#############################################################
##                        mmtbx.xtriage                    ##
##                                                         ##
##      P.H. Zwart, R.W. Grosse-Kunstleve & P.D. Adams     ##
##                                                         ##
#############################################################
#phil __OFF__
  This cryptic code, together with the tags __ON__ and __OFF__
  allows one to use the log file as an input file for xtriage.
  Try : mmtbx.xtriage  <logfile> to give it a try!

Date 2006-08-28 Time 15:27:26 PDT -0700


##-------------------------------------------##
## WARNING:                                  ##
## Number of residues unspecified            ##
##-------------------------------------------##


Effective parameters: 
#phil __ON__
scaling.input {
  parameters {
    asu_contents {
      n_residues = None
      n_bases = 10
      n_copies_per_asu = None
    }
    misc_twin_parameters {
      missing_symmetry {
        tanh_location = 0.08
        tanh_slope = 50
      }
      twinning_with_ncs {
        perform_analyses = False
        n_bins = 7
      }
      twin_test_cuts {
        low_resolution = 10
        high_resolution = None
        isigi_cut = 3
        completeness_cut = 0.85
      }
    }
    reporting {
      verbose = 1
      log = "rna.log"
      ccp4_style_graphs = True
    }
  }
  xray_data {
    file_name = "niggli.mtz"
    obs_labels = None
    calc_labels = None
    unit_cell = 27.9260006 53.02999878 71.96299744 74.5719986 81.36799622 \
                75.60800171
    space_group = "P 1"
    high_resolution = None
    low_resolution = None
    reference {
      data {
        file_name = None
        labels = None
        unit_cell = None
        space_group = None
      }
    }
  }
}
#phil __END__


Symmetry, cell and reflection file content summary

Miller array info: niggli.mtz:niggli,SIGniggli
Observation type: xray.amplitude
Type of data: double, size=12907
Type of sigmas: double, size=12907
Number of Miller indices: 12907
Anomalous flag: False
Unit cell: (27.926, 53.03, 71.963, 74.572, 81.368, 75.608)
Space group: P 1 (No. 1)
Systematic absences: 0
Centric reflections: 0
Resolution range: 25.6734 2.40006
Completeness in resolution range: 0.859206
Completeness with d_max=infinity: 0.858635

##----------------------------------------------------##
##                    Basic statistics                ##
##----------------------------------------------------##


Matthews coefficient and Solvent content statistics
----------------------------------------------------------------
| Copies | Solvent content | Matthews Coef. | P(solvent cont.) |
|--------|-----------------|----------------|------------------|
|      1 |      0.961      |     31.866     |       0.000      |
|      2 |      0.923      |     15.933     |       0.000      |
|      3 |      0.884      |     10.622     |       0.001      |
|      4 |      0.846      |      7.967     |       0.002      |
|      5 |      0.807      |      6.373     |       0.004      |
|      6 |      0.769      |      5.311     |       0.010      |
|      7 |      0.730      |      4.552     |       0.021      |
|      8 |      0.691      |      3.983     |       0.037      |
|      9 |      0.653      |      3.541     |       0.057      |
|     10 |      0.614      |      3.187     |       0.079      |
|     11 |      0.576      |      2.897     |       0.107      |
|     12 |      0.537      |      2.656     |       0.136      |
|     13 |      0.499      |      2.451     |       0.151      |
|     14 |      0.460      |      2.276     |       0.140      |
|     15 |      0.421      |      2.124     |       0.108      |
|     16 |      0.383      |      1.992     |       0.069      |
|     17 |      0.344      |      1.874     |       0.038      |
|     18 |      0.306      |      1.770     |       0.019      |
|     19 |      0.267      |      1.677     |       0.008      |
|     20 |      0.229      |      1.593     |       0.004      |
|     21 |      0.190      |      1.517     |       0.003      |
|     22 |      0.151      |      1.448     |       0.002      |
|     23 |      0.113      |      1.385     |       0.001      |
|     24 |      0.074      |      1.328     |       0.001      |
|     25 |      0.036      |      1.275     |       0.001      |
----------------------------------------------------------------
|              Best guess :   13  copies in the asu            |
----------------------------------------------------------------


Completeness and data strength analyses 

  The following table lists the completeness in various resolution
  ranges, after applying a I/sigI cut. Miller indices for which
  individual I/sigI values are larger than the value specified in
  the top row of the table, are retained, while other intensities
  are discarded. The resulting completeness profiles are an indication
  of the strength of the data.

