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1 : overbeek 1.1
2 :     # Copyright (c) 2003-2006 University of Chicago and Fellowship
3 :     # for Interpretations of Genomes. All Rights Reserved.
4 :     #
5 :     # This file is part of the SEED Toolkit.
6 :     #
7 :     # The SEED Toolkit is free software. You can redistribute
8 :     # it and/or modify it under the terms of the SEED Toolkit
9 :     # Public License.
10 :     #
11 :     # You should have received a copy of the SEED Toolkit Public License
12 :     # along with this program; if not write to the University of Chicago
13 :     # at info@ci.uchicago.edu or the Fellowship for Interpretation of
14 :     # Genomes at veronika@thefig.info or download a copy from
15 :     # http://www.theseed.org/LICENSE.TXT.
16 :     #
17 :    
18 :    
19 :     =pod
20 :    
21 :     =head1 FIGStatisticalModels
22 :    
23 :     A package of Statistical models for applying to FIG data.
24 :    
25 :     At inception this was handling some methods developed by Jon McAuliffe, but feel free to add other methods here
26 :    
27 :     =cut
28 :    
29 :     package FIGStatisticalModels;
30 :     use strict;
31 :     use Exporter;
32 :    
33 :     use vars qw(@EXPORT_OK);
34 :    
35 :     @EXPORT_OK = qw();
36 :    
37 : redwards 1.5 =head1 new
38 : overbeek 1.1
39 :     Instantiate the object. Not necessary, but nice
40 :    
41 :     =cut
42 :    
43 :     sub new {
44 :     my ($class)=@_;
45 :     my $self={};
46 :     return bless $self, $class;
47 :     }
48 :    
49 : redwards 1.5 =head1 mean
50 :    
51 :     Calculate the mean on an array
52 :     e.g. my $mean=FIGStatisticalModels->mean(\@array);
53 :    
54 :     This is the arithmetical mean (i.e. sum/n)
55 :    
56 :     =cut
57 :    
58 :     sub mean {
59 :     my ($self, $arr)=@_;
60 :     return 0 unless ($arr);
61 :     return 0 unless (@$arr);
62 :     my $sum;
63 :     foreach (@$arr) {$sum+=$_};
64 :     return $sum/(scalar @$arr);
65 :     }
66 :    
67 :     =head1 median
68 :    
69 :     Calculate the median for an array
70 :     e.g. my $median=FIGStatisticalModels->median(\@array);
71 :    
72 :     This is the median, the middle number on a list.
73 :    
74 :     =cut
75 :    
76 :     sub median {
77 :     my ($self, $arr)=@_;
78 :     return 0 unless ($arr);
79 :     @$arr=sort {$a <=> $b} @$arr;
80 :     #@$arr=sort @$arr;
81 :     my $mid=$#$arr/2;
82 :     if ($mid == int($mid)) {
83 :     return $$arr[$mid];
84 :     } else {
85 :     return ($$arr[$mid - 0.5] + $$arr[$mid + 0.5])/2;
86 :     }
87 :     }
88 :    
89 :     =head1 min
90 :    
91 :     Return the minimum value in an array.
92 :     e.g. my $min=FIGStatisticalModels->min($array);
93 :    
94 :     =cut
95 :    
96 :     sub min {
97 :     my ($self, $arr)=@_;
98 :     return undef unless ($arr);
99 :     my $min=1e99;
100 :     map {($_ < $min) ? ($min = $_) : 1} @$arr;
101 :     return $min;
102 :     }
103 :    
104 :    
105 :     =head1 max
106 :    
107 :     Return the maximum value in an array.
108 :     e.g. my $max=FIGStatisticalModels->max($array);
109 :    
110 :     =cut
111 :    
112 :     sub max {
113 :     my ($self, $arr)=@_;
114 :     return undef unless ($arr);
115 :     my $max=0;
116 :     map {($_ > $max) ? ($max = $_) : 1} @$arr;
117 :     return $max;
118 :     }
119 :    
120 :     =head1 stdev
121 :    
122 :     Calculate the standard deviation on an array.
123 :    
124 :     e.g. my $stdev=FIGStatisticalModels->stdev(\@array);
125 :     my $stdev=FIGStatisticalModels->stdev([1,2,3,4]);
126 :    
127 :     =cut
128 :    
129 :     sub stdev {
130 :     my ($self, $arr)=@_;
131 :     return 0 unless ($arr);
132 :     my ($n, $sum, $square)=(0,0,0);
133 :     foreach (@$arr) {
134 :     next unless ($_);
135 :     $n++;
136 :     $sum += $_;
137 :     $square += $_ * $_;
138 :     }
139 :    
140 :     #stdev = sqrt(((n.sum(x squared))-((sum x) squared))/n(n-1))
141 :     unless ($n*($n-1)) {die "STDEV at div by zero as $sum, $square, $n\n"; return 0}
142 :     unless ($n*($n-1)) {print STDERR "STDEV at div by zero as $sum, $square, $n\n"; return 0}
143 :     my $stdev = (($n * $square)-($sum * $sum))/($n*($n-1));
144 :     if ($stdev < 0) {die "STdev less than zero ($stdev) as $sum, $square, $n\n"}
145 :     if ($stdev < 0) {print STDERR "STdev less than zero ($stdev) as $sum, $square, $n\n"}
146 :     $stdev=sqrt($stdev);
147 :     return $stdev;
148 :     }
149 : overbeek 1.1
150 :    
151 :     =head1 Functional Coupling Scoring
152 :    
153 :     Scoring direct functional coupling. These are some methods to score the direct functional coupling using a method developed by Jon McAuliffe and Ross Overbeek. It was coded by Rob Edwards. The approach uses a generalized additive model using a thin plate regression spline smoother with 5 degrees of freedom.
