Implementation by Rmosek: Correlated Null

Last updated: 2017-12-21

Code version: 6e42447

Introduction

Being in the development stage, cvxr is not very efficient at the convex optimization in fitting Gaussian derivatives. We are moving to Rmosek.

source("../code/gdash.R")

z.mat = read.table("../output/z_null_liver_777.txt")
fd.bh = scan("../output/fd.bh.0.05_null_liver_777.txt")
pihat0 = scan("../output/pihat0_z_null_liver_777.txt")

Bad

These data sets are unfriendly for both BH and ASH.

Data Set  355 :
 Number of BH false discoveries:  639 ;
 ASH's pihat0 =  0.04750946 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0225585778660563 ; 2 : 1.30307541409737 ; 3 : 0.0529119988564207 ; 4 : 0.86434625413445 ; 5 : 0 ; 6 : 0.115991285668782 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0.012768069558034 ;

Data Set  23 :
 Number of BH false discoveries:  408 ;
 ASH's pihat0 =  0.03139009 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.016039132567407 ; 2 : 1.34993003351774 ; 3 : 0.081263929095916 ; 4 : 0.714486051374311 ; 5 : 0.139320458700019 ; 6 : 0 ; 7 : 0.100823265137825 ; 8 : 0 ; 9 : 0.0608462191710317 ; 10 : 0.000964802431909129 ;

Data Set  122 :
 Number of BH false discoveries:  331 ;
 ASH's pihat0 =  0.04301549 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.00711459304239053 ; 2 : 1.1037821799492 ; 3 : 0 ; 4 : 0.528402132228434 ; 5 : 0 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0.000977244957534518 ; 10 : 0 ;

Data Set  749 :
 Number of BH false discoveries:  114 ;
 ASH's pihat0 =  0.02583276 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0458626422285871 ; 2 : 1.0692347200711 ; 3 : 0.00127673630293041 ; 4 : 0.437875595583542 ; 5 : 0 ; 6 : 0 ; 7 : 0.110561067339144 ; 8 : 0 ; 9 : 0.104817311821469 ; 10 : 0 ;

Data Set  724 :
 Number of BH false discoveries:  79 ;
 ASH's pihat0 =  0.01606004 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0352454327180146 ; 2 : 1.085621208483 ; 3 : 0.072031313135509 ; 4 : 0.313477034131008 ; 5 : 0 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0 ;

Data Set  56 :
 Number of BH false discoveries:  35 ;
 ASH's pihat0 =  0.04829662 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.015603437988185 ; 2 : 0.863695602978145 ; 3 : 0.135238823162622 ; 4 : 0.336622180491709 ; 5 : 0.18630322261983 ; 6 : 0.00268136598387189 ; 7 : 0.0339747362866559 ; 8 : 0.0181307043838827 ; 9 : 0 ; 10 : 0.0344310653486728 ;

Data Set  840 :
 Number of BH false discoveries:  28 ;
 ASH's pihat0 =  0.02316832 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 1.03471486197017 ; 3 : 0 ; 4 : 0.329235042804543 ; 5 : 0 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0.00321461375022215 ; 10 : 0.0593306511691146 ;

Data Set  858 :
 Number of BH false discoveries:  16 ;
 ASH's pihat0 =  0.05110069 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 1.00711281742885 ; 3 : 0 ; 4 : 0.390477482047559 ; 5 : 0 ; 6 : 0 ; 7 : 0.0551377637495718 ; 8 : 0 ; 9 : 0.0479292544081263 ; 10 : 0.0837572023674743 ;

Data Set  771 :
 Number of BH false discoveries:  12 ;
 ASH's pihat0 =  0.04316169 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0.73920141158121 ; 3 : 0.077182005444609 ; 4 : 0.167446690539398 ; 5 : 0.114289140310117 ; 6 : 0 ; 7 : 0.00550636320542386 ; 8 : 0.0464055959765927 ; 9 : 0 ; 10 : 0.0398398101639301 ;

