#############################################################
##                        mmtbx.xtriage                    ##
##                                                         ##
##      P.H. Zwart, R.W. Grosse-Kunstleve & P.D. Adams     ##
##                                                         ##
#############################################################
Date 2006-07-10 Time 23:15:54 PDT -0700


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


Effective parameters: 
scaling.input {
  parameters {
    asu_contents {
      n_residues = None
      n_bases = None
      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 = "logfile.log"
      ccp4_style_graphs = True
    }
  }
  xray_data {
    file_name = "1rfq_refine_001.mtz"
    obs_labels = OBS
    calc_labels = MODEL
    unit_cell = 101.5 101.5 104.1999969 90 90 90
    space_group = "P 43"
    high_resolution = None
    low_resolution = None
  }
}


Symmetry, cell and reflection file content summary

Miller array info: 1rfq_refine_001.mtz:FOBS,SIGFOBS
Observation type: xray.amplitude
Type of data: double, size=20909
Type of sigmas: double, size=20909
Number of Miller indices: 20909
Anomalous flag: False
Unit cell: (101.5, 101.5, 104.2, 90, 90, 90)
Space group: P 43 (No. 78)
Systematic absences: 0
Centric reflections: 850
Resolution range: 46.3505 3.00333
Completeness in resolution range: 0.986599
Completeness with d_max=infinity: 0.986367

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


Matthews coefficient and Solvent content statistics

Number of residues unknown, assuming 50% solvent content

----------------------------------------------------------------
|              Best guess :  982  residues 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 |
----------------------------------------------------------------------------------------
| 46.36 - 7.40 | 99.5%     | 99.5%     | 99.5%     | 99.5%     | 99.1%     | 96.0%     |
|  7.40 - 5.88 | 99.9%     | 99.9%     | 99.9%     | 99.5%     | 98.2%     | 92.7%     |
|  5.88 - 5.13 | 99.9%     | 99.8%     | 99.8%     | 99.3%     | 97.1%     | 88.1%     |
|  5.13 - 4.66 | 99.5%     | 99.4%     | 99.2%     | 98.7%     | 95.0%     | 84.5%     |
|  4.66 - 4.33 | 99.4%     | 99.4%     | 99.4%     | 99.0%     | 94.5%     | 79.6%     |
|  4.33 - 4.08 | 99.7%     | 99.2%     | 98.6%     | 96.9%     | 89.4%     | 64.4%     |
|  4.08 - 3.87 | 99.4%     | 98.8%     | 98.3%     | 95.6%     | 74.2%     | 16.9%     |
|  3.87 - 3.70 | 99.2%     | 98.0%     | 96.7%     | 92.3%     | 64.8%     | 12.6%     |
|  3.70 - 3.56 | 98.4%     | 97.1%     | 94.4%     | 88.7%     | 56.6%     | 9.0%      |
|  3.56 - 3.44 | 96.7%     | 94.5%     | 91.5%     | 80.9%     | 42.6%     | 6.2%      |
|  3.44 - 3.33 | 95.8%     | 92.0%     | 86.7%     | 73.2%     | 28.8%     | 1.8%      |
|  3.33 - 3.24 | 95.0%     | 88.8%     | 81.3%     | 65.3%     | 19.2%     | 0.6%      |
|  3.24 - 3.15 | 90.4%     | 82.2%     | 71.9%     | 49.5%     | 9.0%      | 0.3%      |
|  3.15 - 3.07 | 87.4%     | 75.9%     | 63.2%     | 39.6%     | 7.0%      | 0.1%      |
----------------------------------------------------------------------------------------

  The completeness of data for which I/sig(I)>3.00, exceeds  85% for
  for resolution ranges lower than 3.33A.
  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 1rfq_refine_001.mtz:FOBS,SIGFOBS:
77.14 A**(-2)
Estimated -log of scale factor of 1rfq_refine_001.mtz:FOBS,SIGFOBS:
-1.49


Maximum likelihood anisotropic Wilson scaling 
ML estimate of overall B_cart value of 1rfq_refine_001.mtz:FOBS,SIGFOBS:
82.93,  0.00,  0.00
       82.93,  0.00
              67.65
Equivalent representation as U_cif: 
 1.05, -0.00, -0.00
        1.05,  0.00
               0.86

ML estimate of  -log of scale factor of 1rfq_refine_001.mtz:FOBS,SIGFOBS:
-1.49
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:         - 46.3524 [  0/5  ] 0.000
bin  1: 46.3524 - 10.7334 [484/490] 0.988
bin  2: 10.7334 -  8.5368 [461/463] 0.996
bin  3:  8.5368 -  7.4628 [464/464] 1.000
bin  4:  7.4628 -  6.7827 [464/464] 1.000
bin  5:  6.7827 -  6.2979 [456/456] 1.000
bin  6:  6.2979 -  5.9273 [471/472] 0.998
bin  7:  5.9273 -  5.6310 [459/459] 1.000
bin  8:  5.6310 -  5.3863 [454/455] 0.998
bin  9:  5.3863 -  5.1792 [465/465] 1.000
bin 10:  5.1792 -  5.0007 [472/475] 0.994
unused:  5.0007 -         [  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> |
------------------------------------------------
|    3.892  |   4.86  |   1.00 |     0.835     |
|    3.779  |   4.82  |   1.00 |     0.837     |
------------------------------------------------

 Possible reasons for the presence of the reported
 unexpected low or elevated mean intensity in
 a given resolution bin are : 
 - missing overloaded or weak reflections
 - suboptimal data processing
 - satelite (ice) crystals
 - NCS
 - translational pseudo symmetry (detected elsewhere)
 - outliers (detected elsewhere)
 - ice rings (detected elsewhere)
 - other problems
 Note that the presence of abnormalities 
 in a certain region of reciprocal space might
 confuse the data validation algorithm throughout
 a large region of reciprocal space, even though
 the data is acceptable in those areas. 