154 :    
155 :     Many of these notes have been extracted directly from discussions between Ross, Jon, and Rob. I am including them here for clarity and posterity.
156 :    
157 :     Offline, a matrix was computed whose rows correspond to values of the logscore input on a fine grid. In the ith row, each column corresponds to a basis function of the smoother, evaluated at the ith grid point.
158 :    
159 :     To get the predicted success probability for a given log score, we linearly interpolate between the two nearest grid points, then pass the interpolant and the coefficient vector to a classical linear-logistic prediction routine.
160 :    
161 :     =head2 EQUATION
162 :    
163 :     The formula can be expressed like this:
164 :    
165 :     1
166 :     logistic(x) = -----------
167 :     1 + exp(-x)
168 :    
169 :     then for a given score (s) the probability of success is
170 :    
171 :     P(success) = logistic (1.40 - 4.05 * b1(s) + 1.33 * b2(s) - 14.14 * b3(s) + 6.70 * b4(s) )
172 :    
173 :    
174 :     where b1, b2, b3, and b4 are the basis functions of a thin-plate regression spline with 5 fixed degrees of freedom.
175 :    
176 : redwards 1.5 =head1 fc_score2probability
177 : overbeek 1.1
178 :     Convert a functional coupling score based on E value to a probability
179 :    
180 : overbeek 1.2 use: my $probabilty = $statistic_model->fc_score2probability($score);
181 : overbeek 1.1
182 :     =cut
183 :    
184 : overbeek 1.2 sub fc_score2probability {
185 :     my ($self, $score)=@_;
186 : overbeek 1.1
187 :     # Coefficient vector
188 :     # This vector describes the coefficients in the equation for the probability for success (see EQUATION above).
189 : redwards 1.4 # this is the old coef_vector from version 1 of this analysis
190 :     #my $coef_vector = [1.4038457039268741, -4.0483109464656009, 1.3302963870075235, -14.1387590191618955, 6.6980442362324641];
191 :     # this is the new coef_vector from version 2 of this analysis
192 :     my $coef_vector = [1.047594571110132655, -0.107764949815202224, 0.451326357245773802, 0.739792463755890473, 0.703910832923400731];
193 : overbeek 1.1
194 :     # Matrix
195 :     # This is the precomputed matrix that McAuliffe Provided.
196 :     my ($grid_min, $grid_max, $dgrid, $grid_data)=$self->spline_matrix();
197 :    
198 : overbeek 1.3 my $log_score;
199 :     unless ($score) {print STDERR "Had a score of $score. Can't log it!!\n"; $log_score=$score}
200 :     else {$log_score= log($score)}
201 :    
202 : overbeek 1.2 my $sc = $self->score2success_prob($log_score, $coef_vector, $grid_data, $grid_min, $grid_max, $dgrid);
203 :     return $sc;
204 : overbeek 1.1 }
205 :    
206 :    
207 :     =head1 fc_bitscore2probability
208 :    
209 : overbeek 1.3 Convert a BBH bit score and optional tough call flag to a probability. This uses the fourth column of the BBH data - the bit score normalized by length - as the basis of calculating a probability of being a good match.
210 : overbeek 1.1
211 :     use: my $probability = $statistic_model->fc_bitscore2probability($bitscore, $toughcall);
212 :    
213 :     Toughcall is a boolean. True is if it meets this regexp:
214 :     if ($function =~ /aminotransferase|system|component|oxidase|regulator|cytochrome|specific|permease|transcriptional|transport|dehydrogenase/i)
215 :    
216 :     =cut
217 :    
218 :     sub fc_bitscore2probability {
219 :     my ($self, $bitscore, $toughcall)=@_;
220 :    
221 :     # Coefficient vector
222 :     # This vector describes the coefficients in the equation for the probability for success (see EQUATION above).
223 :     # It is slightly different than the one above since it is for bit-scores not evalues
224 :     my $coef_vector = [3.75350869822815225, 1.73322181593049818, -1.69597537007947863, -0.53396148373755736];
225 :    
226 : overbeek 1.3 my $log_score;
227 :     unless ($bitscore) {print STDERR "Had a bitscore of $bitscore. Can't log it!!\n"; $log_score=$bitscore}
228 :     else {$log_score= log($bitscore)}
229 : overbeek 1.2
230 : overbeek 1.1 my $tc=0; if ($toughcall) {$tc=1} # this just converts it to a 1/0 boolean in case someone uses something weird
231 : redwards 1.6 my $score=$self->linear_logistic_prediction([1, $log_score, $tc, $tc*$log_score], $coef_vector);
232 : overbeek 1.1 return $score;
233 :     }
234 :    
235 :    
236 :    
237 :     =head1 score2success_prob
238 :    
239 :     Convert a score to a probability of success
240 :    
241 :     =cut
242 :    
243 :     sub score2success_prob
244 :     {
245 :     my ($self, $input_score, $coef_vector, $grid_matrix, $grid_min, $grid_max, $dgrid) = @_;
246 :    
247 :     my ($left_nbr_idx, $right_nbr_idx, $weight);
248 :    
249 :     # find the left- and right-hand interpolating values, and the weights { 1-w, w } to combine them for the interpolant
250 :    
251 :     if ($input_score < $grid_min)
252 :     {
253 :     # the score is smaller than the grid. Perhaps we should warn about this.
