[R] Remove highly correlated variables from a data frame or matrix

Ana Marija @okov|c@@n@m@r|j@ @end|ng |rom gm@||@com
Thu Nov 14 23:25:59 CET 2019


HI Jim,

This:
colnames(calc.jim)[colSums(abs(calc.jim)>0.8)<3]

was the master take!

Thank you so much!!!

On Thu, Nov 14, 2019 at 3:39 PM Jim Lemon <drjimlemon using gmail.com> wrote:
>
> I thought you were going to trick us. What I think you are asking now
> is how to get the variable names in the columns that have at most one
> _absolute_ value greater than 0.8. OK:
>
> # I'm not going to try to recreate your correlation matrix
> calc.jim<-matrix(runif(100,min=-1,max=1),nrow=10)
> for(i in 1:10) calc.jim[i,i]<-1
> rownames(calc.jim)<-<-colnames(calc.jim)<-paste0("rs",1:10)
>
> Now that we have a plausible fake correlation matrix, all we have to
> do is extract the column names:
>
> colnames(calc.jim)[colSums(abs(calc.jim)>0.8)<2]
>
> Of course, what you really meant could have been, "I want the column
> names of the variables with at most one absolute value greater than
> 0.8 ignoring the diagonal values because I don't care about those". If
> so:
>
> colnames(calc.jim)[colSums(abs(calc.jim)>0.8)<3]
>
> Any more tricks?
>
> Jim
>
> On Fri, Nov 15, 2019 at 8:17 AM Ana Marija <sokovic.anamarija using gmail.com> wrote:
> >
> > what would be the approach to remove variable that has at least 2
> > correlation coefficients >0.8?
> > this is the whole output of the head()
> >
> > > head(calc.rho)
> >             rs56192520 rs3764410 rs145984817 rs1807401 rs1807402 rs35350506
> > rs56192520       1.000     0.976       0.927     0.927     0.927      0.927
> > rs3764410        0.976     1.000       0.952     0.952     0.952      0.952
> > rs145984817      0.927     0.952       1.000     1.000     1.000      1.000
> > rs1807401        0.927     0.952       1.000     1.000     1.000      1.000
> > rs1807402        0.927     0.952       1.000     1.000     1.000      1.000
> > rs35350506       0.927     0.952       1.000     1.000     1.000      1.000
> >             rs2089177 rs12325677 rs62064624 rs62064631 rs2349295 rs2174369
> > rs56192520      0.927      0.927      0.927      0.927     0.709     0.903
> > rs3764410       0.952      0.952      0.952      0.952     0.728     0.928
> > rs145984817     1.000      1.000      1.000      1.000     0.771     0.975
> > rs1807401       1.000      1.000      1.000      1.000     0.771     0.975
> > rs1807402       1.000      1.000      1.000      1.000     0.771     0.975
> > rs35350506      1.000      1.000      1.000      1.000     0.771     0.975
> >             rs7218554 rs62064634 rs4360974 rs4527060 rs6502526 rs6502527
> > rs56192520      0.903      0.903     0.903     0.903     0.903     0.903
> > rs3764410       0.928      0.928     0.928     0.928     0.928     0.928
> > rs145984817     0.975      0.975     0.975     0.975     0.975     0.975
> > rs1807401       0.975      0.975     0.975     0.975     0.975     0.975
> > rs1807402       0.975      0.975     0.975     0.975     0.975     0.975
> > rs35350506      0.975      0.975     0.975     0.975     0.975     0.975
> >             rs9900318 rs8069906 rs9908521 rs9908336 rs9908870 rs9895995
> > rs56192520      0.903     0.268     0.327     0.327     0.327     0.582
> > rs3764410       0.928     0.276     0.336     0.336     0.336     0.598
> > rs145984817     0.975     0.309     0.371     0.371     0.371     0.638
> > rs1807401       0.975     0.309     0.371     0.371     0.371     0.638
> > rs1807402       0.975     0.309     0.371     0.371     0.371     0.638
> > rs35350506      0.975     0.309     0.371     0.371     0.371     0.638
> >             rs7211086 rs9905280 rs8073305 rs8072086 rs4312350 rs4313843
> > rs56192520      0.880     0.268     0.327     0.880     0.880     0.880
> > rs3764410       0.905     0.276     0.336     0.905     0.905     0.905
> > rs145984817     0.951     0.309     0.371     0.951     0.951     0.951
> > rs1807401       0.951     0.309     0.371     0.951     0.951     0.951
> > rs1807402       0.951     0.309     0.371     0.951     0.951     0.951
