4.55. REGWT: Analyse weights
Authors: B.E. Robertson & H. Wang
Contact: Bev Robertson, Faculty of Science, University of Regina, Regina, Saskatchewan, Canada S4S 0A2
REGWT (Wang et al. 1985) analyzes an existing weighting scheme
by calculating the average value of w
2X
within ranges of |F|,
F2 or I, and sin
/
(=s). REGWT also estimates a
modification to the variance of so that w2X does not
show trends with respect to s or |F(rel)| (Stewart et al. 1976) and applies
either the estimated modification to the variance or weights (Prince,
1983).
4.55.1. Introduction
In weighted least-squares refinement the quantity
minimized is 
w(X) 2X where X is one of the
quantities |F|, F2 or I, and X is |X(rel)-X(cal)|.
The weight, w(X), reflects
the accuracy of the measurement of the intensity of a reflection. A properly
estimated weight will lead to an accurate crystal structure. The correct value
of the weight is the reciprocal of the variance, 
2X. Usually
the contribution of counting statistics to X is easily calculated, but
the contribution from various other sources of error is not.
REGWT provides information to assist in the modification of weights to account for other sources of error. It is equivalent to WTANAL and WTLSQ in XRAY76. REGWT contains an additional feature which allows the calculation of coefficients for the weight modification expression in order to modify the weights. The program does not differentiate between random and systematic error. The procedure is described in detail elsewhere, (Wang and Robertson, 1985). In this program description, w refers to the weight rather than the square root of the weight as it does in some other program descriptions in this manual.
REGWT is used to
examine existing weights based on the distribution of w(X)2X.
The data is divided into blocks bounded by both intervals of X(rel) and
s. By choosing either X(rel) or s the analysis becomes
one-dimensional with respect to X(obs) or s. For each grid point, the
number of data, the average value of
{w(X)2X}1/2
and weighted R index are given.
4.55.2. Normal Weighting Schemes
The weighting schemes used here are based on the program WTLSSQ of the XRAY76 system. Nine weight functions and an opportunity for manipulation of the location of the weights in the bdf are specified on the normal line. The weight functions are specified by a weighting scheme number (0-9) and the coefficients (A to I) corresponding to those functions. The default values of these coefficients are all zero. The full weighting schemes are described in detail below.
Scheme 0:Retrieve the old least-squares weight from one of the least-squares weight locations n901, n902, n903 and place it in the least-squares weight location n900, which is used in least-squares refinement in the logical record lrrefl: when the bdf weights are specified.
| A = 1 : from LSW1(n901) to LSWT(n900) |
| B = 2 : from LSW2(n902) to LSWT(n900) |
| C = 3 : from LSW3(n903) to LSWT(n900) |
Scheme 1: W = 1 / (A +
B2X + C/(oldweight) + D*X(rel) + E*X(rel)2+
G*X(rel)H + I*sin)
Scheme 2: W = 1 / (A +
B*X(rel) + C*X(rel)2/oldweight + D*X(rel)6 +
E*sin)
Scheme 3: W = X * Y
If A*X(rel) is greater than
|X(cal)| then weight is 0.000000001.
Let B = sin limit, then if sin is greater than B, X = 1, else X = sin/B B should not be zero.
Let C = X(rel) limit, then if X(rel) is less than C, Y = 1, else
Y = C/X(rel).
Scheme 4: If X(rel) is less than or equal to A, then W = (X(rel)/A)2, else W = (A/X(rel))2. Reflections with X(rel) = A are given maximum weight.
Scheme 5: If X(rel) is less than or equal to A, then W = 1, else W = (A/X(rel))2. Small reflections are given constant weight. In Hughes' original treatment, A = 4 X(min).
Scheme 6: W = 1 /(1 + ((X(rel) - B)/A)2) Reflections with X(rel) = B are given maximum weight, but if A is very large , the weights are constant.
Scheme 7: W = 1 /( A + X(rel) + B*X(rel)2 + C*X(rel)3 ) Cruickshank suggests that A = 2 X(min), B = 2 / X(max), and C = 0 are useful. A larger value of C, say C = 5 / X(max)2, may help down-weight large X(rel) subject to extinction. These values may prove useful in schemes 1 and 2.
Scheme 8: W = A / max( X(rel), (B*X(rel) + C), (D*G + E) )where G is X(rel)(max).
For a complete description of this (Univ. of Washington) scheme,
see the general section for the program DATRDN of the XRAY76
system.
Scheme 9: W = A This weighting scheme produces constant weights
4.55.3. Estimation Of Weighting Modifications
If the variances of the
structure factor amplitudes are correctly estimated, their average value should
correspond to 2X. (The variance will be assumed to represent
error in both the experiment and the model; i.e. in |X(rel)| and |X(cal)|.) The
process of least squares allows some adjustment of the model to errors in the
experiment, so that
<2X / 2X> = (N-M) / N
rather than unity. N is the number of reflections used for
least-squares refinement and M is the number of least-squares variables.