----------------------------------------------------------------------------------------
| Res. Range   | I/sigI>1  | I/sigI>2  | I/sigI>3  | I/sigI>5  | I/sigI>10 | I/sigI>15 |
----------------------------------------------------------------------------------------
| 25.67 - 5.90 | 87.3%     | 87.3%     | 87.2%     | 87.0%     | 84.9%     | 81.8%     |
|  5.90 - 4.69 | 92.1%     | 91.5%     | 91.4%     | 89.7%     | 87.3%     | 81.9%     |
|  4.69 - 4.10 | 94.2%     | 94.1%     | 93.9%     | 93.2%     | 90.7%     | 84.4%     |
|  4.10 - 3.73 | 94.1%     | 94.0%     | 93.7%     | 92.0%     | 86.7%     | 78.1%     |
|  3.73 - 3.46 | 94.3%     | 93.3%     | 92.7%     | 90.9%     | 82.7%     | 68.2%     |
|  3.46 - 3.26 | 93.2%     | 91.7%     | 90.5%     | 86.8%     | 75.5%     | 54.1%     |
|  3.26 - 3.09 | 92.6%     | 90.9%     | 88.9%     | 84.9%     | 71.3%     | 49.4%     |
|  3.09 - 2.96 | 90.6%     | 87.9%     | 85.4%     | 78.8%     | 58.2%     | 28.8%     |
|  2.96 - 2.85 | 88.4%     | 86.0%     | 80.5%     | 72.0%     | 42.4%     | 7.8%      |
|  2.85 - 2.75 | 86.4%     | 81.6%     | 74.9%     | 62.0%     | 29.4%     | 3.4%      |
|  2.75 - 2.66 | 83.3%     | 76.1%     | 68.2%     | 53.5%     | 20.2%     | 2.4%      |
|  2.66 - 2.59 | 81.1%     | 72.1%     | 63.2%     | 44.9%     | 11.9%     | 0.4%      |
|  2.59 - 2.52 | 78.0%     | 68.3%     | 55.5%     | 38.4%     | 9.7%      | 0.8%      |
|  2.52 - 2.46 | 72.3%     | 61.7%     | 48.7%     | 30.4%     | 4.6%      | 0.0%      |
----------------------------------------------------------------------------------------

  The completeness of data for which I/sig(I)>3.00, exceeds  85% for
  for resolution ranges lower than 2.96A.
  The data is cut at this resolution for the potential twin tests 
  and intensity statistics.


Maximum likelihood isotropic Wilson scaling 
ML estimate of overall B value of niggli.mtz:niggli,SIGniggli:
37.46 A**(-2)
Estimated -log of scale factor of niggli.mtz:niggli,SIGniggli:
-1.84


Maximum likelihood anisotropic Wilson scaling 
ML estimate of overall B_cart value of niggli.mtz:niggli,SIGniggli:
50.28,  3.98, -1.18
       35.33,  1.36
              27.22
Equivalent representation as U_cif: 
 0.60, -0.05, -0.05
        0.43, -0.07
               0.34

ML estimate of  -log of scale factor of niggli.mtz:niggli,SIGniggli:
-1.86
Correcting for anisotropy in the data

Some basic intensity statistics follow.



Low resolution completeness analyses 

 The following table shows the completeness
 of the data to 5 Angstrom.
unused:         - 25.6737 [  0/10 ] 0.000
bin  1: 25.6737 - 10.5435 [121/167] 0.725
bin  2: 10.5435 -  8.4673 [147/167] 0.880
bin  3:  8.4673 -  7.4266 [138/159] 0.868
bin  4:  7.4266 -  6.7612 [147/164] 0.896
bin  5:  6.7612 -  6.2842 [164/178] 0.921
bin  6:  6.2842 -  5.9185 [145/151] 0.960
bin  7:  5.9185 -  5.6253 [149/162] 0.920
bin  8:  5.6253 -  5.3828 [148/163] 0.908
bin  9:  5.3828 -  5.1773 [167/185] 0.903
bin 10:  5.1773 -  5.0000 [154/168] 0.917
unused:  5.0000 -         [  0/0  ]