254 :     $left_nbr_idx = 0;
255 :     $right_nbr_idx = 1;
256 :     $weight = 0;
257 :     }
258 :     elsif ($input_score >= $grid_max)
259 :     {
260 :     # the score is bigger than the grid. Perhaps we should warn about this.
261 :     $right_nbr_idx = $#{$grid_matrix};
262 :     $left_nbr_idx = $right_nbr_idx - 1;
263 :     $weight = 1;
264 :     }
265 :     else
266 :     {
267 :     # just figure out our place in the grid. Note that this is calculated not run through a loop (though we could)
268 :     $left_nbr_idx = int(($input_score - $grid_min) / $dgrid);
269 :     $right_nbr_idx = $left_nbr_idx + 1;
270 :     $weight = ( $input_score - ($grid_min + $left_nbr_idx * $dgrid) ) / $dgrid;
271 :     }
272 :    
273 :     my @interpolant_vector;
274 :    
275 :     for (my $i = 0; $i < scalar(@$coef_vector); $i++)
276 :     {
277 :     $interpolant_vector[$i] =
278 :     (1 - $weight) * ${$grid_matrix}[$left_nbr_idx][$i+1] +
279 :     $weight * ${$grid_matrix}[$right_nbr_idx][$i+1];
280 :     }
281 :    
282 :     return $self->linear_logistic_prediction(\@interpolant_vector, $coef_vector);
283 :     }
284 :    
285 :    
286 :     =head1 linear_logistic_prediction
287 :    
288 :     Given the coefficients of a logistic regression fit and values for the covariates, return the corresponding predicted probability: take the inner product of covariates and coefficients, then pass the result through the inverse link function (the logistic function, h(u) = (1 + exp(-u))^{-1}).
289 :    
290 :     =cut
291 :    
292 :     sub linear_logistic_prediction
293 :     {
294 :     my ($self, $covariate_vector, $coef_vector) = @_;
295 :    
296 :     my $inner_product = 0;
297 :     for (my $i = 0; $i < scalar(@$coef_vector); $i++)
298 :     {
299 :     $inner_product += $covariate_vector->[$i] * $coef_vector->[$i];
300 :     }
301 :    
302 :     my $res = 1 / (1 + exp(-$inner_product));
303 :    
304 :     return $res;
305 :     }
306 :    
307 :    
308 :    
309 :     =head2 spline_matrix
310 :    
311 :     This data is the matrix that was calculated offline by Jon mcAuliffe using the thin plate regression spline smoother. This matrix was provided by Jon, and I just converted it to this array form for easier coding.
312 :    
313 :     =cut
314 :    
315 :     sub spline_matrix {
316 : redwards 1.4 my $self=shift;
317 :     my @matrix=(
318 :     [0, 1, -5.12854483987495, 1.48365029477066, -1.66633483317743, -5.19728628264353],
319 :     [0.001, 1, -5.1276370232775, 1.48319826316280, -1.66565720160812, -5.19610816345883],
320 :     [0.002, 1, -5.12672920668005, 1.48274623155493, -1.66497957003881, -5.19493004427413],
321 :     [0.003, 1, -5.1258213900826, 1.48229419994707, -1.66430193846950, -5.19375192508943],
322 :     [0.004, 1, -5.12491357348514, 1.4818421683392, -1.66362430690020, -5.19257380590473],
323 :     [0.005, 1, -5.12400575688769, 1.48139013673133, -1.66294667533089, -5.19139568672003],
324 :     [0.006, 1, -5.12309794029024, 1.48093810512347, -1.66226904376158, -5.19021756753533],
325 :     [0.007, 1, -5.12219012369279, 1.48048607351560, -1.66159141219227, -5.18903944835064],
326 :     [0.008, 1, -5.12128230709534, 1.48003404190774, -1.66091378062296, -5.18786132916594],
327 :     [0.009, 1, -5.12037449049789, 1.47958201029987, -1.66023614905365, -5.18668320998124],
328 :     [0.01, 1, -5.11946667390043, 1.47912997869201, -1.65955851748434, -5.18550509079654],
329 :     [0.011, 1, -5.11855885730298, 1.47867794708414, -1.65888088591503, -5.18432697161184],
330 :     [0.012, 1, -5.11765104070553, 1.47822591547627, -1.65820325434572, -5.18314885242714],
331 :     [0.013, 1, -5.11674322410808, 1.47777388386841, -1.65752562277641, -5.18197073324244],
332 :     [0.014, 1, -5.11583540751063, 1.47732185226054, -1.6568479912071, -5.18079261405774],
333 :     [0.015, 1, -5.11492759091317, 1.47686982065268, -1.65617035963779, -5.17961449487304],
334 :     [0.016, 1, -5.11401977431572, 1.47641778904481, -1.65549272806848, -5.17843637568834],
335 :     [0.017, 1, -5.11311195771827, 1.47596575743695, -1.65481509649917, -5.17725825650364],
336 :     [0.018, 1, -5.11220414112082, 1.47551372582908, -1.65413746492986, -5.17608013731895],