> > rs35350506      0.951     0.309     0.371     0.951     0.951     0.951
> >             rs8069610 rs883504 rs8072394 rs4280293 rs4465638 rs12602378
> > rs56192520      0.582    0.903     0.582     0.582     0.811      0.302
> > rs3764410       0.598    0.928     0.598     0.598     0.836      0.311
> > rs145984817     0.638    0.975     0.638     0.638     0.879      0.344
> > rs1807401       0.638    0.975     0.638     0.638     0.879      0.344
> > rs1807402       0.638    0.975     0.638     0.638     0.879      0.344
> > rs35350506      0.638    0.975     0.638     0.638     0.879      0.344
> >             rs9899059 rs6502530 rs4380085 rs6502532 rs4792798 rs4792799
> > rs56192520      0.302     0.309     0.834     0.251     0.063     0.063
> > rs3764410       0.311     0.318     0.858     0.259     0.080     0.080
> > rs145984817     0.344     0.352     0.902     0.291     0.086     0.086
> > rs1807401       0.344     0.352     0.902     0.291     0.086     0.086
> > rs1807402       0.344     0.352     0.902     0.291     0.086     0.086
> > rs35350506      0.344     0.352     0.902     0.291     0.086     0.086
> >             rs4316813 rs148563931 rs74751226 rs8068857 rs8069441 rs77397878
> > rs56192520      0.006       0.006      0.006     0.006     0.006      0.006
> > rs3764410       0.006       0.006      0.006     0.006     0.006      0.006
> > rs145984817     0.006       0.006      0.006     0.006     0.006      0.006
> > rs1807401       0.006       0.006      0.006     0.006     0.006      0.006
> > rs1807402       0.006       0.006      0.006     0.006     0.006      0.006
> > rs35350506      0.006       0.006      0.006     0.006     0.006      0.006
> >             rs75339756 rs4608391 rs79569548 rs4275914 rs11870422 rs8075751
> > rs56192520       0.006     0.006      0.006     0.044      0.007     0.004
> > rs3764410        0.006     0.006      0.006     0.042      0.005     0.005
> > rs145984817      0.006     0.006      0.006     0.047      0.002     0.015
> > rs1807401        0.006     0.006      0.006     0.047      0.002     0.015
> > rs1807402        0.006     0.006      0.006     0.047      0.002     0.015
> > rs35350506       0.006     0.006      0.006     0.047      0.002     0.015
> >             rs11658904 rs138437542 rs80344434 rs7222311 rs7221842 rs7223686
> > rs56192520       0.003       0.004      0.004     0.033     0.009     0.000
> > rs3764410        0.004       0.004      0.004     0.031     0.007     0.000
> > rs145984817      0.010       0.004      0.004     0.035     0.011     0.005
> > rs1807401        0.010       0.004      0.004     0.035     0.011     0.005
> > rs1807402        0.010       0.004      0.004     0.035     0.011     0.005
> > rs35350506       0.010       0.004      0.004     0.035     0.011     0.005
> >             rs78013597 rs74965036 rs78063986 rs118106233 rs117345712
> > rs56192520       0.004      0.004      0.004       0.004       0.005
> > rs3764410        0.004      0.004      0.004       0.004       0.006
> > rs145984817      0.004      0.004      0.004       0.004       0.005
> > rs1807401        0.004      0.004      0.004       0.004       0.005
> > rs1807402        0.004      0.004      0.004       0.004       0.005
> > rs35350506       0.004      0.004      0.004       0.004       0.005
> >             rs113004656 rs9898995 rs4985718 rs9893911 rs79110942 rs7208929
> > rs56192520        0.004     0.033     0.033     0.023      0.004     0.023
> > rs3764410         0.004     0.031     0.031     0.021      0.004     0.021
> > rs145984817       0.004     0.035     0.035     0.025      0.004     0.025
> > rs1807401         0.004     0.035     0.035     0.025      0.004     0.025
> > rs1807402         0.004     0.035     0.035     0.025      0.004     0.025
> > rs35350506        0.004     0.035     0.035     0.025      0.004     0.025
> >             rs12601453 rs4078062 rs75129280 rs76664572 rs78961289 rs146364798
> > rs56192520       0.004     0.001      0.004      0.004      0.004       0.004
