This is one of those problems that seems simple at the overview level, but gets more complicated the more you look at it. In principle, the high-school maths treatment is correct: if you have more unknowns than observations, you can come up with an infinite number of solutions that each exactly fit your data - which of course is a dangerous situation to be in. If you have exactly the same number of unknowns as observations, then you can find a single unique solution that exactly fits all your data. In practice, that's not good enough, since experimental data always contains experimental error, an exact fit is often fairly meaningless. You really want the number of observations to significantly outnumber the number of parameters, since you can use the extras to estimate and account for experimental error - you can no longer exactly fit every point, but rather you look for the solution that minimises the residuals.
The other complication is the one that Pavel described: while it's trivially easy to count observations, in the case of crystallography it's far harder to work out how many fitting parameters you actually have. Remember, what is important is the number of *independent* parameters - and in a molecule, no atom is truly independent of its neighbours. Every bond, every angle and every torsion is a restraint that increases the correlation of each atom to its neighbours, and decreases the effective number of independent parameters - but in a way that is very difficult to quantify. Even more difficult to quantify is the effects of non-bonded interactions (Van der Waals forces and electrostatic attraction/repulsion). All of these act to reduce the effective number of degrees of freedom of the system, but in a very fuzzy way.
The way I like to look at it is as a continuum. At the one end, you have the very lucky case of "perfect" atomic-resolution data, where you could in theory fit everything with no knowledge-based restraints - the centroid of every atom is well defined, and each atom can be directly identified from the density. At the other end, you have ab initio structure prediction with no experimental data - here you have to rely entirely on what's already known about molecular structure and interactions, and hope you get close to the real thing. In between, you have a sliding scale where reliance on the experimental data gradually decreases, with increasing reliance on a good force field (and comparison to higher-resolution structures etc.) to handle the details. Where on that continuum any particular structure lies, though? A very difficult question to answer.
Cheers,
Tristan
________________________________________
From: CCP4 bulletin board <[log in to unmask]> on behalf of Debanu Das <[log in to unmask]>
Sent: Wednesday, 12 August 2015 2:47 PM
To: [log in to unmask]
Subject: Re: [ccp4bb] Observations Vs. Parameter Ratio--Original Source?
Hi,
I think to start delving into the origins of this, we can recap high
school algebra of simultaneous equations (or data fitting) as a simple
start. For a system of 2 variables x,y, you will need at least 2
equations to solve for the 2 variables, giving you a data to parameter
ratio of 1. Similarly for 3 unknowns x,y,z, you will need 3 equations
at least to fully solve the system of equations.
If you look through some of the original small
molecule/crystallography books, you can follow up on it. One of the
more recent texts that talks about it is Bernhard Rupp's "Biomolecular
Crystallography": "The basic idea in refinement is that a system of p
independent simultaneous equations is solved." Refinement involves the
fitting of an excess n available experimental data to p model
parameters. So just as in the case of the above simultaneous
equations, this system of simultaneous equations can be solved only if
n >= p (https://books.google.com/books?id=gTAWBAAAQBAJ&lpg=PA622&ots=xZFfmIyGeV&dq=data%20to%20parameter%20ratio%20in%20simultaneous%20equation&pg=PA622#v=onepage&q&f=false).
So the higher the resolution of your diffraction data set, the higher
your 'n', so more detailed the fitting that can be done. When
resolution is low, we use restraints and constraints to get a good
data/parameter ratio.
I assume this is what you were looking for and I hope this helps to
understand the origins of the data/parameter ratio.
Yes, having few thousand more data at I/sig ~2 may help, but really
the only way to tell is if you get better maps and R-values keeping
good statistics.
Regards,
Debanu.
On Tue, Aug 11, 2015 at 8:54 PM, Pavel Afonine <[log in to unmask]> wrote:
> Hi Jacob,
>
> making sense of data-to-parameters ratio is trickier than it may seem. It is
> not as simple as comparing the number of reflections Nreflections with the
> number of parameters, which typically but not always is Natoms * (3 xyz + 1
> Biso or 6 Baniso + some occupancies + some other parameters).
>
> In refinement we use restraints and/or constraints. While constraints reduce
> number of parameters explicitly and can be counted, restraints do so in a
> less obvious way. When restraints are used the a priori information is added
> as a weighted term to the total target that is optimized: T = Tdata + weight
> * Trestraints. The "weight" prescribes the dose of extra information to be
> added (keep in mind: the weight changes during refinement!). This is exactly
> why we can still refine individual coordinates or isotropic B-factors at
> typical "macromolecular" resolutions such as 2-4A or so.
>
> Having said this, and being unable to estimate this number even
> approximately, personally, I could not care less about it for practical
> purposes.
>
> And to answer your very question: no, I do not know where this is discussed
> in great details, apart from being perennially said on mailing lists.
> Perhaps I should have started with this first -;)
>
> All the best,
> Pavel
>
>
> On Tue, Aug 11, 2015 at 8:17 PM, Keller, Jacob <[log in to unmask]>
> wrote:
>>
>> Dear Crystallographers,
>>
>> I've long heard second-hand about the need for favorable
>> observation-to-parameter ratios, but have never really delved into the
>> original literature. Does anyone know of a good source explaining and/or
>> demonstrating this requirement, and perhaps showing how aspects of
>> crystallography per se color the relationship? Ideally this would be an
>> original source (I've found this seems to be the only way to really get to
>> the bottom of things, and is much more efficient than reading digests or
>> reviews, although the "James" might be an exception.)
>>
>> For a crystallographic example, some observations are I/sig of 2, some 50,
>> and it does not seem right to weigh them equally for these purposes. More
>> specifically, I have a case of a truncated dataset which is cut off at 1.7
>> Ang due to the detector, but I think the dataset would have gone to 1.4-1.5
>> with a different setup. Does having a few thousand more measurements at
>> I/sig = ~2 make that much of a difference? How should one think about this?
>>
>> Thanks,
>>
>> Jacob Keller
>>
>> *******************************************
>> Jacob Pearson Keller, PhD
>> Looger Lab/HHMI Janelia Research Campus
>> 19700 Helix Dr, Ashburn, VA 20147
>> email: [log in to unmask]
>> *******************************************
>
>
|