337 :     [0.019, 1, -5.11129632452337, 1.47506169422122, -1.65345983336055, -5.17490201813425],
338 :     [0.02, 1, -5.11038850792592, 1.47460966261335, -1.65278220179124, -5.17372389894955],
339 :     [0.021, 1, -5.10948069132846, 1.47415763100548, -1.65210457022193, -5.17254577976485],
340 :     [0.022, 1, -5.10857287473101, 1.47370559939762, -1.65142693865262, -5.17136766058015],
341 :     [0.023, 1, -5.10766505813356, 1.47325356778975, -1.65074930708331, -5.17018954139545],
342 :     [0.024, 1, -5.10675724153611, 1.47280153618189, -1.65007167551400, -5.16901142221075],
343 :     [0.025, 1, -5.10584942493866, 1.47234950457402, -1.64939404394470, -5.16783330302605],
344 :     [0.026, 1, -5.1049416083412, 1.47189747296616, -1.64871641237539, -5.16665518384135],
345 :     [0.027, 1, -5.10403379174375, 1.47144544135829, -1.64803878080608, -5.16547706465665],
346 :     [0.028, 1, -5.1031259751463, 1.47099340975042, -1.64736114923677, -5.16429894547195],
347 :     [0.029, 1, -5.10221815854885, 1.47054137814256, -1.64668351766746, -5.16312082628725],
348 :     [0.03, 1, -5.10131034195140, 1.47008934653469, -1.64600588609815, -5.16194270710255],
349 :     [0.031, 1, -5.10040252535394, 1.46963731492683, -1.64532825452884, -5.16076458791786],
350 :     [0.032, 1, -5.09949470875649, 1.46918528331896, -1.64465062295953, -5.15958646873316],
351 :     [0.033, 1, -5.09858689215904, 1.46873325171110, -1.64397299139022, -5.15840834954846],
352 :     [0.034, 1, -5.09767907556159, 1.46828122010323, -1.64329535982091, -5.15723023036376],
353 :     [0.035, 1, -5.09677125896414, 1.46782918849536, -1.6426177282516, -5.15605211117906],
354 :     [0.036, 1, -5.09586344236669, 1.4673771568875, -1.64194009668229, -5.15487399199436],
355 :     [0.037, 1, -5.09495562576923, 1.46692512527963, -1.64126246511298, -5.15369587280966],
356 :     [0.038, 1, -5.09404780917178, 1.46647309367177, -1.64058483354367, -5.15251775362496],
357 :     [0.039, 1, -5.09313999257433, 1.46602106206390, -1.63990720197436, -5.15133963444026],
358 :     [0.04, 1, -5.09223217597688, 1.46556903045604, -1.63922957040505, -5.15016151525556],
359 :     [0.041, 1, -5.09132435937943, 1.46511699884817, -1.63855193883574, -5.14898339607086],
360 :     [0.042, 1, -5.09041654278197, 1.46466496724031, -1.63787430726644, -5.14780527688617],
361 :     [0.043, 1, -5.08950872618452, 1.46421293563244, -1.63719667569712, -5.14662715770147],
362 :     [0.044, 1, -5.08860090958707, 1.46376090402458, -1.63651904412782, -5.14544903851677],
363 :     [0.045, 1, -5.08769309298962, 1.46330887241671, -1.63584141255851, -5.14427091933207],
364 :     [0.046, 1, -5.08678527639217, 1.46285684080884, -1.63516378098920, -5.14309280014737],
365 :     [0.047, 1, -5.08587745979471, 1.46240480920098, -1.63448614941989, -5.14191468096267],
366 :     [0.048, 1, -5.08496964319726, 1.46195277759311, -1.63380851785058, -5.14073656177797],
367 :     [0.049, 1, -5.08406182659981, 1.46150074598525, -1.63313088628127, -5.13955844259327],
368 :     [0.05, 1, -5.08315401000236, 1.46104871437738, -1.63245325471196, -5.13838032340857],
369 :     [0.051, 1, -5.08224619340491, 1.46059668276952, -1.63177562314265, -5.13720220422387],
370 :     [0.052, 1, -5.08133837680746, 1.46014465116165, -1.63109799157334, -5.13602408503917],
371 :     [0.053, 1, -5.08043056021, 1.45969261955378, -1.63042036000403, -5.13484596585448],
372 :     [0.054, 1, -5.07952274361255, 1.45924058794592, -1.62974272843472, -5.13366784666978],
373 :     [0.055, 1, -5.0786149270151, 1.45878855633805, -1.62906509686541, -5.13248972748508],
374 :     [0.056, 1, -5.07770711041765, 1.45833652473019, -1.6283874652961, -5.13131160830038],
375 :     [0.057, 1, -5.07679929382019, 1.45788449312232, -1.62770983372679, -5.13013348911568],
376 :     [0.058, 1, -5.07589147722275, 1.45743246151446, -1.62703220215748, -5.12895536993098],
377 :     [0.059, 1, -5.07498366062529, 1.45698042990659, -1.62635457058817, -5.12777725074628],
378 :     [0.06, 1, -5.07407584402784, 1.45652839829872, -1.62567693901886, -5.12659913156158],
379 :     [0.061, 1, -5.07316802743039, 1.45607636669086, -1.62499930744955, -5.12542101237688],
380 :     [0.062, 1, -5.07226021083294, 1.45562433508299, -1.62432167588024, -5.12424289319218],
381 :     [0.063, 1, -5.07135239423548, 1.45517230347513, -1.62364404431094, -5.12306477400748],
