Last updated: 2018-05-15

workflowr checks: (Click a bullet for more information)

  • ✔ R Markdown file: up-to-date

    Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

  • ✔ Environment: empty

    Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

  • ✔ Seed: set.seed(12345)

    The command set.seed(12345) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

  • ✔ Session information: recorded

    Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

  • ✔ Repository version: 388e65e

    Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.

    Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated: 
    
Ignored files:
    Ignored:    .DS_Store
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    analysis/.DS_Store
    Ignored:    analysis/BH_robustness_cache/
    Ignored:    analysis/FDR_Null_cache/
    Ignored:    analysis/FDR_null_betahat_cache/
    Ignored:    analysis/Rmosek_cache/
    Ignored:    analysis/StepDown_cache/
    Ignored:    analysis/alternative2_cache/
    Ignored:    analysis/alternative_cache/
    Ignored:    analysis/ash_gd_cache/
    Ignored:    analysis/average_cor_gtex_2_cache/
    Ignored:    analysis/average_cor_gtex_cache/
    Ignored:    analysis/brca_cache/
    Ignored:    analysis/cash_deconv_cache/
    Ignored:    analysis/cash_fdr_1_cache/
    Ignored:    analysis/cash_fdr_2_cache/
    Ignored:    analysis/cash_fdr_3_cache/
    Ignored:    analysis/cash_fdr_4_cache/
    Ignored:    analysis/cash_fdr_5_cache/
    Ignored:    analysis/cash_fdr_6_cache/
    Ignored:    analysis/cash_plots_cache/
    Ignored:    analysis/cash_sim_1_cache/
    Ignored:    analysis/cash_sim_2_cache/
    Ignored:    analysis/cash_sim_3_cache/
    Ignored:    analysis/cash_sim_4_cache/
    Ignored:    analysis/cash_sim_5_cache/
    Ignored:    analysis/cash_sim_6_cache/
    Ignored:    analysis/cash_sim_7_cache/
    Ignored:    analysis/correlated_z_2_cache/
    Ignored:    analysis/correlated_z_3_cache/
    Ignored:    analysis/correlated_z_cache/
    Ignored:    analysis/create_null_cache/
    Ignored:    analysis/cutoff_null_cache/
    Ignored:    analysis/design_matrix_2_cache/
    Ignored:    analysis/design_matrix_cache/
    Ignored:    analysis/diagnostic_ash_cache/
    Ignored:    analysis/diagnostic_correlated_z_2_cache/
    Ignored:    analysis/diagnostic_correlated_z_3_cache/
    Ignored:    analysis/diagnostic_correlated_z_cache/
    Ignored:    analysis/diagnostic_plot_2_cache/
    Ignored:    analysis/diagnostic_plot_cache/
    Ignored:    analysis/efron_leukemia_cache/
    Ignored:    analysis/fitting_normal_cache/
    Ignored:    analysis/gaussian_derivatives_2_cache/
    Ignored:    analysis/gaussian_derivatives_3_cache/
    Ignored:    analysis/gaussian_derivatives_4_cache/
    Ignored:    analysis/gaussian_derivatives_5_cache/
    Ignored:    analysis/gaussian_derivatives_cache/
    Ignored:    analysis/gd-ash_cache/
    Ignored:    analysis/gd_delta_cache/
    Ignored:    analysis/gd_lik_2_cache/
    Ignored:    analysis/gd_lik_cache/
    Ignored:    analysis/gd_w_cache/
    Ignored:    analysis/knockoff_10_cache/
    Ignored:    analysis/knockoff_2_cache/
    Ignored:    analysis/knockoff_3_cache/
    Ignored:    analysis/knockoff_4_cache/
    Ignored:    analysis/knockoff_5_cache/
    Ignored:    analysis/knockoff_6_cache/
    Ignored:    analysis/knockoff_7_cache/
    Ignored:    analysis/knockoff_8_cache/
    Ignored:    analysis/knockoff_9_cache/
    Ignored:    analysis/knockoff_cache/
    Ignored:    analysis/knockoff_var_cache/
    Ignored:    analysis/marginal_z_alternative_cache/
    Ignored:    analysis/marginal_z_cache/
    Ignored:    analysis/mosek_reg_2_cache/
    Ignored:    analysis/mosek_reg_4_cache/
    Ignored:    analysis/mosek_reg_5_cache/
    Ignored:    analysis/mosek_reg_6_cache/
    Ignored:    analysis/mosek_reg_cache/
    Ignored:    analysis/pihat0_null_cache/
    Ignored:    analysis/plot_diagnostic_cache/
    Ignored:    analysis/poster_obayes17_cache/
    Ignored:    analysis/real_data_simulation_2_cache/
    Ignored:    analysis/real_data_simulation_3_cache/
    Ignored:    analysis/real_data_simulation_4_cache/
    Ignored:    analysis/real_data_simulation_5_cache/
    Ignored:    analysis/real_data_simulation_cache/
    Ignored:    analysis/rmosek_primal_dual_2_cache/
    Ignored:    analysis/rmosek_primal_dual_cache/
    Ignored:    analysis/seqgendiff_cache/
    Ignored:    analysis/simulated_correlated_null_2_cache/
    Ignored:    analysis/simulated_correlated_null_3_cache/
    Ignored:    analysis/simulated_correlated_null_cache/
    Ignored:    analysis/simulation_real_se_2_cache/
    Ignored:    analysis/simulation_real_se_cache/
    Ignored:    analysis/smemo_2_cache/
    Ignored:    data/LSI/
    Ignored:    docs/.DS_Store
    Ignored:    docs/figure/.DS_Store
    Ignored:    output/fig/
    Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
Expand here to see past versions:
    File Version Author Date Message
    html e05bc83 LSun 2018-05-12 Update to 1.0
    rmd cc0ab83 Lei Sun 2018-05-11 update
    html 0f36d99 LSun 2017-12-21 Build site.
    html 853a484 LSun 2017-11-07 Build site.
    html 6639968 LSun 2017-11-05 transfer
    html 603b826 LSun 2017-05-11 writeups
    rmd 29422be LSun 2017-05-11 fitting delta