Mean intensity analyses 
 Analyses of the mean intensity. 
 Inspired by: Morris et al. (2004). J. Synch. Rad.11, 56-59.
 The following resolution shells are worrisome: 
------------------------------------------------
| d_spacing | z_score | compl. | <Iobs>/<Iexp> |
------------------------------------------------
     None
------------------------------------------------

Possible outliers 
  Inspired by: Read, Acta Cryst. (1999). D55, 1759-1764

 Acentric reflections:

-----------------------------------------------------------------
| d_space |      H     K     L |  |E|  | p(wilson) | p(extreme) |
-----------------------------------------------------------------
|   2.666 |     -4,  -14,   12 |  5.33 |  4.54e-13 |   5.81e-09 |
|   3.858 |      0,   -6,   14 |  3.48 |  5.59e-06 |   6.91e-02 |
|   2.654 |      3,  -16,    4 |  3.78 |  6.26e-07 |   7.98e-03 |
-----------------------------------------------------------------

 p(wilson)  : 1-(1-exp[-|E|^2]) 
 p(extreme) : 1-(1-exp[-|E|^2])^(n_acentrics)
 p(wilson) is the probability that an E-value of the specified
 value would be observed when it would selected at random from
 the given data set.
 p(extreme) is the probability that the largest |E| value is 
 larger or equal then the observed largest |E| value.

 Both measures can be used for outlier detection. p(extreme)
 takes into account the size of the dataset.


 Centric reflections:

            None 


Ice ring related problems

 The following statistics were obtained from ice-ring 
 insensitive resolution ranges 
  mean bin z_score      : 0.55
      ( rms deviation   : 0.39 )
  mean bin completeness : 0.86
     ( rms deviation   : 0.12 )

 The following table shows the z-scores 
 and completeness in ice-ring sensitive areas.
 Large z-scores and high completeness in these 
 resolution ranges might be a reason to re-assess
 your data processsing if ice rings were present.

------------------------------------------------
| d_spacing | z_score | compl. | Rel. Ice int. |
------------------------------------------------
|    3.897  |   0.29  |   0.95 |     1.000     |
|    3.669  |   0.98  |   0.95 |     0.750     |
|    3.441  |   1.04  |   0.95 |     0.530     |
|    2.671  |   0.80  |   0.85 |     0.170     |
------------------------------------------------

 Abnormalities in mean intensity or completeness at
 resolution ranges with a relative ice ring intensity
 lower then 0.10 will be ignored.

 No ice ring related problems detected.
 If ice rings were present, the data does not look
 worse at ice ring related d_spacings as compared
 to the rest of the data set 




Basic analyses completed

##----------------------------------------------------##
##                   Twinning Analyses                ##
##----------------------------------------------------##



Using data between 10.00 to 2.96 Angstrom.

Determining possible twin laws.

The following twin laws have been found:

--------------------------------------------------------------------------------
| Type | Axis   | R metric (%) | delta (le Page) | delta (Lebedev) | Twin law  |
--------------------------------------------------------------------------------
|  PM  | 2-fold | 5.268        | 2.559           | 0.037           | h,h-k,h-l |
--------------------------------------------------------------------------------
M:  Merohedral twin law
PM: Pseudomerohedral twin law

  0 merohedral twin operators found
  1 pseudo-merohedral twin operators found
In total,   1 twin operator were found


Number of centrics  : 0
Number of acentrics : 7250

 Largest patterson peak with length larger then 15 Angstrom 

 Frac. coord.        :   0.243    0.463    0.070
 Distance to origin  :  28.853
 Height (origin=100) :   7.276
 p_value(height)     :   4.495e-01

   The reported p_value has the following meaning: 
     The probability that a peak of the specified height 
     or larger is found in a Patterson function of a 
     macro molecule that does not have any translational
     pseudo symmetry is equal to  4.495e-01 
     p_values smaller then 0.05 might indicate 
     weak translation pseudo symmetry, or the self vector of 
     a large anomalous scatterer such as Hg, whereas values 
     smaller then 1e-3 are a very strong indication for 
     the presence of translational pseudo symmetry.



