Hi Pavel, I think your email highlights one of the differences between us and one of the reasons for this discussion:
I am a scientist, not a mathematician - I want to improve crystallographic methods because people who solve crystal structures want an answer to a biological or chemical or physical question rather than because they enjoy watching the realisation of a mathematical definition. I like Ken Follett's definition of a physicist, for whom reality is a poor approximation to theory, but the motivation for my research runs the other way round. Cheers, Tim On 11/29/2014 05:12 AM, Pavel Afonine wrote: > Hi Tim, > > your examples are valid and valuable, and clearly exemplify existing > problems, limitations as well as common misconceptions. > > However, if you follow mathematics and strict definitions thereof, then > crystallographic structure refinement is nothing but an optimization > problem that, fundamentally, to be defined requires: a) definition of model > parameterization, b) definition of a function that relates experimental > data and model parameters, and c) definition of a method that changes model > parameters in a such a way that optimizes (most of the time minimizes) the > chosen (at step "b") function. > > Please don't think that I've just made up or invented these "a)-b)-c)" > steps above.. In fact, this has been published, for example, in > *Acta Cryst.* (1985). A*41*, 327-333, > and then reiterated using modern jargon, for example, in > *Acta Cryst.* (2012). D*68*, 352-367. > > (I say "for example" above just to stick to the context and also point out > that you can find more examples in crystallographic literature as well as > in totally different disciplines such as economics, aerospace science etc.) > > Anyways, once all the above (a-b-c) are set and defined, then your only > goal is as "simple" as finding the global minimum of the function that you > have chosen to optimize. > > Anything else beyond that are either technical details or various > inefficiencies related to improper model parameterization, improper target > choice or using limited optimization tool. > > All the best, > Pavel > > > On Fri, Nov 28, 2014 at 11:40 AM, Tim Gruene <[email protected]> > wrote: > >> Dear Pavel, >> >> there is a beautiful paper called 'Where freedom is given, liberties are >> taken' by Kleywegt and Jones, but also a wide variety of articles that >> (fortunately) fought hard for the introduction of Rfree to the >> (macro-)crystallographic community. >> >> In there is mentioned the threading of an amino acid chain backwards >> into the density achieving (by refinement) a lower R-value than the >> original one. Since this was achieved with refinement, the former >> structure was closer to the global minimum than the latter one. >> Apparently none of these authors had an idea how to modify the target >> function so that this would not happen - whyfore they suggested to use >> cross validation to avoid it. >> >> If you don't like this line of thought, I can offer a different one: >> >> there is a vast number of sets of parameters that ideally fit your data: >> fill your asymmetric unit randomly with atoms so that your data to >> parameter ratio is 1 or lower. Refine unrestrained and your are going to >> end up with an R-value of 0. For unrestrained refinement, the formula >> for the R-value corresponds (maybe not for maximum likelhood based >> target functions, you may have to do some translation here) to the >> target function, which usually has a lower bound of zero, hence this >> vast number of "structures" all reached the global minimum. Note that >> the deposited structure has an R value much greater than 0, i.e. it is >> far away from the global minimum. >> >> In order to improve the situation, one modifies the target function by >> adding restraints. They increase the target value of all "structures", >> but in general those for the arbitrary solutions increase so much more >> than that for an acceptable solution that most of those are lifted above >> that of an acceptable solution. >> As an example, one of the structures for the yeast polymerase I contains >> about 34,500 atoms, i.e. the target function is minimised in a 138,000 >> dimensional space. I don't think there is a proof that any set of >> restraints is ever so ideal that all false solutions are lifted above >> the target value of the accepted solution. In fact, without being able >> to proove it, I doubt that this the case, which lead me to the below >> claim that we don,t necessarily want to reach the global minimum of the >> target function. >> >> Of course an acceptable structure actually may have a target value >> representing a global minimum, but I don't think this is always true. >> >> Best, >> Tim >> >> On 11/28/2014 05:42 PM, Pavel Afonine wrote: >>> Hi Tim, >>> >>> you don't necessarily want to find the global minimum (...) >>> >>> >>> this contradicts the definition of crystallographic structure refinement. >>> If finding the global minimum is not what you ultimately want then either >>> the refinement target or model parameterization are poor. >>> >>> Clearly, given complexity of refinement target function profile (in case >> of >>> macromolecules) we unlikely to reach the global minimum; however, >> reaching >>> it is what we aim for (by definition and construction of refinement >>> program) . >>> >>> Pavel >>> >> >> -- >> Dr Tim Gruene >> Institut fuer anorganische Chemie >> Tammannstr. 4 >> D-37077 Goettingen >> >> GPG Key ID = A46BEE1A >> >> > -- Dr Tim Gruene Institut fuer anorganische Chemie Tammannstr. 4 D-37077 Goettingen GPG Key ID = A46BEE1A
signature.asc
Description: OpenPGP digital signature