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

 Acentric reflections:

            None 

 Centric reflections:

            None 


Ice ring related problems

 The following statistics were obtained from ice-ring 
 insensitive resolution ranges 
  mean bin z_score      : 1.97
      ( rms deviation   : 1.36 )
  mean bin completeness : 0.98
     ( rms deviation   : 0.04 )

 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  |   4.86  |   1.00 |     1.000     |
|    3.669  |   3.10  |   1.00 |     0.750     |
|    3.441  |   1.72  |   0.99 |     0.530     |
------------------------------------------------

 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 3.33 Angstrom.

Determining possible twin laws.

The following twin laws have been found:

-------------------------------------------------------------------------------
| Type | Axis   | R metric (%) | delta (le Page) | delta (Lebedev) | Twin law |
-------------------------------------------------------------------------------
|   M  | 2-fold | 0.000        | 0.000           | 0.000           | -h,k,-l  |
|  PM  | 2-fold | 3.531        | 1.504           | 0.021           | -h,-l,-k |
|  PM  | 4-fold | 3.531        | 0.000           | 0.021           | h,-l,k   |
|  PM  | 3-fold | 3.531        | 1.418           | 0.021           | k,l,h    |
|  PM  | 3-fold | 3.531        | 1.418           | 0.021           | -k,l,-h  |
-------------------------------------------------------------------------------
M:  Merohedral twin law
PM: Pseudomerohedral twin law

  1 merohedral twin operators found
  4 pseudo-merohedral twin operators found
In total,   5 twin operator were found


Number of centrics  : 627
Number of acentrics : 14321

 Largest patterson peak with length larger then 15 Angstrom 

 Frac. coord.        :   0.500    0.500    0.011
 Distance to origin  :  71.780
 Height (origin=100) :   3.627
 p_value(height)     :   9.997e-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  9.997e-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    :1.611   (untwinned: 2.000; perfect twin 1.500)
   <F>^2/<F^2>    :0.867   (untwinned: 0.785; perfect twin 0.885)
   <|E^2 - 1|>    :0.576   (untwinned: 0.736; perfect twin 0.541)


Centric reflections 
   <I^2>/<I>^2    :2.165   (untwinned: 3.000; perfect twin 2.000)
   <F>^2/<F^2>    :0.782   (untwinned: 0.637; perfect twin 0.785)
   <|E^2 - 1|>    :0.861   (untwinned: 0.968; perfect twin 0.736)



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 |   0.022 |    0.095 |  0.069 |   0.248 |
| 0.2 |   0.073 |    0.181 |  0.144 |   0.345 |
| 0.3 |   0.146 |    0.259 |  0.219 |   0.419 |
| 0.4 |   0.219 |    0.330 |  0.282 |   0.474 |
| 0.5 |   0.298 |    0.394 |  0.324 |   0.520 |
| 0.6 |   0.369 |    0.451 |  0.388 |   0.561 |
| 0.7 |   0.439 |    0.503 |  0.437 |   0.597 |
| 0.8 |   0.503 |    0.551 |  0.494 |   0.629 |
| 0.9 |   0.561 |    0.593 |  0.534 |   0.657 |
| 1.0 |   0.613 |    0.632 |  0.563 |   0.683 |
-----------------------------------------------
| Maximum deviation acentric      :  0.113    |
| Maximum deviation centric       :  0.202    |
|                                             |
| <NZ(obs)-NZ(twinned)>_acentric  : -0.068    |
| <NZ(obs)-NZ(twinned)>_centric   : -0.153    |
-----------------------------------------------


 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|   :0.387  (untwinned: 0.500; perfect twin: 0.375)
  Mean  L^2  :0.212  (untwinned: 0.333; perfect twin: 0.200)

  The distribution of |L| values indicates a twin fraction of
  0.30. 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,k,-l
---------------------------------------------


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

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

mean |H| : 0.125   (0.50: untwinned; 0.0: 50% twinned)
mean H^2 : 0.025   (0.33: untwinned; 0.0: 50% twinned)
Estimation of twin fraction via mean |H|: 0.375
Estimation of twin fraction via cum. dist. of H: 0.368



Britton analyses

  Extrapolation performed on  0.39 < alpha < 0.495 
  Estimated twin fraction: 0.364
  Correlation: 0.9957

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   : 0.135
   R_abs_twin calculated data : 0.439

  R_sq_twin = <(I1-I2)^2>/<(I1+I2)^2>
   R_sq_twin observed data    : 0.026
   R_sq_twin calculated data  : 0.236



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


   The estimated twin fraction is equal to 0.358


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


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

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

mean |H| : 0.387   (0.50: untwinned; 0.0: 50% twinned)
mean H^2 : 0.218   (0.33: untwinned; 0.0: 50% twinned)
Estimation of twin fraction via mean |H|: 0.113
Estimation of twin fraction via cum. dist. of H: 0.089



Britton analyses

  Extrapolation performed on  0.08 < alpha < 0.495 
  Estimated twin fraction: 0.089
  Correlation: 0.9956

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   : 0.404
   R_abs_twin calculated data : 0.507

  R_sq_twin = <(I1-I2)^2>/<(I1+I2)^2>
   R_sq_twin observed data    : 0.241
   R_sq_twin calculated data  : 0.321



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


   The estimated twin fraction is equal to 0.022


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


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

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

mean |H| : 0.391   (0.50: untwinned; 0.0: 50% twinned)
mean H^2 : 0.219   (0.33: untwinned; 0.0: 50% twinned)
Estimation of twin fraction via mean |H|: 0.109
Estimation of twin fraction via cum. dist. of H: 0.088



Britton analyses

  Extrapolation performed on  0.10 < alpha < 0.495 
  Estimated twin fraction: 0.099
  Correlation: 0.9953

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   : 0.395
   R_abs_twin calculated data : 0.512

  R_sq_twin = <(I1-I2)^2>/<(I1+I2)^2>
   R_sq_twin observed data    : 0.233
   R_sq_twin calculated data  : 0.345