Data Set  389 :
 Number of BH false discoveries:  11 ;
 ASH's pihat0 =  0.03156173 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0104278129211743 ; 2 : 0.81975416226431 ; 3 : 0.175093847880514 ; 4 : 0.237691296370699 ; 5 : 0.0928656721296627 ; 6 : 0 ; 7 : 0 ; 8 : 0.0957748191090392 ; 9 : 0 ; 10 : 0.14869448917214 ;

Data Set  485 :
 Number of BH false discoveries:  9 ;
 ASH's pihat0 =  0.04794909 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0.688297974677999 ; 3 : 0.102717821163846 ; 4 : 0.1348683673964 ; 5 : 0.0774735270815003 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0 ;

Data Set  77 :
 Number of BH false discoveries:  7 ;
 ASH's pihat0 =  0.05151585 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0316698203536559 ; 2 : 0.900083565443527 ; 3 : 0.369105500650609 ; 4 : 0.348606888712178 ; 5 : 0.225564147471124 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0 ;

Data Set  984 :
 Number of BH false discoveries:  7 ;
 ASH's pihat0 =  0.04567195 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0.699043214591173 ; 3 : 0.100302581737407 ; 4 : 0.145190970625763 ; 5 : 0.0474679405200551 ; 6 : 0 ; 7 : 0 ; 8 : 0.134635889035747 ; 9 : 0 ; 10 : 0.110318005619196 ;

Data Set  360 :
 Number of BH false discoveries:  6 ;
 ASH's pihat0 =  0.0308234 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0370729985554797 ; 2 : 0.965843584122268 ; 3 : 0.113777617637278 ; 4 : 0.182892304104783 ; 5 : 0.0985758556513691 ; 6 : 0 ; 7 : 0 ; 8 : 0.175602983077514 ; 9 : 0 ; 10 : 0.128160801611689 ;

Data Set  522 :
 Number of BH false discoveries:  4 ;
 ASH's pihat0 =  0.02083846 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0.836683104968565 ; 3 : 0 ; 4 : 0.0380324164945556 ; 5 : 0 ; 6 : 0 ; 7 : 0 ; 8 : 0.299307885307397 ; 9 : 0 ; 10 : 0.0990067134691387 ;

Data Set  22 :
 Number of BH false discoveries:  1 ;
 ASH's pihat0 =  0.03271534 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.011182140702874 ; 2 : 0.637669154664922 ; 3 : 0.0954740117610049 ; 4 : 0.0145362179577236 ; 5 : 0.0384875211132818 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0.0366040528004269 ;

Data Set  269 :
 Number of BH false discoveries:  1 ;
 ASH's pihat0 =  0.02787007 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0350127752264117 ; 2 : 0.665763246213506 ; 3 : 0 ; 4 : 0 ; 5 : 0.360745065018111 ; 6 : 0 ; 7 : 0.660778015759139 ; 8 : 0 ; 9 : 0.341451404165077 ; 10 : 0 ;

Data Set  403 :
 Number of BH false discoveries:  1 ;
 ASH's pihat0 =  0.0531628 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.00287620650301181 ; 2 : 0.547191548824974 ; 3 : 0 ; 4 : 0 ; 5 : 0.0381682188675178 ; 6 : 0.000237362926646135 ; 7 : 0.0760788952836512 ; 8 : 0.155507732499129 ; 9 : 0.0557968950438762 ; 10 : 0.00176813778552284 ;

Data Set  923 :
 Number of BH false discoveries:  1 ;
 ASH's pihat0 =  0.03239883 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0237509340668928 ; 2 : 0.920816163763449 ; 3 : 0.126877447333694 ; 4 : 0.158793996465808 ; 5 : 0.0298935777348498 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0 ;

Data Set  3 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  0.0465218 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0232314826078553 ; 2 : 0.491281828479503 ; 3 : 0.165633823951823 ; 4 : 0 ; 5 : 0 ; 6 : 0 ; 7 : 0 ; 8 : 0 ; 9 : 0 ; 10 : 0 ;

Good

These data sets are friendly for both BH and ASH.