> > rs3764410        0.004     0.002      0.004      0.004      0.004       0.004
> > rs145984817      0.004     0.001      0.004      0.004      0.004       0.004
> > rs1807401        0.004     0.001      0.004      0.004      0.004       0.004
> > rs1807402        0.004     0.001      0.004      0.004      0.004       0.004
> > rs35350506       0.004     0.001      0.004      0.004      0.004       0.004
> >             rs76715413 rs4078534 rs79457460 rs74369938 rs76423171 rs74668400
> > rs56192520           0     0.004      0.004      0.002      0.004      0.004
> > rs3764410            0     0.004      0.004      0.001      0.004      0.004
> > rs145984817          0     0.004      0.004      0.005      0.004      0.004
> > rs1807401            0     0.004      0.004      0.005      0.004      0.004
> > rs1807402            0     0.004      0.004      0.005      0.004      0.004
> > rs35350506           0     0.004      0.004      0.005      0.004      0.004
> >             rs75146120 rs1135237 rs9914671 rs117759512 rs4985696 rs16961340
> > rs56192520       0.004     0.003     0.009       0.004     0.009      0.004
> > rs3764410        0.004     0.003     0.007       0.004     0.007      0.004
> > rs145984817      0.004     0.003     0.011       0.004     0.011      0.004
> > rs1807401        0.004     0.003     0.011       0.004     0.011      0.004
> > rs1807402        0.004     0.003     0.011       0.004     0.011      0.004
> > rs35350506       0.004     0.003     0.011       0.004     0.011      0.004
> >             rs17794159 rs4247118 rs78572469 rs12601193 rs2349646 rs2090018
> > rs56192520       0.001     0.033      0.002      0.004     0.020     0.033
> > rs3764410        0.002     0.031      0.001      0.004     0.019     0.031
> > rs145984817      0.001     0.035      0.005      0.004     0.022     0.035
> > rs1807401        0.001     0.035      0.005      0.004     0.022     0.035
> > rs1807402        0.001     0.035      0.005      0.004     0.022     0.035
> > rs35350506       0.001     0.035      0.005      0.004     0.022     0.035
> >             rs12601424 rs4985701 rs8064550 rs2271521 rs2271520 rs11078374
> > rs56192520       0.004     0.033     0.033     0.004     0.033      0.014
> > rs3764410        0.004     0.031     0.031     0.004     0.031      0.012
> > rs145984817      0.004     0.035     0.035     0.004     0.035      0.016
> > rs1807401        0.004     0.035     0.035     0.004     0.035      0.016
> > rs1807402        0.004     0.035     0.035     0.004     0.035      0.016
> > rs35350506       0.004     0.035     0.035     0.004     0.035      0.016
> >             rs4985702 rs1124961 rs11652674 rs3924340 rs112450164 rs7208973
> > rs56192520      0.033     0.003      0.002     0.001       0.004     0.033
> > rs3764410       0.031     0.003      0.001     0.002       0.004     0.031
> > rs145984817     0.035     0.003      0.005     0.001       0.004     0.035
> > rs1807401       0.035     0.003      0.005     0.001       0.004     0.035
> > rs1807402       0.035     0.003      0.005     0.001       0.004     0.035
> > rs35350506      0.035     0.003      0.005     0.001       0.004     0.035
> >             rs9910857 rs78574480 rs8072184 rs12602196 rs6502563 rs3744135
> > rs56192520      0.006      0.004     0.014      0.004     0.033     0.004
> > rs3764410       0.005      0.004     0.012      0.004     0.031     0.004
> > rs145984817     0.002      0.004     0.016      0.004     0.035     0.004
> > rs1807401       0.002      0.004     0.016      0.004     0.035     0.004
> > rs1807402       0.002      0.004     0.016      0.004     0.035     0.004
> > rs35350506      0.002      0.004     0.016      0.004     0.035     0.004
> >             rs148779543 rs77689691 rs41319048 rs117340532 rs78647096 rs77712968
> > rs56192520            0      0.004      0.004       0.002      0.004      0.004
> > rs3764410             0      0.004      0.004       0.001      0.004      0.004
> > rs145984817           0      0.004      0.004       0.005      0.004      0.004
> > rs1807401             0      0.004      0.004       0.005      0.004      0.004
