On 11/09/2015 09:46 PM, Keller, Jacob wrote:
> Derivatives of what?
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> JPK
Derivatives of the target function with respect to the parameters.
The minimum (or maximum) of the target function is where all the first derivatives are zero. So finding that minimum is solving for the point where first derivatives are zero, based on the derivatives at the current point in parameter space. This is a nonlinear problem and not directly solvable. However if you you take the approximation that in a small enough region around the current point, the first derivatives are linear functions of the parameters, the dependencies being the derivatives of each first derivative wrt each parameter, then this simple problem is solvable by matrix inversion. The derivatives of each first 1st derivative wrt each parameter are just the 2nd derivatives (including mixed), so you are inverting a matrix of 2nd derivatives of the taret function wrt each pair of parameters. Initially you are far from the optimum and the answer is wrong, but hopefully closer, so you iterate. Sometimes it is better to start with a gradient search when you are far from the optim
um. Marquardt-Meiron is an algorithm which starts out as nearly a gradient search and switches toward the (newton-raphson?) approach described above as the search converges. Once you are very near the optimum, the linear approximation gets better and better and convergence is supposedly rapid.
And now the point being discussed here involves estimating the uncertainty of the value of the optimized parameters, given the uncertainty in the data (Fobs), translating into uncertainty in the target function? and I am even more out of my depth here, so I better stop. But in general error propagation depends on derivatives wrt whatever the error is propagating from, and you can imagine mixed second derivatives would affect the ability of parameters to compensate for each other?
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> *From:*Ethan Merritt [mailto:[log in to unmask]]
> *Sent:* Monday, November 09, 2015 9:36 PM
> *To:* Keller, Jacob
> *Cc:* [log in to unmask]
> *Subject:* Re: [ccp4bb] precision of bond lengths and angles
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> On Tuesday, 10 November 2015 02:31:35 AM Keller, Jacob wrote:
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>> I have heard this subject bandied about over the years, and it sounds interesting. I wonder whether someone can point to the original source for the procedure? Particularly I am curious what matrix is being inverted and what that means. What are the numbers in the matrix?
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> The Hessian matrix of partial derivatives
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> https://en.wikipedia.org/wiki/Hessian_matrix
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> Ethan
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>> Thanks,
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>> Jacob Keller
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>> Ps I did do a quick Google, but found nothing fundamental-only applications.
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>> From: CCP4 bulletin board [mailto:[log in to unmask]] On Behalf Of George Sheldrick
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>> Sent: Monday, November 09, 2015 6:19 PM
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>> To: [log in to unmask] <mailto:[log in to unmask]>
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>> Subject: Re: [ccp4bb] precision of bond lengths and angles
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>> Dear Dale,
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>> There are a number of strategies for getting (almost) full-matrix esds without causing the program to blow up, for example you can selectively remove the geometric restraints on the parts of the structure that you are interested in and are relatively well-behaved, or you can block out the ADP refinement etc. Of course there may be flexible regions for which it is not possible to get meaningful esds, but they can still be included in the refinement with suitable restraints. An adequate resolution (i.e. data to parameter ratio) is of course essential.
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>> Best wishes, George
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>> On 11/09/2015 10:23 PM, Dale Tronrud wrote:
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>> The key issue is inverting the matrix. The Normal matrix is almost
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>> always singular unless you have huge amounts of x-ray data and a well
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>> ordered molecule (which often go hand in hand). My understanding is
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>> that George prefers to calculate esds from a matrix calculated with no
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>> influence from the restraints, so the problem is even worst. In Shelxl
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>> you can turn on the restraints to stabilize the inversion but you have
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>> to realize that the uncertainty in a bond length will be determined
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>> almost entirely by the restraint and not the X-ray data and for those
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>> uncertainties it would certainly be quicker to simply look in your
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>> restraint file to find the "sigma". Even then at lower resolutions (and
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>> George just posted that this limit is somewhere around 1.6 A) the matrix
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>> will be singular and you will have to bolster it with more information.
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>> Remember those atoms being discussed in another thread today? The
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>> ones being build into little to no density because we know they have to
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>> be somewhere in that area? They will certainly blow up your inversion.
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>> If you can move a parameter about and it makes no change to the fit to
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>> any of your data (x-ray or stereochemical) then your first derivative is
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>> constant at zero and your second derivative is zero and your inversion
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>> is toast. More interesting, but harder to identify, are the cases where
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>> you can move a subset of parameters in a coordinated fashion without
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>> changing the fit. This also creates a singularity.
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>> An alternative to finding more data (or trimming the invisible atoms
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>> from your model) is to filter out the singularities during the
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>> inversion. This can be done with eigenvalue filtering or single value
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>> decomposition, but these methods are very time consuming for large
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>> matrices like these and I'm not aware that anyone has implemented them
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>> in macromolecular refinement.
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>> Dale Tronrud
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>> On 11/9/2015 9:59 AM, Bernhard Rupp (Hofkristallrat a.D.) wrote:
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>> inverting the full matrix of second derivatives, and I believe that only
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>> Shelxl can do that and then only in the presence of very high resolution
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>> x-ray data.
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>> This I do not completely understand: As long as I can set up a LSQ design
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>> (normal equation) matrix, I can make a second derivative matrix, and invert
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>> it. So in principle as long as I am determined, I can do this. If it makes
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>> sense in absence of a limited number of data points to refine against, is a
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>> different question, and the O(n^p) with p > 2.4 at best adds a
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>> computational problem for large n as in macro.
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>> So where reasonably would be the data/parameter ratio (or resolution) where
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>> an inversion makes sense as far as the accuracy of the (co)variances go,
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>> irrespective of the time requirements for inversion?
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>> Thx, BR
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>> Dale Tronrud
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>> On 11/9/2015 2:15 AM, Zhenyao Luo wrote:
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>> Dear CCP4 community,
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>> I would like to ask a question regarding determining the precision of
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>> bond lengths and angles in a protein crystal structure.
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>> I am currently analysing the bond lengths and angles of the Zn-ligand
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>> coordination bonds in some Zn-binding protein crystal structures. For
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>> the bond length/angle values, I intended to keep two decimal places
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>> for the bond lengths and one decimal place for the bond angles. But is
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>> it really justified to quote the values to that precision? Also, how
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>> do you usually determine the appropriate precision (how many decimal
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>> places to keep) when reporting bond lengths/angles values at a given
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>> resolution?
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>> Any advice would be greatly appreciated. Thanks very much in advance.
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>> Best wishes, Zhen
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>> --
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>>
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>> Prof. George M. Sheldrick FRS
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>> Dept. Structural Chemistry,
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>> University of Goettingen,
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>> Tammannstr. 4,
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>> D37077 Goettingen, Germany
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>> Tel. +49-551-39-33021 or -33068
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>> Fax. +49-551-39-22582
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>>
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> --
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> mail: Biomolecular Structure Center, K-428 Health Sciences Bldg
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> MS 357742, University of Washington, Seattle 98195-7742
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