Wilson ratio and moments 

Acentric reflections 
   <I^2>/<I>^2    :2.143   (untwinned: 2.000; perfect twin 1.500)
   <F>^2/<F^2>    :0.783   (untwinned: 0.785; perfect twin 0.885)
   <|E^2 - 1|>    :0.743   (untwinned: 0.736; perfect twin 0.541)



NZ test (0<=z<1) to detect twinning and possible translational NCS


-----------------------------------------------
|  Z  | Nac_obs | Nac_theo | Nc_obs | Nc_theo |
-----------------------------------------------
| 0.0 |   0.000 |    0.000 |  0.000 |   0.000 |
| 0.1 |   0.086 |    0.095 |  0.000 |   0.248 |
| 0.2 |   0.180 |    0.181 |  0.000 |   0.345 |
| 0.3 |   0.263 |    0.259 |  0.000 |   0.419 |
| 0.4 |   0.339 |    0.330 |  0.000 |   0.474 |
| 0.5 |   0.404 |    0.394 |  0.000 |   0.520 |
| 0.6 |   0.462 |    0.451 |  0.000 |   0.561 |
| 0.7 |   0.515 |    0.503 |  0.000 |   0.597 |
| 0.8 |   0.562 |    0.551 |  0.000 |   0.629 |
| 0.9 |   0.606 |    0.593 |  0.000 |   0.657 |
| 1.0 |   0.646 |    0.632 |  0.000 |   0.683 |
-----------------------------------------------
| Maximum deviation acentric      :  0.014    |
| Maximum deviation centric       :  0.683    |
|                                             |
| <NZ(obs)-NZ(twinned)>_acentric  : +0.007    |
| <NZ(obs)-NZ(twinned)>_centric   : -0.467    |
-----------------------------------------------


 L test for acentric data

 using difference vectors (dh,dk,dl) of the form: 
(2hp,2kp,2lp)
  where hp, kp, and lp are random signed integers such that 
  2 <= |dh| + |dk| + |dl| <= 8 

  Mean |L|   :0.483  (untwinned: 0.500; perfect twin: 0.375)
  Mean  L^2  :0.313  (untwinned: 0.333; perfect twin: 0.200)

  The distribution of |L| values indicates a twin fraction of
  0.02. Note that this estimate is not as reliable as obtained
  via a Britton plot or H-test if twin laws are available. 



---------------------------------------------
 Analysing possible twin law :  h,h-k,h-l
---------------------------------------------


Results of the H-test on a-centric data: 

 (Only 50.0% of the strongest twin pairs were used)

mean |H| : 0.447   (0.50: untwinned; 0.0: 50% twinned)
mean H^2 : 0.286   (0.33: untwinned; 0.0: 50% twinned)
Estimation of twin fraction via mean |H|: 0.053
Estimation of twin fraction via cum. dist. of H: 0.044



Britton analyses

  Extrapolation performed on  0.00 < alpha < 0.495 
  Estimated twin fraction: 0.021
  Correlation: 0.9994

R vs R statistic:
  R_abs_twin = <|I1-I2|>/<|I1+I2|>
  Lebedev, Vagin, Murshudov. Acta Cryst. (2006). D62, 83-95

   R_abs_twin observed data   : 0.472

  R_sq_twin = <(I1-I2)^2>/<(I1+I2)^2>
   R_sq_twin observed data    : 0.289
  No calculated data available.
  R_twin for calculated data not determined.


Maximum Likelihood twin fraction determination
    Zwart, Read, Grosse-Kunstleve & Adams, to be published.


   The estimated twin fraction is equal to 0.022



Exploring higher metric symmetry

The point group of data as dictated by the space group is P 1
  the point group in the niggli setting is P 1
The point group of the lattice is C 1 2 1 (-x+y,z,2*x-z)
A summary of R values for various possible point groups follow.