> > rs1807402             0      0.004      0.004       0.005      0.004      0.004
> > rs35350506            0      0.004      0.004       0.005      0.004      0.004
> >             rs16961396 rs80054920 rs7206981 rs4985740 rs3803762 rs77103270
> > rs56192520       0.004      0.004     0.033     0.023     0.004      0.002
> > rs3764410        0.004      0.004     0.031     0.021     0.004      0.001
> > rs145984817      0.004      0.004     0.035     0.025     0.004      0.005
> > rs1807401        0.004      0.004     0.035     0.025     0.004      0.005
> > rs1807402        0.004      0.004     0.035     0.025     0.004      0.005
> > rs35350506       0.004      0.004     0.035     0.025     0.004      0.005
> >             rs7207485 rs77342773 rs3826304 rs3744126 rs7210879 rs7211576
> > rs56192520      0.029      0.004     0.004     0.004     0.023     0.006
> > rs3764410       0.027      0.004     0.004     0.004     0.021     0.005
> > rs145984817     0.031      0.004     0.004     0.004     0.025     0.002
> > rs1807401       0.031      0.004     0.004     0.004     0.025     0.002
> > rs1807402       0.031      0.004     0.004     0.004     0.025     0.002
> > rs35350506      0.031      0.004     0.004     0.004     0.025     0.002
> >             rs117967362 rs75978745 rs6502564 rs9894565 rs36079048 rs8076621
> > rs56192520        0.004      0.004     0.007     0.017          0     0.004
> > rs3764410         0.004      0.004     0.005     0.015          0     0.004
> > rs145984817       0.004      0.004     0.009     0.019          0     0.004
> > rs1807401         0.004      0.004     0.009     0.019          0     0.004
> > rs1807402         0.004      0.004     0.009     0.019          0     0.004
> > rs35350506        0.004      0.004     0.009     0.019          0     0.004
> >             rs7218795 rs3803761 rs12602675 rs7208065 rs4985705 rs8080386
> > rs56192520      0.026     0.032          0     0.018     0.014     0.003
> > rs3764410       0.024     0.029          0     0.015     0.011     0.002
> > rs145984817     0.028     0.034          0     0.021     0.016     0.004
> > rs1807401       0.028     0.034          0     0.021     0.016     0.004
> > rs1807402       0.028     0.034          0     0.021     0.016     0.004
> > rs35350506      0.028     0.034          0     0.021     0.016     0.004
> >             rs8065832 rs2018781 rs1736221 rs1736220 rs1736217 rs1708620
> > rs56192520      0.008     0.039     0.003     0.003     0.021     0.009
> > rs3764410       0.006     0.037     0.002     0.002     0.019     0.007
> > rs145984817     0.010     0.042     0.004     0.004     0.024     0.011
> > rs1807401       0.010     0.042     0.004     0.004     0.024     0.011
> > rs1807402       0.010     0.042     0.004     0.004     0.024     0.011
> > rs35350506      0.010     0.042     0.004     0.004     0.024     0.011
> >             rs1708619 rs1736216 rs76319098 rs1736215 rs1736214 rs1708617
> > rs56192520      0.009     0.024      0.017     0.012     0.019     0.029
> > rs3764410       0.007     0.021      0.016     0.009     0.016     0.026
> > rs145984817     0.011     0.026      0.018     0.014     0.022     0.031
> > rs1807401       0.011     0.026      0.018     0.014     0.022     0.031
> > rs1807402       0.011     0.026      0.018     0.014     0.022     0.031
> > rs35350506      0.011     0.026      0.018     0.014     0.022     0.031
> >             rs12602831 rs12602871 rs1736213 rs1736212 rs76045368 rs34518797
> > rs56192520       0.000      0.000     0.015     0.029      0.001      0.001
> > rs3764410        0.001      0.001     0.013     0.026      0.001      0.001
> > rs145984817      0.000      0.000     0.018     0.031      0.000      0.000
> > rs1807401        0.000      0.000     0.018     0.031      0.000      0.000
> > rs1807402        0.000      0.000     0.018     0.031      0.000      0.000
> > rs35350506       0.000      0.000     0.018     0.031      0.000      0.000
> >             rs11078378 rs8079562 rs8065774 rs8066090 rs41337846 rs1736209