<A> is the average value of A. The quantity (N-M)/N will be referred to
in what follows as the "freedom factor". The calculated variance,
2X(cal), is the variance obtained from a knowledge of the
experimental conditions and is usually based entirely on counting statistics. A
quantity 2X(mod), the modified variance, may be added to
2X(cal) to give an improved value of 2X
or 1/w(X). The correction, 2X(mod), commonly called the
"ignorance factor", may be entered by the user as some function such as
E*|F(rel)|2 + I*sin
(see Scheme 1, normal weighting schemes)
or it may be estimated automatically. An approximate expression for
2X(mod) is
2X(mod) =
{(2X + VC) - 2X(cal)} / (freedom factor) (1)
where VC is a correction term including variance (VAR) and covariance (COV) terms.
This term results from replacing <
2X / 2X > by < 2X
>/< 2X >, and the user may choose to not use it.
Also, the "freedom factor" may be replaced by unity.
The independent variables
are normalized by dividing by |X(rel)|max and
smax. The new independent variables are:
V(X) = |X(rel)| / |X(rel)|max and V(S) = s /smax
The correction, 2X(mod), is estimated by least-squares fitting of
the following expression to equation (1)
< 2X(mod)> =
p(q {[A(p-q,q)] [V(X)p-q] [V(S)q]})
(2)
If p=0, a constant term is determined.
If p=1, the coefficients in the expression of the form
A(0,0) + A(1,0)*V(F) + A(0,1)*V(S)
are determined; etc. The coefficients A(p,q) can then be used to
calculate 2X(mod) for an individual reflection. The weight
for a reflection is then calculated as:
1/w(X) = 2X = 2X(cal) +
2X(mod) (3)
The structure is refined by normal least squares using the modified weights and new
values of 2X are created. If the individual reflection option
is chosen, the variance and covariance correction is not used or calculated.
The process is iterated until the coefficients do not change. The change in the
standard deviations of least-squares variables on the first iteration will
probably be between 0 and 40% of their initial value with non-modified weights.
The improvement on the second iteration will typically be 10% of the
improvement of the previous one. A third iteration would seldom seem justified.
The quantity 2X is required for calculating the variance and
covariance correction (VC), but is not well known until after first iteration.
Therefore 2X(cal) must be used instead of
2X in the first iteration, if the VC correction is applied.
In
the REGWT calculation, the value of 2X(cal) in (3) above is
obtained from IDN 1900 in logical record lrrefl: on the bdf. If IDN 1900
is empty, the value of 2X(cal) is then obtained from IDN 130n
(n=1, 3 or 5 for I, F2 or F, respectively). After the modified
weight w(X) is calculated, its value will be stored in IDN 1900, replacing the
old value of 2X(cal).
The square root of the number of reflections in a grid point is used to weight the grid points when fitting equation (2) to equation (1). Not surprisingly, grid points with high V(X) and high V(S) may be empty and others may have few reflections. The option also exits to not average but treat every reflection as a separate grid point. Since averaging to form grid points involves first order differences, the influence of reflections whose contribution to the average in equation (1) deviate far from the mean will be enhanced if individual reflections are used to find the A(p,q). If the individual reflection option is chosen, the variance and covariance correction is not used or calculated.
4.55.4. File Assignments
Reads reflection data from the input archive bdf
Optionally writes reflection data to the output archive bdf
4.55.5. Examples
REGWT lst wta maxhkl 12 12 12 0.0 0.824 fgrid 2.3 4.3 7.6 12.4 fgrid 13.4 17.5 20.5 25.5 30.5 35.5 fgrid 45.0 55.5 65. 80. 100. 99999. sgrid .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 .65 sgrid .7 .75 .80 .85 |
In this example, the weight-analysis process is called to analyze the weights which have been assigned for each reflection. The analysis maps are specified as 16x16 by the fgrid and sgrid lines. The program does not update the archive bdf. The reflection data will be printed.
REGWT nolist wfc 10.0 rel cnt 99.0 regina pow 2 ind fac 1. avc archiv 1902 -1903 fgrid 3.0 4.0 5.0 6.0 7.0 8.0 fgrid 9.50 12.0 17.0 21.0 27.0 30.0 50.0 fgrid 70.0 999.0 sgrid 0.150 0.200 0.250 0.300 0.350 0.400 0.450 sgrid 0.500 0.515 0.530 0.550 1.000 |
The weight modification function with 2 as the highest power is specified. The estimation of the weights is based on the individual reflection mode. The variance and covariance correction is applied and the least-squares freedom factor is 1. The weight in location 1903 will be deleted from the binary data file and the old weight will be stored in the weight location 1902. Only the observed reflections are used excluding the reflections for which the value of ?(F) is greater then 99.0. The output analysis map (15x12) will be scaled by the factor 10.
A normal weighting scheme is indicated in the REGWT line. The scheme number is specified as 0, which transfers the weight saved at location 1902 to location 1900 where it will be used for weighted least-squares refinement. The old weights at location 1903, if they exist, will be erased from the bdf.
References
Wang, Hong and Robertson, B.E. 1985. Structure and Statistics in Crystallography.Ed. A.J.C. Wilson. Adenine Press: New York.
Stewart, J.M., Machin, P.A., Dickinson, C.W., Ammon, H.L., Heck, H. and Flack, H.D. 1976. The XRAY76 SYSTEM. Tech. Report. TR446, Computer Science Center, University of Maryland.
Prince, E. 1982. Mathematical Techniques in Crystallography and Materials Science. Springer-Verlag: New York.
Johnston, R.L. 1982. Numerical Methods -- A software approach. John Wiley: New York.