---------------------------------------------------------------------------------------------
| Point group            | mean R_used | max R_used | mean R_unused | min R_unused | choice |
---------------------------------------------------------------------------------------------
| C 1 2 1 (-x+y,z,2*x-z) | 0.472       | 0.472      | None          | None         |        |
| P 1                    | None        | None       | 0.472         | 0.472        | <---   |
---------------------------------------------------------------------------------------------

R_used: mean and maximum R value for symmetry operators *used* in this point group
R_unused: mean and minimum R value for symmetry operators *not used* in this point group
 The likely point group of the data is:  P 1

Possible space groups in this point groups are:
   Unit cell: (27.926, 53.03, 71.963, 74.572, 81.368, 75.608)
   Space group: P 1 (No. 1)

Note that this analyses does not take into account the effects of twinning.
If the data is (allmost) perfectly twinned, the symmetry will appear to be
higher than it actually is.





-------------------------------------------------------------------------------
Twinning and intensity statistics summary (acentric data):

Statistics independent of twin laws
  - <I^2>/<I>^2 : 2.143
  - <F>^2/<F^2> : 0.783
  - <|E^2-1|>   : 0.743
  - 
, <L^2>: 0.483, 0.313
       Multivariate Z score L-test: 0.769 
       The multivariate Z score is a quality measure of the given
       spread in intensities. Good to reasonable data is expected
       to have a Z score lower than 3.5. 
       Large values can indicate twinning, but small values do not
       neccesarily exclude it. 


Statistics depending on twin laws
------------------------------------------------------------------
| Operator  | type | R obs. | Britton alpha | H alpha | ML alpha |
------------------------------------------------------------------
| h,h-k,h-l |  PM  | 0.472  | 0.021         | 0.044   | 0.022    |
------------------------------------------------------------------

Patterson analyses
  - Largest peak height   : 7.276
   (correpsonding p value : 4.495e-01)


The largest off-origin peak in the Patterson function is 7.28% of the 
height of the origin peak. No significant pseudotranslation is detected.

The results of the L-test indicate that the intensity statistics
behave as expected. No twinning is suspected.
Even though no twinning is suspected, it might be worthwhile carrying out 
a refinement using a dedicated twin target anyway, as twinned structures with
low twin fractions are difficult to distinguish from non-twinned structures.


-------------------------------------------------------------------------------


$TABLE: Intensity plots:
$GRAPHS
:Intensity plots 
:A:1,2,3,4:
$$
1/resol^2   <I>_smooth_approximation   <I>_via_binning   <I>_expected   $$ 
$$
0.010048   8.679734   44596.120664   32641.576268
0.014048   8.508878   30641.619328   33783.301726
0.018048   8.455946   33813.327991   30764.088011
0.022048   8.346651   28971.296629   28812.264852
0.026048   8.435485   30279.810236   32953.624297
0.030048   8.588338   37579.749758   39329.983665
0.034048   8.649575   41588.610074   43161.810137
0.038048   8.656055   37326.025119   44090.526829
0.042048   8.698797   43021.920676   44689.124424
0.046048   8.792547   44230.463277   47019.626842
0.050048   8.892718   52909.533809   51061.217209
0.054048   8.966749   53597.323629   55367.583643
0.058048   9.019122   60056.228091   58521.487961
0.062048   9.064542   54388.305458   60076.037514
0.066048   9.099541   62879.029269   60568.416988
0.070048   9.106219   67674.223613   60919.828702
0.074048   9.077858   57218.022885   61765.613034
0.078048   9.034675   55006.040398   63101.861999
0.082048   9.011914   58997.599644   64342.741970
0.086048   9.030927   55204.274286   64654.919657
0.090048   9.079366   66139.492156   63350.661861
0.094048   9.117689   57662.625515   60160.430496
0.098048   9.106280   63770.464269   55305.641832
0.102048   9.030865   57459.668102   49390.536396
0.106048   8.906686   49294.587607   43191.885249
0.110048   8.760895   45282.299089   37438.254392
0.114048   8.611179   34019.161709   32648.408825
0.118048   8.460783   31707.967208   29060.629349
0.122048   8.313243   28764.406389   26648.796762
0.126048   8.188701   24843.608065   25197.649178
0.130048   8.118534   21455.226477   24401.516851
0.134048   8.115412   23118.751284   23955.284937
0.138048   8.146866   24097.301969   23617.836551
0.142048   8.149733   25247.003058   23241.234226
0.146048   8.088194   20675.588600   22769.416617
0.150048   8.000033   20698.430132   22216.328478
0.154048   7.960861   19761.097586   21635.108063
0.158048   7.981767   20297.826112   21088.329975
0.162048   7.985107   20388.115274   20625.879561
0.166048   7.953728   18957.983193   20273.231028
0.170048   7.971961   20373.073271   20029.703784
0.173602   7.961141   18934.678425   19888.101884
$$


