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Warning: package 'reticulate' was built under R version 3.5.3
Warning: package 'plotrix' was built under R version 3.5.3
Warning: package 'robustbase' was built under R version 3.5.1

In previous section we have shown MLE estimates variability for datasets generated with the same parameters. Now we would want to investigate the sampling distribution of MLE using bootsrapp.

Bootstrap procedure:

  1. Generate one time 500 cells with given \(k_{on}\), \(k_{off}\) and \(k_r\) parameters

  2. For 500 iterations, bootstrap the sample with replacement

  3. Estimate parameters by maximum likelihood

bootstrap<-function(boot_number,k_on,k_off,kr,n_cells){
  est=matrix(, nrow = boot_number, ncol = 3)
  x=generate_data(k_on,k_off,kr,n_cells)
  for (i in 1:boot_number){
    boot_x = as.matrix(x[sample(nrow(x),nrow(x),replace=TRUE)])
    est[i,]=MaximumLikelihood(boot_x)}
  return(est)}

I) Small \(k_{on}\) and \(k_{off}\) case:

#True k_on=k_off=0.1
boot_number=500
k_on=as.numeric(0.1)
k_off=as.numeric(0.1)
kr=as.numeric(100)
n_cells=as.integer(500)
est<-bootstrap(boot_number,k_on,k_off,kr,n_cells)

hist(log(est[,1]),breaks=30, main="MLE estimate of k_on parameter,true k_on=0.1")
abline(v=log(0.1),col='red')

paste("true k_on is 0.1;","k_on mean;",round(mean(est[,1],na.rm=TRUE),3),"k_on median", round(median(est[,1],na.rm=TRUE),3),"k_on st.deviation",round(sqrt(var(est[,1],na.rm=TRUE)),3))
[1] "true k_on is 0.1; k_on mean; 0.119 k_on median 0.119 k_on st.deviation 0.01"
hist(log(est[,2]),breaks=30, main="MLE estimate of k_off parameter, true k_off=0.1")
abline(v=log(0.1),col='red')

paste("true k_off is 0.1;","k_off mean;",round(mean(est[,2],na.rm=TRUE),3),"k_off median", round(median(est[,2],na.rm=TRUE),3),"k_off st.deviation",round(sqrt(var(est[,2],na.rm=TRUE)),3))
[1] "true k_off is 0.1; k_off mean; 0.13 k_off median 0.129 k_off st.deviation 0.014"
hist(log(est[,3]),breaks=30, main="MLE estimate of k_r parameter,true k_r=100")
abline(v=log(100),col='red')

paste("true k_r is 100;","k_r mean;",round(mean(est[,3],na.rm=TRUE),3),"k_r median", round(median(est[,3],na.rm=TRUE),3),"k_r st.deviation",round(sqrt(var(est[,3],na.rm=TRUE)),3))
[1] "true k_r is 100; k_r mean; 100.114 k_r median 100.115 k_r st.deviation 0.93"

As we saw in previous vignette, this small \(k_{on}\), \(k_{off}\) scenario has clearly identifiable solutions which is confirmed by boorstrap as we see again mean, median are close to the true value and standart deviation is small as well.

II) Large \(k_{on}\) and \(k_{off}\) case:

[1] "true k_on is 50; k_on mean; 72.132 k_on median 79.435 k_on st.deviation 32.035"

[1] "true k_off is 50; k_off mean; 140.46 k_off median 128.02 k_off st.deviation 125.1"

[1] "true k_r is 100; k_r mean; 128.336 k_r median 120.557 k_r st.deviation 62.507"

Large \(k_{on}\) and \(k_{off}\) as said earlier produces a dataset with unidentifiable parameters as we do not approach true values with neigher mean or median with huge standart deviation.

III) Large value of \(k_{off}\) and small \(k_{on}\) case:

[1] "true k_on is 1; k_on mean; 0.954 k_on median 0.951 k_on st.deviation 0.088"

[1] "true k_off is 10; k_off mean; 10.538 k_off median 6.585 k_off st.deviation 15.538"

[1] "true k_r is 100; k_r mean; 106.493 k_r median 74.295 k_r st.deviation 127.793"

Big \(k_{off}\) and small \(k_{on}\) values allow us to identify \(k_{on}\) parameter, but prevent correct estimation of \(k_{off}\) and \(k_{r}\). We see high standart deviation of the parameters as well as some huge outliers.

IV) Small value of \(k_{off}\) and big \(k_{on}\)

[1] "true k_on is 10; k_on mean; 10.326 k_on median 9.308 k_on st.deviation 5.637"

[1] "true k_off is 1; k_off mean; 1.286 k_off median 0.753 k_off st.deviation 3.991"

[1] "true k_r is 100; k_r mean; 100.143 k_r median 99.106 k_r st.deviation 5.758"

We again observe that parameters get estimated quite accurately, even though not as good as in case I. Standart deviation is reasonable, while mean and median approach true value.


sessionInfo()
R version 3.5.0 (2018-04-23)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 17763)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

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

other attached packages:
[1] robustbase_0.93-3 plotrix_3.7-5     reticulate_1.13  

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2      knitr_1.20      magrittr_1.5    workflowr_1.5.0
 [5] lattice_0.20-38 R6_2.3.0        stringr_1.3.1   highr_0.7      
 [9] tools_3.5.0     grid_3.5.0      git2r_0.26.1    htmltools_0.3.6
[13] yaml_2.2.0      rprojroot_1.3-2 digest_0.6.17   Matrix_1.2-14  
[17] later_0.8.0     fs_1.3.1        promises_1.0.1  glue_1.3.0     
[21] evaluate_0.11   rmarkdown_1.10  stringi_1.1.7   DEoptimR_1.0-8 
[25] compiler_3.5.0  backports_1.1.2 jsonlite_1.5    httpuv_1.5.1