Introduction

Theoretically, the Dirac delta function \(\delta_z\) can be approximated by infinite orders of Gaussian derivatives as follows,

\[ \delta_z = \sum\limits_{l = 0}^\infty \frac{1}{\sqrt{l!}}h_l\left(z\right) \left(\frac{1}{\sqrt{l!}}h_l\left(x\right)\varphi(x)\right) \ . \] Previously with the EQL::hermite function, it takes a very long time to evaluate an Hermite polynomial of a high degree, yet the PolynomF package provides a computationally efficient way to check if higher order Gaussian derivatives indeed approximate \(\delta_z\) at any \(z\).

With finite \(L\), we are looking at

\[ f_L\left(x\right) := \sum\limits_{l = 0}^L \frac{1}{\sqrt{l!}}h_l\left(z\right) \left( \frac{1}{\sqrt{l!}}h_l\left(x\right) \varphi\left(x\right) \right) \] and

\[ F_L\left(x\right) := \Phi\left(x\right) - \sum\limits_{l = 1}^L \frac{1}{\sqrt{l}} \frac{1}{\sqrt{l!}}h_l\left(z\right) \left( \frac{1}{\sqrt{\left(l - 1\right)!}}h_{l - 1}\left(x\right)\varphi\left(x\right)\right) \ . \] Given any \(z\), \(f_L\) should get closer to \(\delta_z\) and \(F_L\) to the \(0\)-\(1\) step function, as \(L\to\infty\).

\(z = -2\)

\(z = 0\)

\(z = 4\)

\(z = 6\)

Numerical instability.

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 10.13.4

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     

loaded via a namespace (and not attached):
 [1] workflowr_1.0.1   Rcpp_0.12.16      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.21.0      magrittr_1.5      evaluate_0.10.1  
[10] stringi_1.1.6     whisker_0.3-2     R.oo_1.21.0      
[13] R.utils_2.6.0     rmarkdown_1.9     tools_3.4.3      
[16] stringr_1.3.0     yaml_2.1.18       compiler_3.4.3   
[19] htmltools_0.3.6   knitr_1.20

This reproducible R Markdown analysis was created with workflowr 1.0.1