$TABLE: Z scores and completeness:
$GRAPHS
:Data sanity and completeness check 
:A:1,2,3:
$$
1/resol^2   Z_score   Completeness   $$ 
$$
0.010048   1.356271   0.857143
0.014048   0.414952   0.830189
0.018048   0.485491   0.939394
0.022048   0.028846   0.877005
0.026048   0.483007   0.960000
0.030048   0.283389   0.921951
0.034048   0.235289   0.904959
0.038048   0.988089   0.914729
0.042048   0.234672   0.935622
0.046048   0.397275   0.951493
0.050048   0.254856   0.930556
0.054048   0.239321   0.951724
0.058048   0.195931   0.930403
0.062048   0.741852   0.957055
0.066048   0.291284   0.951368
0.070048   0.833764   0.921283
0.074048   0.567849   0.927711
0.078048   0.978003   0.950000
0.082048   0.613982   0.949602
0.086048   1.040545   0.946381
0.090048   0.292243   0.920000
0.094048   0.269967   0.932642
0.098048   0.955465   0.930851
0.102048   1.029932   0.927083
0.106048   0.937889   0.922717
0.110048   1.477002   0.905569
0.114048   0.338099   0.910377
0.118048   0.809665   0.888631
0.122048   0.762817   0.867612
0.126048   0.143282   0.886747
0.130048   1.263880   0.843220
0.134048   0.347168   0.863539
0.138048   0.192027   0.820961
0.142048   0.801303   0.854077
0.146048   0.896184   0.808642
0.150048   0.672758   0.815965
0.154048   0.880654   0.806383
0.158048   0.384783   0.767355
0.162048   0.115958   0.761341
0.166048   0.630558   0.643287
0.170048   0.154355   0.537428
0.173602   0.344683   0.423256
$$


$TABLE: <I/sigma_I>:
$GRAPHS
:Signal to noise 
:N:1,2:
$$
1/resol^2   <I/sigma_I>;_Signal_to_noise   $$ 
$$
0.003517   22.510961
0.007517   24.689290
0.011517   24.111021
0.015517   24.490168
0.019517   24.344330
0.023517   23.918365
0.027517   24.466378
0.031517   23.683284
0.035517   22.928196
0.039517   23.562665
0.043517   23.533916
0.047517   23.574894
0.051517   23.140869
0.055517   22.955285
0.059517   22.971036
0.063517   22.044784
0.067517   20.975396
0.071517   21.000889
0.075517   19.810832
0.079517   17.331339
0.083517   15.456616
0.087517   14.879946
0.091517   15.185623
0.095517   14.427043
0.099517   14.741062
0.103517   13.021334
0.107517   12.024707
0.111517   11.393169
0.115517   10.007593
0.119517   9.464005
0.123517   8.641250
0.127517   8.160513
0.131517   7.719907
0.135517   7.137649
0.139517   6.445382
0.143517   6.141164
0.147517   5.902341
0.151517   5.761205
0.155517   5.248673
0.159517   4.942808
0.163517   4.622733
0.167517   4.500523
0.171517   4.391574
0.175517   3.746674
$$