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


   The estimated twin fraction is equal to 0.022


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


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

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

mean |H| : 0.386   (0.50: untwinned; 0.0: 50% twinned)
mean H^2 : 0.213   (0.33: untwinned; 0.0: 50% twinned)
Estimation of twin fraction via mean |H|: 0.114
Estimation of twin fraction via cum. dist. of H: 0.094



Britton analyses

  Extrapolation performed on  0.11 < alpha < 0.495 
  Estimated twin fraction: 0.105
  Correlation: 0.9956

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   : 0.395
   R_abs_twin calculated data : 0.511

  R_sq_twin = <(I1-I2)^2>/<(I1+I2)^2>
   R_sq_twin observed data    : 0.232
   R_sq_twin calculated data  : 0.345



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


   The estimated twin fraction is equal to 0.022


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


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

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

mean |H| : 0.382   (0.50: untwinned; 0.0: 50% twinned)
mean H^2 : 0.211   (0.33: untwinned; 0.0: 50% twinned)
Estimation of twin fraction via mean |H|: 0.118
Estimation of twin fraction via cum. dist. of H: 0.096



Britton analyses

  Extrapolation performed on  0.08 < alpha < 0.495 
  Estimated twin fraction: 0.095
  Correlation: 0.9951

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   : 0.403
   R_abs_twin calculated data : 0.515

  R_sq_twin = <(I1-I2)^2>/<(I1+I2)^2>
   R_sq_twin observed data    : 0.239
   R_sq_twin calculated data  : 0.343



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 4
  the point group in the niggli setting is P 4
The point group of the lattice is P 4 3 2
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 |
----------------------------------------------------------------------------------
| P 4 3 2     | 0.348       | 0.404      | None          | None         |        |
| P 4 2 2     | 0.135       | 0.135      | 0.399         | 0.395        |        |
| P 4         | None        | None       | 0.348         | 0.135        | <---   |
----------------------------------------------------------------------------------

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 4

Possible space groups in this point groups are:
   Unit cell: (101.5, 101.5, 104.2, 90, 90, 90)
   Space group: P 41 (No. 76)

   Unit cell: (101.5, 101.5, 104.2, 90, 90, 90)
   Space group: P 43 (No. 78)

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 : 1.611
  - <F>^2/<F^2> : 0.867
  - <|E^2-1|>   : 0.576
  - 
, <L^2>: 0.387, 0.212
       Multivariate Z score L-test: 10.253 
       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_abs obs. | R_abs calc. | Britton alpha | H alpha | ML alpha |
-----------------------------------------------------------------------------------
| -h,k,-l  |   M  | 0.135      | 0.439       | 0.364         | 0.368   | 0.358    |
| -h,-l,-k |  PM  | 0.404      | 0.507       | 0.089         | 0.089   | 0.022    |
| h,-l,k   |  PM  | 0.395      | 0.512       | 0.099         | 0.088   | 0.022    |
| k,l,h    |  PM  | 0.395      | 0.511       | 0.105         | 0.094   | 0.022    |
| -k,l,-h  |  PM  | 0.403      | 0.515       | 0.095         | 0.096   | 0.022    |
-----------------------------------------------------------------------------------

Patterson analyses
  - Largest peak height   : 3.627
   (correpsonding p value : 9.997e-01)


The largest off-origin peak in the Patterson function is 3.63% of the 
height of the origin peak. No significant pseudotranslation is detected.

The results of the L-test indicate that the intensity statistics
are significantly different then is expected from good to reasonable,
untwinned data.
As there are twin laws possible given the crystal symmetry, twinning could
be the reason for the departure of the intensity statistics from normality.
It might be worthwhile carrying refinement with a twin specific target function.

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


$TABLE: Intensity plots:
$GRAPHS
:Intensity plots 
:A:1,2,3,4:
$$
1/resol^2   <I>_smooth_approximation   <I>_via_binning   <I>_expected   $$ 
$$
0.010007   652380.446715   597069.956706   403421.726747
0.014007   381115.442393   391136.170928   317745.228230
0.018007   288289.551798   281795.363165   233341.320490
0.022007   192105.610359   192677.020765   199945.774786
0.026007   195989.300727   204558.890271   199856.767997
0.030007   228177.786019   224425.061918   219950.757244
0.034007   239403.576258   243351.006765   267200.539389
0.038007   283422.463811   287559.222813   342391.437884
0.042007   376957.466816   378883.239198   427530.750493
0.046007   482789.414264   482735.455758   497428.489292
0.050007   543389.570832   543310.255808   536574.955984
0.054007   530904.552540   524715.620023   545681.926114
0.058007   482214.335766   483668.383217   536523.971172
0.062007   449670.485041   458170.478161   522307.629455
0.066007   435941.863463   426613.337432   510728.647935
0.070007   423376.661832   420403.862707   502472.151623
0.074007   424399.570591   436007.305374   493847.042520
0.078007   439959.008322   429302.475734   480627.502783
0.082007   430729.306119   428476.401277   460744.565756
0.086007   402288.801445   409151.457245   434966.177023
0.090007   406298.100356   406482.508796   405993.158158
0.094007   400625.987032   389786.650172   376963.099230
0.098007   352798.501792   354817.688608   350241.706797
0.102007   361723.903961   361185.723749   326922.305295
0.106007   348981.624273   355545.045328   306990.545303
0.110865   338981.899299   331687.210403   286462.863949
$$


$TABLE: Z scores and completeness:
$GRAPHS
:Data sanity and completeness check 
:A:1,2,3:
$$
1/resol^2   Z_score   Completeness   $$ 
$$
0.010007   4.319748   0.994253
0.014007   2.845306   0.997653
0.018007   2.916125   1.000000
0.022007   0.697867   1.000000
0.026007   0.438064   0.998168
0.030007   0.384362   1.000000
0.034007   1.887273   0.998413
0.038007   3.703212   0.995608
0.042007   2.518325   0.997143
0.046007   0.621011   1.000000
0.050007   0.262834   0.998684
0.054007   0.879560   0.997500
0.058007   2.526176   0.998804
0.062007   3.393588   0.998833
0.066007   4.857739   0.997738
0.070007   4.823555   0.996659
0.074007   3.331594   0.997821
0.078007   3.099435   0.996988
0.082007   1.981569   0.992784
0.086007   1.718326   0.988083
0.090007   0.034107   0.993177
0.094007   0.955420   0.985929
0.098007   0.365448   0.981518
0.102007   2.798911   0.971556
0.106007   3.996646   0.969863
0.110865   3.351014   0.843447
$$