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1903 :     [1.585, 1, -3.68900120267827, 0.766168678226175, -0.592221988862935, -3.32996737489541],
1904 :     [1.586, 1, -3.68808345739346, 0.765701298027931, -0.591543343577044, -3.32878925571071],
1905 :     [1.587, 1, -3.68716561217586, 0.76523376334528, -0.590864688088039, -3.32761113652602],
1906 :     [1.588, 1, -3.68624766652507, 0.764766073404661, -0.590186022344831, -3.32643301734132],
1907 :     [1.589, 1, -3.68532961994069, 0.764298227432514, -0.589507346296328, -3.32525489815662],
1908 :     [1.59, 1, -3.68441147192233, 0.763830224655278, -0.588828659891439, -3.32407677897192],
1909 :     [1.591, 1, -3.68349322196958, 0.76336206429939, -0.588149963079074, -3.32289865978722],
1910 :     [1.592, 1, -3.68257486958205, 0.762893745591293, -0.587471255808142, -3.32172054060252],
1911 :     [1.593, 1, -3.68165641425933, 0.762425267757425, -0.586792538027552, -3.32054242141782],
1912 :     [1.594, 1, -3.68073785550102, 0.761956630024226, -0.586113809686213, -3.31936430223312],
1913 :     [1.595, 1, -3.67981919280672, 0.761487831618134, -0.585435070733036, -3.31818618304842],
1914 :     [1.596, 1, -3.67890042567603, 0.76101887176559, -0.584756321116928, -3.31700806386372],
1915 :     [1.597, 1, -3.67798155360855, 0.760549749693032, -0.584077560786799, -3.31582994467902],
1916 :     [1.598, 1, -3.67706257610388, 0.7600804646269, -0.583398789691559, -3.31465182549432],
1917 :     [1.599, 1, -3.67614349266162, 0.759611015793634, -0.582720007780116, -3.31347370630963],
1918 :     [1.6, 1, -3.67522430278138, 0.759141402419673, -0.58204121500138, -3.31229558712493],
1919 :     [1.601, 1, -3.67430500596274, 0.758671623731457, -0.581362411304261, -3.31111746794023],
1920 :     [1.602, 1, -3.67338560170531, 0.758201678955425, -0.580683596637666, -3.30993934875553],
1921 :     [1.603, 1, -3.67246608950868, 0.757731567318016, -0.580004770950507, -3.30876122957083],
1922 :     [1.604, 1, -3.67154646887247, 0.75726128804567, -0.57932593419169, -3.30758311038613],
1923 :     [1.605, 1, -3.67062673929626, 0.756790840364825, -0.578647086310128, -3.30640499120143],
1924 :     [1.606, 1, -3.66970690027966, 0.756320223501923, -0.577968227254727, -3.30522687201673],
1925 :     [1.607, 1, -3.66878695132226, 0.755849436683401, -0.577289356974398, -3.30404875283203],
1926 :     [1.608, 1, -3.66786689192368, 0.7553784791357, -0.576610475418050, -3.30287063364733],
1927 :     [1.609, 1, -3.66694672158349, 0.754907350085259, -0.575931582534592, -3.30169251446263],
1928 :     [1.61, 1, -3.66602643979227, 0.754436048749889, -0.57525267827343, -3.30051439527793],
1929 :     [1.611, 1, -3.66510604588271, 0.753964574196697, -0.574573762592668, -3.29933627609324],
1930 :     [1.612, 1, -3.66418553905323, 0.753492925364649, -0.573894835457799, -3.29815815690854],
1931 :     [1.613, 1, -3.66326491849801, 0.753021101188633, -0.573215896834555, -3.29698003772384],
1932 :     [1.614, 1, -3.66234418341121, 0.752549100603531, -0.572536946688666, -3.29580191853914],
1933 :     [1.615, 1, -3.66142333298700, 0.752076922544229, -0.57185798498586, -3.29462379935444],
1934 :     [1.616, 1, -3.66050236641955, 0.751604565945613, -0.571179011691871, -3.29344568016974],
1935 :     [1.617, 1, -3.65958128290303, 0.751132029742568, -0.570500026772427, -3.29226756098504],
1936 :     [1.618, 1, -3.65866008163159, 0.750659312869977, -0.569821030193259, -3.29108944180034],
1937 :     [1.619, 1, -3.65773876179941, 0.750186414262728, -0.569142021920098, -3.28991132261564],
1938 :     [1.62, 1, -3.65681732260065, 0.749713332855706, -0.568463001918673, -3.28873320343094],
1939 :     [1.621, 1, -3.65589576322947, 0.749240067583793, -0.567783970154716, -3.28755508424624],
1940 :     [1.622, 1, -3.65497408288006, 0.748766617381877, -0.567104926593956, -3.28637696506155],
1941 :     [1.623, 1, -3.65405228074656, 0.748292981184843, -0.566425871202124, -3.28519884587685],
1942 :     [1.624, 1, -3.65313035602315, 0.747819157927576, -0.56574680394495, -3.28402072669215],
1943 :     [1.625, 1, -3.652208307904, 0.747345146544959, -0.565067724788165, -3.28284260750745],
1944 :     [1.626, 1, -3.65128613558326, 0.74687094597188, -0.5643886336975, -3.28166448832275],