$TABLE: NZ test:
$GRAPHS
:NZ test, acentric and centric data 
:A:1,2,3,4,5:
$$
z   Acentric_observed   Acentric_untwinned   Centric_observed   Centric_untwinned   $$ 
$$
0.000000   0.000000   0.000000   0.000000   0.000000
0.100000   0.085655   0.095200   0.000000   0.248100
0.200000   0.179724   0.181300   0.000000   0.345300
0.300000   0.262759   0.259200   0.000000   0.418700
0.400000   0.338621   0.329700   0.000000   0.473800
0.500000   0.404000   0.393500   0.000000   0.520500
0.600000   0.462483   0.451200   0.000000   0.561400
0.700000   0.514897   0.503400   0.000000   0.597200
0.800000   0.562345   0.550700   0.000000   0.628900
0.900000   0.606069   0.593400   0.000000   0.657200
1.000000   0.646069   0.632100   0.000000   0.683300
$$


$TABLE: L test,acentric data:
$GRAPHS
:L test, acentric data 
:A:1,2,3,4:
$$
|l|   Observed   Acentric_theory   Acentric_theory,_perfect_twin   $$ 
$$
0.000000   0.000000   0.000000   0.000000
0.020000   0.021006   0.020000   0.029996
0.040000   0.042012   0.040000   0.059968
0.060000   0.063018   0.060000   0.089892
0.080000   0.085180   0.080000   0.119744
0.100000   0.106957   0.100000   0.149500
0.120000   0.128927   0.120000   0.179136
0.140000   0.149162   0.140000   0.208628
0.160000   0.169975   0.160000   0.237952
0.180000   0.191559   0.180000   0.267084
0.200000   0.210060   0.200000   0.296000
0.220000   0.233764   0.220000   0.324676
0.240000   0.254192   0.240000   0.353088
0.260000   0.273849   0.260000   0.381212
0.280000   0.294662   0.280000   0.409024
0.300000   0.316631   0.300000   0.436500
0.320000   0.337637   0.320000   0.463616
0.340000   0.357680   0.340000   0.490348
0.360000   0.377144   0.360000   0.516672
0.380000   0.396994   0.380000   0.542564
0.400000   0.417614   0.400000   0.568000
0.420000   0.437078   0.420000   0.592956
0.440000   0.457314   0.440000   0.617408
0.460000   0.476007   0.460000   0.641332
0.480000   0.496049   0.480000   0.664704
0.500000   0.516284   0.500000   0.687500
0.520000   0.535941   0.520000   0.709696
0.540000   0.557911   0.540000   0.731268
0.560000   0.574484   0.560000   0.752192
0.580000   0.595683   0.580000   0.772444
0.600000   0.613606   0.600000   0.792000
0.620000   0.636924   0.620000   0.810836
0.640000   0.659857   0.640000   0.828928
0.660000   0.676816   0.660000   0.846252
0.680000   0.698015   0.680000   0.862784
0.700000   0.719792   0.700000   0.878500
0.720000   0.739642   0.720000   0.893376
0.740000   0.758335   0.740000   0.907388
0.760000   0.782810   0.760000   0.920512
0.780000   0.803430   0.780000   0.932724
0.800000   0.827712   0.800000   0.944000
0.820000   0.851031   0.820000   0.954316
0.840000   0.872615   0.840000   0.963648
0.860000   0.891887   0.860000   0.971972
0.880000   0.909616   0.880000   0.979264
0.900000   0.931008   0.900000   0.985500
0.920000   0.945654   0.920000   0.990656
0.940000   0.963770   0.940000   0.994708
0.960000   0.980536   0.960000   0.997632
0.980000   0.993255   0.980000   0.999404
$$