$TABLE: <I/sigma_I>:
$GRAPHS
:Signal to noise 
:N:1,2:
$$
1/resol^2   <I/sigma_I>;_Signal_to_noise   $$ 
$$
0.002465   23.094384
0.006465   26.386800
0.010465   27.315903
0.014465   27.669998
0.018465   26.810417
0.022465   25.556973
0.026465   21.316799
0.030465   21.040816
0.034465   20.437086
0.038465   19.947142
0.042465   19.710939
0.046465   19.793928
0.050465   17.895036
0.054465   16.703995
0.058465   15.306013
0.062465   12.083335
0.066465   11.518181
0.070465   10.957901
0.074465   10.497571
0.078465   9.850786
0.082465   8.750734
0.086465   8.004656
0.090465   7.149559
0.094465   6.347974
0.098465   5.311986
0.102465   4.747573
0.106465   4.346215
0.110465   3.615850
$$


$TABLE: NZ test:
$GRAPHS
:NZ test, acentric and centric data 
:A:1,2,3,4,5:
$$
z   Acentric_observed   Acentric_untwinned   Centric_observed   Centric_untwinned   $$ 
$$
0.000000   0.000000   0.000000   0.000000   0.000000
0.100000   0.021507   0.095200   0.068581   0.248100
0.200000   0.072900   0.181300   0.143541   0.345300
0.300000   0.145800   0.259200   0.218501   0.418700
0.400000   0.218560   0.329700   0.282297   0.473800
0.500000   0.298233   0.393500   0.323764   0.520500
0.600000   0.369388   0.451200   0.387560   0.561400
0.700000   0.438657   0.503400   0.437002   0.597200
0.800000   0.503317   0.550700   0.494418   0.628900
0.900000   0.560645   0.593400   0.534290   0.657200
1.000000   0.612527   0.632100   0.562998   0.683300
$$


$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   0.003583   0.000000   0.000000
0.020000   0.031292   0.020000   0.029996
0.040000   0.059798   0.040000   0.059968
0.060000   0.085755   0.060000   0.089892
0.080000   0.115455   0.080000   0.119744
0.100000   0.145792   0.100000   0.149500
0.120000   0.173979   0.120000   0.179136
0.140000   0.199936   0.140000   0.208628
0.160000   0.228681   0.160000   0.237952
0.180000   0.255036   0.180000   0.267084
0.200000   0.283621   0.200000   0.296000
0.220000   0.312047   0.220000   0.324676
0.240000   0.338005   0.240000   0.353088
0.260000   0.365793   0.260000   0.381212
0.280000   0.393582   0.280000   0.409024
0.300000   0.421690   0.300000   0.436500
0.320000   0.447727   0.320000   0.463616
0.340000   0.473366   0.340000   0.490348
0.360000   0.500597   0.360000   0.516672
0.380000   0.526634   0.380000   0.542564
0.400000   0.550044   0.400000   0.568000
0.420000   0.574409   0.420000   0.592956
0.440000   0.598853   0.440000   0.617408
0.460000   0.624333   0.460000   0.641332
0.480000   0.645991   0.480000   0.664704
0.500000   0.667649   0.500000   0.687500
0.520000   0.689068   0.520000   0.709696
0.540000   0.709690   0.540000   0.731268
0.560000   0.731109   0.560000   0.752192
0.580000   0.750856   0.580000   0.772444
0.600000   0.769807   0.600000   0.792000
0.620000   0.787722   0.620000   0.810836
0.640000   0.806911   0.640000   0.828928
0.660000   0.825543   0.660000   0.846252
0.680000   0.842026   0.680000   0.862784
0.700000   0.859623   0.700000   0.878500
0.720000   0.875866   0.720000   0.893376
0.740000   0.891870   0.740000   0.907388
0.760000   0.906680   0.760000   0.920512
0.780000   0.921968   0.780000   0.932724
0.800000   0.935823   0.800000   0.944000
0.820000   0.948085   0.820000   0.954316
0.840000   0.958038   0.840000   0.963648
0.860000   0.966478   0.860000   0.971972
0.880000   0.973963   0.880000   0.979264
0.900000   0.982005   0.900000   0.985500
0.920000   0.987738   0.920000   0.990656
0.940000   0.992038   0.940000   0.994708
0.960000   0.996019   0.960000   0.997632
0.980000   0.999045   0.980000   0.999404
$$


$TABLE: Britton plot for twin law -h,k,-l:
$GRAPHS
:percentage negatives 
:A:1,2,3:
$$
alpha   percentage_negatives   fit   $$ 
$$
0.000000   0.000000   0.000000
0.009901   0.000000   0.000000
0.019802   0.000000   0.000000
0.029703   0.000210   0.000000
0.039604   0.000210   0.000000
0.049505   0.000280   0.000000
0.059406   0.000350   0.000000
0.069307   0.000350   0.000000
0.079208   0.000350   0.000000
0.089109   0.000350   0.000000
0.099010   0.000420   0.000000
0.108911   0.000420   0.000000
0.118812   0.000490   0.000000
0.128713   0.000629   0.000000
0.138614   0.000909   0.000000
0.148515   0.000979   0.000000
0.158416   0.001329   0.000000
0.168317   0.001469   0.000000
0.178218   0.001678   0.000000
0.188119   0.001748   0.000000
0.198020   0.001818   0.000000
0.207921   0.001958   0.000000
0.217822   0.002308   0.000000
0.227723   0.002448   0.000000
0.237624   0.002657   0.000000
0.247525   0.003357   0.000000
0.257426   0.003776   0.000000
0.267327   0.004406   0.000000
0.277228   0.005035   0.000000
0.287129   0.005664   0.000000
0.297030   0.006713   0.000000
0.306931   0.008671   0.000000
0.316832   0.012238   0.000000
0.326733   0.017972   0.000000
0.336634   0.025175   0.000000
0.346535   0.034196   0.000000
0.356436   0.044126   0.000000
0.366337   0.057622   0.008955
0.376238   0.072448   0.040865
0.386139   0.090769   0.072776
0.396040   0.109231   0.104686
0.405941   0.133986   0.136596
0.415842   0.159790   0.168506
0.425743   0.190490   0.200417
0.435644   0.222587   0.232327
0.445545   0.257413   0.264237
0.455446   0.291538   0.296147
0.465347   0.328741   0.328057
0.475248   0.365455   0.359968
0.485149   0.405594   0.391878
$$