1945 :     [1.627, 1, -3.65036383825511, 0.746396555143223, -0.563709530638683, -3.28048636913805],
1946 :     [1.628, 1, -3.64944141511372, 0.745921972993872, -0.563030415577446, -3.27930824995335],
1947 :     [1.629, 1, -3.64851886535324, 0.745447198458715, -0.56235128847952, -3.27813013076865],
1948 :     [1.63, 1, -3.64759618816785, 0.744972230472634, -0.561672149310634, -3.27695201158395],
1949 :     [1.631, 1, -3.64667338275171, 0.744497067970515, -0.560992998036518, -3.27577389239925],
1950 :     [1.632, 1, -3.64575044829899, 0.744021709887244, -0.560313834622904, -3.27459577321455],
1951 :     [1.633, 1, -3.64482738400386, 0.743546155157706, -0.559634659035522, -3.27341765402985],
1952 :     [1.634, 1, -3.64390418906047, 0.743070402716786, -0.558955471240102, -3.27223953484516],
1953 :     [1.635, 1, -3.64298086266301, 0.742594451499369, -0.558276271202375, -3.27106141566046],
1954 :     [1.636, 1, -3.64205740400562, 0.74211830044034, -0.55759705888807, -3.26988329647576],
1955 :     [1.637, 1, -3.64113381228249, 0.741641948474586, -0.556917834262918, -3.26870517729106],
1956 :     [1.638, 1, -3.64021008668778, 0.741165394536988, -0.55623859729265, -3.26752705810636],
1957 :     [1.639, 1, -3.63928622641564, 0.740688637562434, -0.555559347942995, -3.26634893892166],
1958 :     [1.64, 1, -3.63836223066026, 0.740211676485808, -0.554880086179685, -3.26517081973696],
1959 :     [1.641, 1, -3.63743809861580, 0.739734510241998, -0.554200811968449, -3.26399270055226],
1960 :     [1.642, 1, -3.63651382947641, 0.739257137765886, -0.553521525275018, -3.26281458136756],
1961 :     [1.643, 1, -3.63558942243627, 0.738779557992359, -0.552842226065124, -3.26163646218286],
1962 :     [1.644, 1, -3.63466487668955, 0.7383017698563, -0.552162914304494, -3.26045834299816],
1963 :     [1.645, 1, -3.63374019143041, 0.737823772292595, -0.551483589958861, -3.25928022381347],
1964 :     [1.646, 1, -3.63281536585302, 0.737345564236131, -0.550804252993955, -3.25810210462877],
1965 :     [1.647, 1, -3.63189039915154, 0.736867144621791, -0.550124903375505, -3.25692398544407],
1966 :     [1.648, 1, -3.63096529052015, 0.73638851238446, -0.549445541069242, -3.25574586625937],
1967 :     [1.649, 1, -3.630040039153, 0.735909666459026, -0.548766166040898, -3.25456774707467],
1968 :     [1.65, 1, -3.62911464424426, 0.73543060578037, -0.5480867782562, -3.25338962788997],
1969 :     [1.651, 1, -3.6281891049881, 0.73495132928338, -0.547407377680882, -3.25221150870527],
1970 :     [1.652, 1, -3.62726342057869, 0.73447183590294, -0.546727964280672, -3.25103338952057],
1971 :     [1.653, 1, -3.62633759021019, 0.733992124573936, -0.546048538021301, -3.24985527033587],
1972 :     [1.654, 1, -3.62541161307677, 0.733512194231252, -0.5453690988685, -3.24867715115117],
1973 :     [1.655, 1, -3.6244854883726, 0.733032043809774, -0.544689646787999, -3.24749903196647],
1974 :     [1.656, 1, -3.62355921529183, 0.732551672244387, -0.544010181745528, -3.24632091278177],
1975 :     [1.657, 1, -3.62263279302865, 0.732071078469977, -0.543330703706818, -3.24514279359708],
1976 :     [1.658, 1, -3.62170622077720, 0.731590261421426, -0.542651212637598, -3.24396467441238],
1977 :     [1.659, 1, -3.62077949773167, 0.731109220033624, -0.541971708503601, -3.24278655522768],
1978 :     [1.66, 1, -3.61985262308622, 0.730627953241451, -0.541292191270554, -3.24160843604298],
1979 :     [1.661, 1, -3.61892559603501, 0.730146459979796, -0.54061266090419, -3.24043031685828],
1980 :     [1.662, 1, -3.61799841577221, 0.729664739183542, -0.539933117370239, -3.23925219767358],
1981 :     [1.663, 1, -3.61707108149198, 0.729182789787574, -0.53925356063443, -3.23807407848888],
1982 :     [1.664, 1, -3.6161435923885, 0.72870061072678, -0.538573990662494, -3.23689595930418],
1983 :     [1.665, 1, -3.61521594765593, 0.728218200936042, -0.537894407420162, -3.23571784011948],
1984 :     [1.666, 1, -3.61428814648843, 0.727735559350246, -0.537214810873164, -3.23453972093478],
1985 :     [1.667, 1, -3.61336018808018, 0.727252684904279, -0.536535200987231, -3.23336160175008],
1986 :     [1.668, 1, -3.61243207162533, 0.726769576533023, -0.535855577728092, -3.23218348256539],
1987 :     [1.669, 1, -3.61150379631805, 0.726286233171365, -0.535175941061478, -3.23100536338069],