> > rs56192520       0.043     0.001     0.001     0.029      0.000     0.029
> > rs3764410        0.041     0.001     0.001     0.026      0.001     0.026
> > rs145984817      0.046     0.000     0.000     0.031      0.000     0.031
> > rs1807401        0.046     0.000     0.000     0.031      0.000     0.031
> > rs1807402        0.046     0.000     0.000     0.031      0.000     0.031
> > rs35350506       0.046     0.000     0.000     0.031      0.000     0.031
> >             rs1736208 rs12949822 rs76246042 rs12600635 rs55689224 rs1736207
> > rs56192520      0.043      0.043      0.000      0.000      0.000     0.015
> > rs3764410       0.041      0.041      0.001      0.001      0.001     0.013
> > rs145984817     0.046      0.046      0.000      0.000      0.000     0.018
> > rs1807401       0.046      0.046      0.000      0.000      0.000     0.018
> > rs1807402       0.046      0.046      0.000      0.000      0.000     0.018
> > rs35350506      0.046      0.046      0.000      0.000      0.000     0.018
> >             rs1708626 rs1736206 rs9896078 rs16961474 rs1708627 rs1736205
> > rs56192520      0.015     0.015     0.001      0.001     0.017     0.021
> > rs3764410       0.013     0.013     0.001      0.001     0.014     0.019
> > rs145984817     0.018     0.018     0.000      0.001     0.020     0.024
> > rs1807401       0.018     0.018     0.000      0.001     0.020     0.024
> > rs1807402       0.018     0.018     0.000      0.001     0.020     0.024
> > rs35350506      0.018     0.018     0.000      0.001     0.020     0.024
> >             rs1708628 rs7220577 rs2294155 rs1736204 rs1736203 rs1736202
> > rs56192520      0.011     0.011     0.000     0.021     0.014     0.014
> > rs3764410       0.009     0.009     0.000     0.019     0.011     0.011
> > rs145984817     0.013     0.013     0.001     0.024     0.016     0.016
> > rs1807401       0.013     0.013     0.001     0.024     0.016     0.016
> > rs1807402       0.013     0.013     0.001     0.024     0.016     0.016
> > rs35350506      0.013     0.013     0.001     0.024     0.016     0.016
> >             rs12937908 rs1736200 rs1708623 rs1708624 rs9894884 rs9901894
> > rs56192520       0.009     0.009     0.008     0.007     0.009     0.009
> > rs3764410        0.007     0.007     0.006     0.005     0.007     0.007
> > rs145984817      0.011     0.011     0.010     0.008     0.011     0.011
> > rs1807401        0.011     0.011     0.010     0.008     0.011     0.011
> > rs1807402        0.011     0.011     0.010     0.008     0.011     0.011
> > rs35350506       0.011     0.011     0.010     0.008     0.011     0.011
> >             rs9903294 rs2472689 rs1630656 rs111478970 rs3182911 rs7219012
> > rs56192520      0.008     0.011     0.007       0.007     0.008     0.000
> > rs3764410       0.006     0.009     0.005       0.005     0.006     0.000
> > rs145984817     0.010     0.013     0.008       0.008     0.010     0.001
> > rs1807401       0.010     0.013     0.008       0.008     0.010     0.001
> > rs1807402       0.010     0.013     0.008       0.008     0.010     0.001
> > rs35350506      0.010     0.013     0.008       0.008     0.010     0.001
> >             rs9890657 rs12453455 rs12947291 rs150267386 rs16961493 rs11652745
> > rs56192520      0.009      0.007      0.008       0.013      0.000      0.009
> > rs3764410       0.007      0.005      0.006       0.012      0.000      0.007
> > rs145984817     0.011      0.008      0.010       0.014      0.001      0.011
> > rs1807401       0.011      0.008      0.010       0.014      0.001      0.011
> > rs1807402       0.011      0.008      0.010       0.014      0.001      0.011
> > rs35350506      0.011      0.008      0.010       0.014      0.001      0.011
> >             rs9907107 rs8070574 rs4985759 rs3866959 rs7219248 rs6502568
> > rs56192520      0.009     0.009     0.009     0.011     0.009     0.011
> > rs3764410       0.007     0.007     0.007     0.009     0.007     0.009
> > rs145984817     0.011     0.011     0.011     0.013     0.011     0.013