$TABLE: H test for possible twin law -h,k,-l:
$GRAPHS
:H test for Acentric data 
:A:1,2,3:
$$
H   Observed_S(H)   Fitted_S(H)   $$ 
$$
0.000000   0.073287   0.000000
0.020000   0.152168   0.075547
0.040000   0.235245   0.151094
0.060000   0.318042   0.226642
0.080000   0.396923   0.302189
0.100000   0.467133   0.377736
0.120000   0.540420   0.453283
0.140000   0.609790   0.528830
0.160000   0.674126   0.604377
0.180000   0.728112   0.679925
0.200000   0.775664   0.755472
0.220000   0.820979   0.831019
0.240000   0.857622   0.906566
0.260000   0.888392   0.982113
0.280000   0.917203   1.000000
0.300000   0.937902   1.000000
0.320000   0.956084   1.000000
0.340000   0.970909   1.000000
0.360000   0.983217   1.000000
0.380000   0.994685   1.000000
0.400000   0.997203   1.000000
0.420000   0.998601   1.000000
0.440000   1.000000   1.000000
0.460000   1.000000   1.000000
0.480000   1.000000   1.000000
0.500000   1.000000   1.000000
0.520000   1.000000   1.000000
0.540000   1.000000   1.000000
0.560000   1.000000   1.000000
0.580000   1.000000   1.000000
0.600000   1.000000   1.000000
0.620000   1.000000   1.000000
0.640000   1.000000   1.000000
0.660000   1.000000   1.000000
0.680000   1.000000   1.000000
0.700000   1.000000   1.000000
0.720000   1.000000   1.000000
0.740000   1.000000   1.000000
0.760000   1.000000   1.000000
0.780000   1.000000   1.000000
0.800000   1.000000   1.000000
0.820000   1.000000   1.000000
0.840000   1.000000   1.000000
0.860000   1.000000   1.000000
0.880000   1.000000   1.000000
0.900000   1.000000   1.000000
0.920000   1.000000   1.000000
0.940000   1.000000   1.000000
0.960000   1.000000   1.000000
0.980000   1.000000   1.000000
$$


$TABLE: Likelihood based twin fraction estimation for possible twin law -h,k,-l:
$GRAPHS
:Likelihood based twin fraction estimate 
:A:1,2:
$$
alpha   NLL_(acentric_data)   $$ 
$$
0.021739   27731.184580
0.043478   27075.638273
0.065217   26391.074403
0.086957   25671.359832
0.108696   24914.378777
0.130435   24117.453906
0.152174   23275.284806
0.173913   22381.627698
0.195652   21429.440459
0.217391   20412.460342
0.239130   19323.305054
0.260870   18157.094891
0.282609   16920.508058
0.304348   15660.662446
0.326087   14545.143865
0.347826   13938.469756
0.369565   14515.603864
0.391304   17626.874612
0.413043   25354.308999
0.434783   42244.089240
0.456522   68356.269511
0.478261   104782.649735
$$


$TABLE: Britton plot for twin law -h,-l,-k:
$GRAPHS
:percentage negatives 
:A:1,2,3:
$$
alpha   percentage_negatives   fit   $$ 
$$
0.000000   0.000000   0.000000
0.009901   0.000998   0.000000
0.019802   0.002352   0.000000
0.029703   0.003564   0.000000
0.039604   0.006130   0.000000
0.049505   0.009052   0.000000
0.059406   0.012901   0.000000
0.069307   0.016393   0.000000
0.079208   0.021240   0.000000
0.089109   0.026158   0.000000
0.099010   0.031219   0.010683
0.108911   0.037491   0.021711
0.118812   0.043763   0.032740
0.128713   0.050606   0.043768
0.138614   0.059159   0.054796
0.148515   0.066928   0.065825
0.158416   0.074269   0.076853
0.168317   0.082680   0.087882
0.178218   0.091304   0.098910
0.188119   0.100499   0.109938
0.198020   0.111547   0.120967
0.207921   0.119815   0.131995
0.217822   0.129865   0.143024
0.227723   0.140627   0.154052
0.237624   0.152031   0.165080
0.247525   0.163435   0.176109
0.257426   0.174626   0.187137
0.267327   0.186386   0.198166
0.277228   0.196935   0.209194
0.287129   0.208268   0.220222
0.297030   0.220955   0.231251
0.306931   0.230934   0.242279
0.316832   0.243051   0.253308
0.326733   0.254811   0.264336
0.336634   0.267997   0.275364
0.346535   0.281468   0.286393
0.356436   0.292017   0.297421
0.366337   0.303136   0.308450
0.376238   0.315538   0.319478
0.386139   0.327014   0.330507
0.396040   0.340271   0.341535
0.405941   0.353100   0.352563
0.415842   0.366999   0.363592
0.425743   0.380898   0.374620
0.435644   0.393942   0.385649
0.445545   0.407341   0.396677
0.455446   0.422381   0.407705
0.465347   0.434783   0.418734
0.475248   0.449679   0.429762
0.485149   0.462580   0.440791
$$