1988 :     [1.67, 1, -3.61057536135252, 0.72580265375419, -0.53449629095312, -3.22982724419599],
1989 :     [1.671, 1, -3.60964676592289, 0.725318837216382, -0.533816627368747, -3.22864912501129],
1990 :     [1.672, 1, -3.60871800922334, 0.724834782492828, -0.533136950274092, -3.22747100582659],
1991 :     [1.673, 1, -3.60778909044803, 0.724350488518412, -0.532457259634882, -3.22629288664189],
1992 :     [1.674, 1, -3.60686000879112, 0.72386595422802, -0.531777555416849, -3.22511476745719],
1993 :     [1.675, 1, -3.60593076344679, 0.723381178556535, -0.531097837585724, -3.22393664827249],
1994 :     [1.676, 1, -3.60500135360920, 0.722896160438845, -0.530418106107236, -3.22275852908779],
1995 :     [1.677, 1, -3.60407177847251, 0.722410898809833, -0.529738360947117, -3.22158040990309],
1996 :     [1.678, 1, -3.60314203723089, 0.721925392604386, -0.529058602071096, -3.22040229071839],
1997 :     [1.679, 1, -3.60221212907851, 0.721439640757386, -0.528378829444904, -3.21922417153369],
1998 :     [1.68, 1, -3.60128205320954, 0.720953642203721, -0.527699043034271, -3.21804605234900],
1999 :     [1.681, 1, -3.60035180881813, 0.720467395878276, -0.527019242804928, -3.2168679331643],
2000 :     [1.682, 1, -3.59942139509847, 0.719980900715936, -0.526339428722604, -3.2156898139796],
2001 :     [1.683, 1, -3.59849081124471, 0.719494155651585, -0.525659600753032, -3.2145116947949],
2002 :     [1.684, 1, -3.59756005645101, 0.719007159620108, -0.524979758861939, -3.2133335756102],
2003 :     [1.685, 1, -3.59662912991156, 0.718519911556391, -0.524299903015058, -3.2121554564255],
2004 :     [1.686, 1, -3.59569803082051, 0.718032410395319, -0.523620033178118, -3.2109773372408],
2005 :     [1.687, 1, -3.59476675837202, 0.717544655071777, -0.522940149316851, -3.2097992180561],
2006 :     [1.688, 1, -3.59383531176027, 0.717056644520651, -0.522260251396985, -3.20862109887140],
2007 :     [1.689, 1, -3.59290369017943, 0.716568377676825, -0.521580339384252, -3.20744297968670],
2008 :     [1.69, 1, -3.59197189282365, 0.716079853475185, -0.520900413244381, -3.20626486050200],
2009 :     [1.691, 1, -3.59103991888711, 0.715591070850615, -0.520220472943105, -3.20508674131731],
2010 :     [1.692, 1, -3.59010776756396, 0.715102028738001, -0.519540518446152, -3.20390862213261],
2011 :     [1.693, 1, -3.58917543804839, 0.714612726072228, -0.518860549719253, -3.20273050294791],
2012 :     [1.694, 1, -3.58824292953455, 0.714123161788181, -0.518180566728138, -3.20155238376321],
2013 :     [1.695, 1, -3.58731024121661, 0.713633334820745, -0.517500569438538, -3.20037426457851],
2014 :     [1.696, 1, -3.58637737228873, 0.713143244104806, -0.516820557816184, -3.19919614539381],
2015 :     [1.697, 1, -3.58544432194509, 0.712652888575248, -0.516140531826805, -3.19801802620911],
2016 :     [1.698, 1, -3.58451108937985, 0.712162267166957, -0.515460491436132, -3.19683990702441],
2017 :     [1.699, 1, -3.58357767378718, 0.711671378814817, -0.514780436609895, -3.19566178783971],
2018 :     [1.7, 1, -3.58264407436123, 0.711180222453714, -0.514100367313825, -3.19448366865501],
2019 :     [1.701, 1, -3.58171029029619, 0.710688797018534, -0.513420283513652, -3.19330554947031],
2020 :     [1.702, 1, -3.58077632078621, 0.71019710144416, -0.512740185175106, -3.19212743028562],
2021 :     [1.703, 1, -3.57984216502546, 0.709705134665479, -0.512060072263918, -3.19094931110092],
2022 :     [1.704, 1, -3.57890782220811, 0.709212895617375, -0.511379944745819, -3.18977119191622],
2023 :     [1.705, 1, -3.57797329152832, 0.708720383234735, -0.510699802586538, -3.18859307273152],
2024 :     [1.706, 1, -3.57703857218026, 0.708227596452442, -0.510019645751806, -3.18741495354682],
2025 :     [1.707, 1, -3.5761036633581, 0.707734534205381, -0.509339474207353, -3.18623683436212],
2026 :     [1.708, 1, -3.575168564256, 0.707241195428438, -0.50865928791891, -3.18505871517742],
2027 :     [1.709, 1, -3.57423327406814, 0.706747579056499, -0.507979086852207, -3.18388059599272],
2028 :     [1.71, 1, -3.57329779198866, 0.706253684024447, -0.507298870972974, -3.18270247680802],
2029 :     [1.711, 1, -3.57236211721175, 0.70575950926717, -0.506618640246942, -3.18152435762332],