> > rs1807401       0.011     0.011     0.011     0.013     0.011     0.013
> > rs1807402       0.011     0.011     0.011     0.013     0.011     0.013
> > rs35350506      0.011     0.011     0.011     0.013     0.011     0.013
> >             rs7220275 rs12450037 rs7225876 rs9892352 rs4985760 rs6502569
> > rs56192520      0.009      0.008     0.007     0.011     0.011     0.011
> > rs3764410       0.007      0.006     0.005     0.009     0.009     0.009
> > rs145984817     0.011      0.010     0.008     0.013     0.013     0.013
> > rs1807401       0.011      0.010     0.008     0.013     0.013     0.013
> > rs1807402       0.011      0.010     0.008     0.013     0.013     0.013
> > rs35350506      0.011      0.010     0.008     0.013     0.013     0.013
> >             rs1029830 rs2012954 rs1029832 rs2270180 rs8072402 rs7221553
> > rs56192520      0.009     0.011     0.008     0.000     0.009     0.011
> > rs3764410       0.007     0.009     0.006     0.000     0.007     0.009
> > rs145984817     0.011     0.013     0.010     0.001     0.011     0.013
> > rs1807401       0.011     0.013     0.010     0.001     0.011     0.013
> > rs1807402       0.011     0.013     0.010     0.001     0.011     0.013
> > rs35350506      0.011     0.013     0.010     0.001     0.011     0.013
> >             rs145597919 rs150772017 rs2041393 rs6502578 rs11078382 rs9912109
> > rs56192520        0.013       0.013     0.005     0.005          0     0.005
> > rs3764410         0.012       0.012     0.004     0.004          0     0.004
> > rs145984817       0.014       0.014     0.006     0.006          0     0.006
> > rs1807401         0.014       0.014     0.006     0.006          0     0.006
> > rs1807402         0.014       0.014     0.006     0.006          0     0.006
> > rs35350506        0.014       0.014     0.006     0.006          0     0.006
> >             rs12601631 rs11869054 rs11869079 rs9912599 rs7220057 rs9896970
> > rs56192520           0          0          0         0         0         0
> > rs3764410            0          0          0         0         0         0
> > rs145984817          0          0          0         0         0         0
> > rs1807401            0          0          0         0         0         0
> > rs1807402            0          0          0         0         0         0
> > rs35350506           0          0          0         0         0         0
> >             rs34121330 rs34668117 rs67773570 rs242252 rs955893 rs28583584
> > rs56192520       0.000      0.000          0    0.002    0.001      0.013
> > rs3764410        0.001      0.001          0    0.003    0.001      0.005
> > rs145984817      0.000      0.000          0    0.002    0.001      0.004
> > rs1807401        0.000      0.000          0    0.002    0.001      0.004
> > rs1807402        0.000      0.000          0    0.002    0.001      0.004
> > rs35350506       0.000      0.000          0    0.002    0.001      0.004
> >             rs9944423 rs7217764 rs11651957 rs73978990 rs8071007 rs56044345
> > rs56192520      0.013     0.011      0.011      0.011     0.011      0.011
> > rs3764410       0.005     0.004      0.004      0.004     0.004      0.004
> > rs145984817     0.004     0.003      0.003      0.003     0.003      0.003
> > rs1807401       0.004     0.003      0.003      0.003     0.003      0.003
> > rs1807402       0.004     0.003      0.003      0.003     0.003      0.003
> > rs35350506      0.004     0.003      0.003      0.003     0.003      0.003
> >             rs17804843
> > rs56192520           0
> > rs3764410            0
> > rs145984817          0
> > rs1807401            0
> > rs1807402            0
> > rs35350506           0
> >
> >
> > On Thu, Nov 14, 2019 at 2:59 PM Abby Spurdle <spurdle.a using gmail.com> wrote:
> > >
> > > That's assuming your data was returned by head().
> >
> > ______________________________________________
> > R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> > and provide commented, minimal, self-contained, reproducible code.



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