$TABLE: H test for possible twin law -h,-l,-k:
$GRAPHS
:H test for Acentric data 
:A:1,2,3:
$$
H   Observed_S(H)   Fitted_S(H)   $$ 
$$
0.000000   0.025802   0.000000
0.020000   0.055738   0.024326
0.040000   0.084533   0.048651
0.060000   0.113329   0.072977
0.080000   0.137277   0.097303
0.100000   0.168639   0.121628
0.120000   0.196864   0.145954
0.140000   0.221953   0.170279
0.160000   0.247327   0.194605
0.180000   0.274127   0.218931
0.200000   0.299786   0.243256
0.220000   0.327156   0.267582
0.240000   0.349964   0.291908
0.260000   0.371062   0.316233
0.280000   0.396436   0.340559
0.300000   0.417819   0.364885
0.320000   0.444048   0.389210
0.340000   0.471133   0.413536
0.360000   0.495652   0.437861
0.380000   0.518460   0.462187
0.400000   0.542409   0.486513
0.420000   0.564647   0.510838
0.440000   0.588311   0.535164
0.460000   0.606842   0.559490
0.480000   0.631076   0.583815
0.500000   0.653314   0.608141
0.520000   0.675552   0.632467
0.540000   0.699216   0.656792
0.560000   0.722309   0.681118
0.580000   0.742552   0.705443
0.600000   0.758803   0.729769
0.620000   0.781896   0.754095
0.640000   0.801853   0.778420
0.660000   0.818959   0.802746
0.680000   0.838061   0.827072
0.700000   0.852316   0.851397
0.720000   0.866287   0.875723
0.740000   0.884248   0.900049
0.760000   0.899073   0.924374
0.780000   0.911903   0.948700
0.800000   0.926158   0.973026
0.820000   0.937277   0.997351
0.840000   0.950392   1.000000
0.860000   0.959800   1.000000
0.880000   0.968068   1.000000
0.900000   0.976907   1.000000
0.920000   0.984034   1.000000
0.940000   0.990592   1.000000
0.960000   0.994298   1.000000
0.980000   0.997434   1.000000
$$


$TABLE: Likelihood based twin fraction estimation for possible twin law -h,-l,-k:
$GRAPHS
:Likelihood based twin fraction estimate 
:A:1,2:
$$
alpha   NLL_(acentric_data)   $$ 
$$
0.021739   28098.976736
0.043478   28822.276080
0.065217   32302.981989
0.086957   38462.458098
0.108696   47029.247937
0.130435   58606.023998
0.152174   72049.526087
0.173913   88257.005034
0.195652   107674.318518
0.217391   129390.892366
0.239130   154778.442608
0.260870   183974.978782
0.282609   213410.201091
0.304348   246117.865441
0.326087   283680.948318
0.347826   323768.056540
0.369565   361499.629612
0.391304   400087.098080
0.413043   442813.820569
0.434783   488587.548091
0.456522   535896.877206
0.478261   584596.125238
$$


$TABLE: Britton plot for twin law h,-l,k:
$GRAPHS
:percentage negatives 
:A:1,2,3:
$$
alpha   percentage_negatives   fit   $$ 
$$
0.000000   0.000000   0.000000
0.009901   0.000927   0.000000
0.019802   0.002852   0.000000
0.029703   0.004206   0.000000
0.039604   0.006202   0.000000
0.049505   0.009909   0.000000
0.059406   0.013189   0.000000
0.069307   0.016112   0.000000
0.079208   0.020532   0.000000
0.089109   0.025593   0.000000
0.099010   0.031582   0.000422
0.108911   0.037285   0.012013
0.118812   0.043915   0.023604
0.128713   0.050973   0.035195
0.138614   0.058601   0.046786
0.148515   0.066229   0.058377
0.158416   0.073501   0.069968
0.168317   0.081201   0.081559
0.178218   0.090255   0.093150
0.188119   0.099879   0.104741
0.198020   0.110145   0.116332
0.207921   0.120268   0.127923
0.217822   0.129821   0.139514
0.227723   0.139018   0.151105
0.237624   0.149569   0.162696
0.247525   0.160405   0.174287
0.257426   0.171598   0.185879
0.267327   0.183004   0.197470
0.277228   0.194482   0.209061
0.287129   0.205817   0.220652
0.297030   0.218222   0.232243
0.306931   0.229985   0.243834
0.316832   0.244528   0.255425
0.326733   0.256790   0.267016
0.336634   0.268625   0.278607
0.346535   0.282455   0.290198
0.356436   0.294432   0.301789
0.366337   0.308049   0.313380
0.376238   0.319527   0.324971
0.386139   0.332430   0.336562
0.396040   0.344978   0.348153
0.405941   0.357453   0.359744
0.415842   0.373423   0.371335
0.425743   0.389606   0.382926
0.435644   0.405646   0.394517
0.445545   0.418978   0.406108
0.455446   0.432737   0.417700
0.465347   0.446710   0.429291
0.475248   0.460327   0.440882
0.485149   0.475440   0.452473
$$


