
>  be exhaustive, within its declared bounds
 
Lets take "c) Grid search", again we have the qualifier:
>Grid search methods are formally exhaustive for the particular
>grid-point array used, and become fully exhaustive (within their 
 >various other bounds, such as limits on cell volume,
>  Miller indices, etc.) if the step size is made sufficiently small.
>Grid search is inherently a relatively inefficient method (though its
>calculations sometimes incorporate sophisticated
>optimisations), but it can be very robust.
 
Grid search is truly an exhaustive method. With infinitesimally small 
steps all of parameter space can be traversed. I hope that I don't 
offend but is this really categorized as a "method".
 
>I'd add that there are circumstances in which least-squares
>itself can become unstable. In such situations one can still
>fall back on parabolic refinement against a merit surface
>(the 3-point fitting of a parabola cyclically to each
>variable parameter in turn, until convergence is reached).
>This is a slower but incredibly robust and
>general method,which can often succeed when others fail.
 
I am a bit lost here. What I can say is that iterative least squares 
(ie. SVD-Index), with a "trivial" qualifier regarding the number of 
missing lines, has found a particular solution to 1000s of simulated 
data sets without fail with relatively large simulated errors. I would 
however restrain from claiming it is exhaustive in the true sense as the 
correct solution is often ranked low. What can be extrapolated from this 
is that for cases where an unambiguous solution does exist (a 
"reasonable" rank lets say) then iterative least squares find its.
 
I stress that the reason why I know that the solution was found is that 
simulated data was used where the solution is known. Not investigating a 
new method's behavior with respect to carefully generated simulated data 
is in my opinion a hit and miss affair when it comes to accessing its 
success rate. The more I think about this issue the more I think that it 
would be worthwhile to create a set of test simulated data sets. This is 
especially true for methods that operate on d-spacings only. Thus if any 
UPPW users would like to see this then let me know and I will consider 
generating such data sets.
 
thanks again Robin,
cheers
alan
 
 


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