$TABLE: Britton plot for twin law h,h-k,h-l:
$GRAPHS
:percentage negatives 
:A:1,2,3:
$$
alpha   percentage_negatives   fit   $$ 
$$
0.000000   0.000000   0.000000
0.009901   0.002839   0.000000
0.019802   0.006275   0.000000
0.029703   0.012550   0.008141
0.039604   0.020021   0.017621
0.049505   0.028089   0.027102
0.059406   0.037352   0.036582
0.069307   0.044674   0.046063
0.079208   0.052293   0.055543
0.089109   0.061557   0.065024
0.099010   0.071268   0.074504
0.108911   0.079337   0.083984
0.118812   0.088451   0.093465
0.128713   0.096668   0.102945
0.138614   0.107874   0.112426
0.148515   0.117137   0.121906
0.158416   0.127596   0.131387
0.168317   0.136859   0.140867
0.178218   0.144629   0.150348
0.188119   0.156880   0.159828
0.198020   0.164500   0.169309
0.207921   0.174062   0.178789
0.217822   0.184222   0.188270
0.227723   0.194382   0.197750
0.237624   0.205140   0.207230
0.247525   0.216495   0.216711
0.257426   0.227103   0.226191
0.267327   0.235470   0.235672
0.277228   0.243837   0.245152
0.287129   0.253698   0.254633
0.297030   0.263260   0.264113
0.306931   0.273569   0.273594
0.316832   0.282683   0.283074
0.326733   0.291499   0.292555
0.336634   0.299866   0.302035
0.346535   0.311669   0.311516
0.356436   0.321082   0.320996
0.366337   0.329747   0.330477
0.376238   0.339907   0.339957
0.386139   0.350366   0.349437
0.396040   0.359629   0.358918
0.405941   0.370238   0.368398
0.415842   0.381294   0.377879
0.425743   0.389810   0.387359
0.435644   0.399821   0.396840
0.445545   0.409831   0.406320
0.455446   0.419842   0.415801
0.465347   0.428507   0.425281
0.475248   0.437472   0.434762
0.485149   0.447184   0.444242
$$


$TABLE: H test for possible twin law h,h-k,h-l:
$GRAPHS
:H test for Acentric data 
:A:1,2,3:
$$
H   Observed_S(H)   Fitted_S(H)   $$ 
$$
0.000000   0.054692   0.000000
0.020000   0.079199   0.021928
0.040000   0.099522   0.043855
0.060000   0.118649   0.065783
0.080000   0.139570   0.087711
0.100000   0.159892   0.109639
0.120000   0.187388   0.131566
0.140000   0.205320   0.153494
0.160000   0.220861   0.175422
0.180000   0.243574   0.197349
0.200000   0.263897   0.219277
0.220000   0.283622   0.241205
0.240000   0.305738   0.263133
0.260000   0.325463   0.285060
0.280000   0.342499   0.306988
0.300000   0.361626   0.328916
0.320000   0.381949   0.350843
0.340000   0.407053   0.372771
0.360000   0.422594   0.394699
0.380000   0.439331   0.416627
0.400000   0.455469   0.438554
0.420000   0.481769   0.460482
0.440000   0.501494   0.482410
0.460000   0.516438   0.504337
0.480000   0.533772   0.526265
0.500000   0.554692   0.548193
0.520000   0.576210   0.570120
0.540000   0.601315   0.592048
0.560000   0.619247   0.613976
0.580000   0.636581   0.635904
0.600000   0.658697   0.657831
0.620000   0.675433   0.679759
0.640000   0.696354   0.701687
0.660000   0.714883   0.723614
0.680000   0.729827   0.745542
0.700000   0.753138   0.767470
0.720000   0.772863   0.789398
0.740000   0.795577   0.811325
0.760000   0.810520   0.833253
0.780000   0.832038   0.855181
0.800000   0.845786   0.877108
0.820000   0.866109   0.899036
0.840000   0.881650   0.920964
0.860000   0.898984   0.942892
0.880000   0.912732   0.964819
0.900000   0.931261   0.986747
0.920000   0.950389   1.000000
0.940000   0.965929   1.000000
0.960000   0.983861   1.000000
0.980000   0.992230   1.000000
$$


$TABLE: Likelihood based twin fraction estimation for possible twin law h,h-k,h-l:
$GRAPHS
:Likelihood based twin fraction estimate 
:A:1,2:
$$
alpha   NLL_(acentric_data)   $$ 
$$
0.021739   15094.589178
0.043478   19592.863265
0.065217   28124.522053
0.086957   38845.769047
0.108696   50722.196460
0.130435   63409.231928
0.152174   78534.417673
0.173913   94617.032773
0.195652   110133.704864
0.217391   126194.657857
0.239130   143176.053734
0.260870   160764.845117
0.282609   178996.248456
0.304348   197447.113571
0.326087   215659.760193
0.347826   234964.091231
0.369565   255483.053448
0.391304   273577.925036
0.413043   292351.284401
0.434783   312795.362946
0.456522   334382.556194
0.478261   356164.138121
$$