$TABLE: H test for possible twin law h,-l,k:
$GRAPHS
:H test for Acentric data 
:A:1,2,3:
$$
H   Observed_S(H)   Fitted_S(H)   $$ 
$$
0.000000   0.001426   0.000000
0.020000   0.034080   0.024267
0.040000   0.061172   0.048533
0.060000   0.089976   0.072800
0.080000   0.123770   0.097067
0.100000   0.154000   0.121333
0.120000   0.182090   0.145600
0.140000   0.209468   0.169867
0.160000   0.239983   0.194133
0.180000   0.268787   0.218400
0.200000   0.297162   0.242667
0.220000   0.322544   0.266933
0.240000   0.347640   0.291200
0.260000   0.370170   0.315467
0.280000   0.396264   0.339733
0.300000   0.417796   0.364000
0.320000   0.445886   0.388267
0.340000   0.468558   0.412533
0.360000   0.491373   0.436800
0.380000   0.518038   0.461067
0.400000   0.540995   0.485333
0.420000   0.561671   0.509600
0.440000   0.585484   0.533867
0.460000   0.608869   0.558133
0.480000   0.633538   0.582400
0.500000   0.654214   0.606667
0.520000   0.676743   0.630933
0.540000   0.698845   0.655200
0.560000   0.717952   0.679467
0.580000   0.739341   0.703733
0.600000   0.760587   0.728000
0.620000   0.781121   0.752267
0.640000   0.800513   0.776533
0.660000   0.820334   0.800800
0.680000   0.837587   0.825067
0.700000   0.855126   0.849333
0.720000   0.870811   0.873600
0.740000   0.885784   0.897867
0.760000   0.898902   0.922133
0.780000   0.913161   0.946400
0.800000   0.922430   0.970667
0.820000   0.936832   0.994933
0.840000   0.948382   1.000000
0.860000   0.957507   1.000000
0.880000   0.966063   1.000000
0.900000   0.975046   1.000000
0.920000   0.983602   1.000000
0.940000   0.989163   1.000000
0.960000   0.993726   1.000000
0.980000   0.997576   1.000000
$$


$TABLE: Likelihood based twin fraction estimation for possible twin law h,-l,k:
$GRAPHS
:Likelihood based twin fraction estimate 
:A:1,2:
$$
alpha   NLL_(acentric_data)   $$ 
$$
0.021739   28211.815848
0.043478   29235.096242
0.065217   33006.010129
0.086957   39437.656906
0.108696   47818.229710
0.130435   58384.758708
0.152174   71758.536479
0.173913   89104.621338
0.195652   107567.335307
0.217391   129299.375750
0.239130   155085.122968
0.260870   182883.608616
0.282609   212773.929310
0.304348   245793.885486
0.326087   280699.159772
0.347826   319111.210817
0.369565   358952.636174
0.391304   399793.992388
0.413043   443876.491528
0.434783   487644.722931
0.456522   537397.399829
0.478261   586484.773188
$$


$TABLE: Britton plot for twin law k,l,h:
$GRAPHS
:percentage negatives 
:A:1,2,3:
$$
alpha   percentage_negatives   fit   $$ 
$$
0.000000   0.000000   0.000000
0.009901   0.000896   0.000000
0.019802   0.001642   0.000000
0.029703   0.003135   0.000000
0.039604   0.005374   0.000000
0.049505   0.007463   0.000000
0.059406   0.011195   0.000000
0.069307   0.015076   0.000000
0.079208   0.018957   0.000000
0.089109   0.023136   0.000000
0.099010   0.029107   0.000000
0.108911   0.034406   0.004986
0.118812   0.040003   0.016751
0.128713   0.046421   0.028516
0.138614   0.052765   0.040280
0.148515   0.060602   0.052045
0.158416   0.069781   0.063810
0.168317   0.078289   0.075575
0.178218   0.086574   0.087339
0.188119   0.096425   0.099104
0.198020   0.106650   0.110869
0.207921   0.117546   0.122634
0.217822   0.127845   0.134399
0.227723   0.137697   0.146163
0.237624   0.148444   0.157928
0.247525   0.159863   0.169693
0.257426   0.170013   0.181458
0.267327   0.181879   0.193223
0.277228   0.193970   0.204987
0.287129   0.205463   0.216752
0.297030   0.216285   0.228517
0.306931   0.227704   0.240282
0.316832   0.240839   0.252046
0.326733   0.252183   0.263811
0.336634   0.263452   0.275576
0.346535   0.276812   0.287341
0.356436   0.288753   0.299106
0.366337   0.301217   0.310870
0.376238   0.315397   0.322635
0.386139   0.327786   0.334400
0.396040   0.341593   0.346165
0.405941   0.354579   0.357929
0.415842   0.369580   0.369694
0.425743   0.383835   0.381459
0.435644   0.399806   0.393224
0.445545   0.416150   0.404989
0.455446   0.431674   0.416753
0.465347   0.447571   0.428518
0.475248   0.464811   0.440283
0.485149   0.477498   0.452048
$$


$TABLE: H test for possible twin law k,l,h:
$GRAPHS
:H test for Acentric data 
:A:1,2,3:
$$
H   Observed_S(H)   Fitted_S(H)   $$ 
$$
0.000000   0.000597   0.000000
0.020000   0.033139   0.024625
0.040000   0.059710   0.049250
0.060000   0.088969   0.073875
0.080000   0.123153   0.098499
0.100000   0.153904   0.123124
0.120000   0.182415   0.147749
0.140000   0.210927   0.172374
0.160000   0.242275   0.196999
0.180000   0.271235   0.221624
0.200000   0.299746   0.246248
0.220000   0.325571   0.270873
0.240000   0.351246   0.295498
0.260000   0.374534   0.320123
0.280000   0.400358   0.344748
0.300000   0.422451   0.369373
0.320000   0.450963   0.393998
0.340000   0.473504   0.418622
0.360000   0.497089   0.443247
0.380000   0.524108   0.467872
0.400000   0.547544   0.492497
0.420000   0.568742   0.517122
0.440000   0.593372   0.541747
0.460000   0.617555   0.566371
0.480000   0.641738   0.590996
0.500000   0.662188   0.615621
0.520000   0.685177   0.640246
0.540000   0.707718   0.664871
0.560000   0.727273   0.689496
0.580000   0.748619   0.714121
0.600000   0.769966   0.738745
0.620000   0.789819   0.763370
0.640000   0.809822   0.787995
0.660000   0.828780   0.812620
0.680000   0.845649   0.837245
0.700000   0.862517   0.861870
0.720000   0.878191   0.886494
0.740000   0.893268   0.911119
0.760000   0.906702   0.935744
0.780000   0.919839   0.960369
0.800000   0.928945   0.984994
0.820000   0.943723   1.000000
0.840000   0.954919   1.000000
0.860000   0.962980   1.000000
0.880000   0.970742   1.000000
0.900000   0.979251   1.000000
0.920000   0.986117   1.000000
0.940000   0.991043   1.000000
0.960000   0.995223   1.000000
0.980000   0.998059   1.000000
$$