Data Set  4 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0220506376197135 ; 4 : 0 ;

Data Set  5 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.00299571447314808 ; 2 : 0 ; 3 : 0 ; 4 : 0 ;

Data Set  7 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0 ; 4 : 0.0154789111560678 ;

Data Set  8 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0448250444179864 ; 2 : 0 ; 3 : 0 ; 4 : 0.143559202208159 ;

Data Set  9 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0.00733790864173648 ; 3 : 0.0163403533804334 ; 4 : 0 ;

Data Set  12 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0 ; 4 : 0.0221346391841642 ;

Data Set  13 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0 ; 4 : 0 ;

Data Set  14 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0319220622489123 ; 4 : 0.0192425975845108 ;

Data Set  16 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0132467627058809 ; 2 : 0 ; 3 : 0.0050633052630629 ; 4 : 0.0460280223671055 ;

Data Set  17 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.019759781277875 ; 2 : 0 ; 3 : 0 ; 4 : 0.0663856267063882 ;

Data Set  18 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0334151016576349 ; 4 : 0.00459370714209619 ;

Data Set  25 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0110557207434819 ; 4 : 0.165219572806289 ;

Data Set  28 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0200505087249252 ; 2 : 0 ; 3 : 0 ; 4 : 0 ;

Data Set  30 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0118483507943389 ; 4 : 0 ;

Data Set  34 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.101371139384381 ; 4 : 0 ;

Data Set  35 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0165529780499778 ; 2 : 0 ; 3 : 0 ; 4 : 0.102360725261069 ;

Data Set  36 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.00780175544762885 ; 2 : 0 ; 3 : 0 ; 4 : 0.0653975820808355 ;

Data Set  38 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0 ; 4 : 0.0643337534695275 ;

Data Set  42 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0117929353952078 ; 4 : 0.14259210767113 ;

Data Set  43 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0266591670097924 ; 4 : 0.119934975949713 ;

Data Set  45 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0197168202529947 ; 2 : 0 ; 3 : 0.00535774384797348 ; 4 : 0.0981326520063197 ;

Data Set  46 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0142593356797949 ; 4 : 0 ;

Data Set  47 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0553713103730739 ; 2 : 0 ; 3 : 0 ; 4 : 0.0424140129177028 ;

Data Set  49 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0399557075219927 ; 4 : 0.0203389185614439 ;

Data Set  52 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0717118497321021 ; 2 : 0 ; 3 : 0 ; 4 : 0 ;

Data Set  53 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0 ; 2 : 0 ; 3 : 0.0203994359033729 ; 4 : 0 ;

Data Set  54 :
 Number of BH false discoveries:  0 ;
 ASH's pihat0 =  1 ;
Normalized coefficients of Gaussian derivatives:
 1 : 0.0226491329325567 ; 2 : 0 ; 3 : 0 ; 4 : 0.0369913458520336 ;

Remarks

The implementation by Rmosek is more convenient, cleaner, yet more complicated. Need to understand different issues such as initialization, optimal orders, controlling parameters, dual construction, etc.

Session information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.2

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] Rmosek_8.0.69     PolynomF_1.0-1    CVXR_0.94-4       REBayes_1.2      
[5] Matrix_1.2-12     SQUAREM_2017.10-1 EQL_1.0-0         ttutils_1.0-1    

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.14      knitr_1.17        magrittr_1.5     
 [4] bit_1.1-12        lattice_0.20-35   R6_2.2.2         
 [7] stringr_1.2.0     tools_3.4.3       grid_3.4.3       
[10] R.oo_1.21.0       git2r_0.20.0      scs_1.1-1        
[13] htmltools_0.3.6   bit64_0.9-7       yaml_2.1.16      
[16] rprojroot_1.3-1   digest_0.6.13     gmp_0.5-13.1     
[19] ECOSolveR_0.3-2   R.utils_2.6.0     evaluate_0.10.1  
[22] rmarkdown_1.8     stringi_1.1.6     compiler_3.4.3   
[25] Rmpfr_0.6-1       backports_1.1.2   R.methodsS3_1.7.1

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Implementation by `Rmosek`: Correlated Null

Lei Sun

2017-05-02

Introduction

Bad

Good

Remarks

Session information