2030 :     [1.712, 1, -3.57142624893157, 0.70526505371955, -0.505938394639842, -3.18034623843862],
2031 :     [1.713, 1, -3.57049018634229, 0.704770316316475, -0.505258134117403, -3.17916811925392],
2032 :     [1.714, 1, -3.56955392863806, 0.704275295992828, -0.504577858645356, -3.17799000006923],
2033 :     [1.715, 1, -3.56861747501306, 0.703779991683496, -0.503897568189431, -3.17681188088453],
2034 :     [1.716, 1, -3.56768082466146, 0.703284402323362, -0.503217262715358, -3.17563376169983],
2035 :     [1.717, 1, -3.56674397677742, 0.702788526847313, -0.50253694218887, -3.17445564251513],
2036 :     [1.718, 1, -3.56580693055510, 0.702292364190233, -0.501856606575694, -3.17327752333043],
2037 :     [1.719, 1, -3.56486968518868, 0.701795913287007, -0.501176255841562, -3.17209940414573],
2038 :     [1.72, 1, -3.56393223987232, 0.70129917307252, -0.500495889952204, -3.17092128496103],
2039 :     [1.721, 1, -3.56299459380018, 0.70080214248166, -0.499815508873351, -3.16974316577633],
2040 :     [1.722, 1, -3.56205674616644, 0.700304820449308, -0.499135112570733, -3.16856504659163],
2041 :     [1.723, 1, -3.56111869616525, 0.699807205910352, -0.49845470101008, -3.16738692740693],
2042 :     [1.724, 1, -3.56018044299079, 0.699309297799675, -0.497774274157122, -3.16620880822223],
2043 :     [1.725, 1, -3.55924198583723, 0.698811095052165, -0.497093831977591, -3.16503068903754],
2044 :     [1.726, 1, -3.55830332389871, 0.698312596602705, -0.496413374437216, -3.16385256985284],
2045 :     [1.727, 1, -3.55736445636943, 0.69781380138618, -0.495732901501727, -3.16267445066814],
2046 :     [1.728, 1, -3.55642538244353, 0.697314708337475, -0.495052413136857, -3.16149633148344],
2047 :     [1.729, 1, -3.55548610131519, 0.696815316391477, -0.494371909308333, -3.16031821229874],
2048 :     [1.73, 1, -3.55454661217858, 0.69631562448307, -0.493691389981888, -3.15914009311404],
2049 :     [1.731, 1, -3.55360691422785, 0.69581563154714, -0.49301085512325, -3.15796197392934],
2050 :     [1.732, 1, -3.55266700665718, 0.69531533651857, -0.492330304698152, -3.15678385474464],
2051 :     [1.733, 1, -3.55172688866074, 0.694814738332246, -0.491649738672322, -3.15560573555994],
2052 :     [1.734, 1, -3.55078655943268, 0.694313835923054, -0.490969157011492, -3.15442761637524],
2053 :     [1.735, 1, -3.54984601816718, 0.693812628225879, -0.490288559681391, -3.15324949719054],
2054 :     [1.736, 1, -3.54890526405840, 0.693311114175606, -0.489607946647751, -3.15207137800585],
2055 :     [1.737, 1, -3.5479642963005, 0.69280929270712, -0.488927317876301, -3.15089325882115],
2056 :     [1.738, 1, -3.54702311408766, 0.692307162755305, -0.488246673332772, -3.14971513963645],
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2203 :     [1.885, 1, -3.40587846809888, 0.614563586540987, -0.388009470978786, -2.97653161948568],
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3377 :     [3.059, 1, -1.77115945632434, -0.439709829237747, 0.358614288096981, -1.5934196966489],
3378 :     [3.06, 1, -1.76924274094723, -0.440749079800723, 0.359085874781399, -1.5922415774642],
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3385 :     [3.067, 1, -1.75580328011893, -0.448006610876086, 0.362370643561529, -1.58399474317131],
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3387 :     [3.069, 1, -1.75195625503729, -0.450074554866533, 0.36330386852898, -1.58163850480191],
3388 :     [3.07, 1, -1.75003155230229, -0.451107570742998, 0.363769596130517, -1.58046038561721],
3389 :     [3.071, 1, -1.74810605779187, -0.452139944744922, 0.364234732526381, -1.57928226643251],
3390 :     [3.072, 1, -1.74617977277154, -0.453171673525781, 0.364699276753411, -1.57810414724781],
3391 :     [3.073, 1, -1.74425269850683, -0.45420275373905, 0.365163227848445, -1.57692602806311],
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3398 :     [3.08, 1, -1.73074118611485, -0.461401874231287, 0.368394196912300, -1.56867919377022],
3399 :     [3.081, 1, -1.72880784345381, -0.462427645428926, 0.368853368277579, -1.56750107458552],
3400 :     [3.082, 1, -1.72687372293808, -0.463452737940247, 0.369311937842414, -1.56632295540082],
3401 :     [3.083, 1, -1.72493882583316, -0.464477148418726, 0.369769904643645, -1.56514483621612],
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3778 :