$TABLE: Likelihood based twin fraction estimation for possible twin law k,l,h:
$GRAPHS
:Likelihood based twin fraction estimate 
:A:1,2:
$$
alpha   NLL_(acentric_data)   $$ 
$$
0.021739   26300.170866
0.043478   26940.315612
0.065217   29249.011872
0.086957   34235.549777
0.108696   41955.857416
0.130435   50921.451107
0.152174   62815.731205
0.173913   77273.823342
0.195652   95022.846385
0.217391   115440.035337
0.239130   138694.912923
0.260870   164789.820118
0.282609   194407.605607
0.304348   224356.221206
0.326087   257215.378306
0.347826   292474.775598
0.369565   330523.993449
0.391304   371401.267227
0.413043   415218.926322
0.434783   459020.610793
0.456522   506124.574793
0.478261   552159.030946
$$


$TABLE: Britton plot for twin law -k,l,-h:
$GRAPHS
:percentage negatives 
:A:1,2,3:
$$
alpha   percentage_negatives   fit   $$ 
$$
0.000000   0.000000   0.000000
0.009901   0.000746   0.000000
0.019802   0.001791   0.000000
0.029703   0.002985   0.000000
0.039604   0.005075   0.000000
0.049505   0.007910   0.000000
0.059406   0.010896   0.000000
0.069307   0.014179   0.000000
0.079208   0.018209   0.000000
0.089109   0.022687   0.000000
0.099010   0.028507   0.005061
0.108911   0.033209   0.016243
0.118812   0.040075   0.027425
0.128713   0.046493   0.038607
0.138614   0.052985   0.049789
0.148515   0.060448   0.060971
0.158416   0.069328   0.072153
0.168317   0.078806   0.083335
0.178218   0.087015   0.094517
0.188119   0.096418   0.105699
0.198020   0.107015   0.116881
0.207921   0.116343   0.128063
0.217822   0.124552   0.139245
0.227723   0.133358   0.150427
0.237624   0.144254   0.161609
0.247525   0.156269   0.172791
0.257426   0.169104   0.183973
0.267327   0.180970   0.195155
0.277228   0.192388   0.206337
0.287129   0.205224   0.217519
0.297030   0.216567   0.228701
0.306931   0.227761   0.239883
0.316832   0.239925   0.251065
0.326733   0.252388   0.262247
0.336634   0.265597   0.273429
0.346535   0.278955   0.284611
0.356436   0.291791   0.295793
0.366337   0.304254   0.306975
0.376238   0.316716   0.318157
0.386139   0.329179   0.329339
0.396040   0.342537   0.340521
0.405941   0.354030   0.351703
0.415842   0.368060   0.362885
0.425743   0.380970   0.374067
0.435644   0.394776   0.385249
0.445545   0.409328   0.396431
0.455446   0.421791   0.407613
0.465347   0.433881   0.418795
0.475248   0.447090   0.429977
0.485149   0.461716   0.441159
$$


$TABLE: H test for possible twin law -k,l,-h:
$GRAPHS
:H test for Acentric data 
:A:1,2,3:
$$
H   Observed_S(H)   Fitted_S(H)   $$ 
$$
0.000000   0.026866   0.000000
0.020000   0.052836   0.024747
0.040000   0.081194   0.049493
0.060000   0.110149   0.074240
0.080000   0.137612   0.098986
0.100000   0.164478   0.123733
0.120000   0.189552   0.148480
0.140000   0.218209   0.173226
0.160000   0.248955   0.197973
0.180000   0.272836   0.222719
0.200000   0.298209   0.247466
0.220000   0.325672   0.272213
0.240000   0.352239   0.296959
0.260000   0.377015   0.321706
0.280000   0.401791   0.346452
0.300000   0.426866   0.371199
0.320000   0.451940   0.395946
0.340000   0.477015   0.420692
0.360000   0.502985   0.445439
0.380000   0.525373   0.470185
0.400000   0.548060   0.494932
0.420000   0.569552   0.519679
0.440000   0.593731   0.544425
0.460000   0.618806   0.569172
0.480000   0.641791   0.593918
0.500000   0.664776   0.618665
0.520000   0.694627   0.643412
0.540000   0.714030   0.668158
0.560000   0.731343   0.692905
0.580000   0.748060   0.717651
0.600000   0.770746   0.742398
0.620000   0.789552   0.767145
0.640000   0.807761   0.791891
0.660000   0.828358   0.816638
0.680000   0.846269   0.841384
0.700000   0.863881   0.866131
0.720000   0.880000   0.890878
0.740000   0.894030   0.915624
0.760000   0.909254   0.940371
0.780000   0.920896   0.965117
0.800000   0.933134   0.989864
0.820000   0.946269   1.000000
0.840000   0.956418   1.000000
0.860000   0.965075   1.000000
0.880000   0.972537   1.000000
0.900000   0.978806   1.000000
0.920000   0.987463   1.000000
0.940000   0.992537   1.000000
0.960000   0.995224   1.000000
0.980000   0.997910   1.000000
$$


$TABLE: Likelihood based twin fraction estimation for possible twin law -k,l,-h:
$GRAPHS
:Likelihood based twin fraction estimate 
:A:1,2:
$$
alpha   NLL_(acentric_data)   $$ 
$$
0.021739   26364.947018
0.043478   27452.826538
0.065217   29512.908243
0.086957   33973.438190
0.108696   40838.889164
0.130435   51734.447433
0.152174   62823.588945
0.173913   77470.029928
0.195652   94681.144047
0.217391   115648.574454
0.239130   136676.408930
0.260870   162533.192035
0.282609   191697.405738
0.304348   221603.205661
0.326087   255257.188968
0.347826   291504.427259
0.369565   329970.668823
0.391304   369684.069507
0.413043   412103.297473
0.434783   457576.139438
0.456522   505474.579313
0.478261   551511.087130
$$