Last updated: 2024-07-30

Checks: 7 0

Knit directory: muse/

This reproducible R Markdown analysis was created with workflowr (version 1.7.1). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.

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(20200712)

The command set.seed(20200712) 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.

Cache: none

Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.

File paths: relative

Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.

Repository version: 67813b6

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.

The results in this page were generated with repository version 67813b6. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

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:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    r_packages_4.3.3/
    Ignored:    r_packages_4.4.0/

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.

These are the previous versions of the repository in which changes were made to the R Markdown (analysis/poisson.Rmd) and HTML (docs/poisson.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

File Version Author Date Message
Rmd 67813b6 Dave Tang 2024-07-30 Fix LaTeX
html e49a794 Dave Tang 2024-07-30 Build site.
Rmd 2dceb55 Dave Tang 2024-07-30 The Poisson distribution

Introduction

A Poisson distribution is the probability distribution that results from a Poisson experiment. A probability distribution assigns a probability to possible outcomes of a random experiment. A Poisson experiment has the following properties:

  1. The outcomes of the experiment can be classified as either successes or failures.
  2. The average number of successes that occurs in a specified region is known.
  3. The probability that a success will occur is proportional to the size of the region.
  4. The probability that a success will occur in an extremely small region is virtually zero.

A Poisson random variable is the number of successes that result from a Poisson experiment. Given the mean number of successes that occur in a specified region, we can compute the Poisson probability based on the following formula:

\[ P(x; \mu) = \frac{(e^{-\mu})(\mu^x)}{x!} \]

which is also written as:

\[ Pr(X = k) = e^{-\lambda} \frac{\lambda^k}{k!} \ \ k = 0, 1, 2, \dotsc \]

Examples

The average number of homes sold is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

\[ P(3; 2) = \frac{(e^{-2}) (2^3)}{3!} \]

Calculating this manually in R:

e <- exp(1)
((e^-2)*(2^3))/factorial(3)
[1] 0.180447

Using dpois():

dpois(x = 3, lambda = 2)
[1] 0.180447

RNA-seq

The Poisson distribution can be used to estimate the technical variance in high-throughput sequencing experiments. My basic understanding is that the variance between technical replicates can be modelled using the Poisson distribution. Check out Why Does Rna-Seq Read Count Fit Poisson Distribution? on Biostars.

From Chris Miller:

Picture a process whereby you take the genome and choose a location at random to produce a read. This is a Poisson process. If you plot the depth of sequence along this theoretical genome, it will be a poisson distribution.

Calculating confidence intervals

Calculate the confidence intervals using R. Create data with 1,000,000 values that follow a Poisson distribution with lambda = 20.

set.seed(1984)
n <- 1000000
data <- rpois(n, 20)

Functions for calculating the lower and upper tails.

poisson_lower_tail <- function(n) {
   qchisq(0.025, 2*n)/2
}
poisson_upper_tail <- function(n) {
   qchisq(0.975, 2*(n+1))/2
}

Lower limit for lambda = 20.

poisson_lower_tail(20)
[1] 12.21652

Upper limit for lambda = 20.

poisson_upper_tail(20)
[1] 30.88838

How many values in data are lower than the lower limit?

table(data<poisson_lower_tail(20))

 FALSE   TRUE 
961213  38787

How many values in data are higher than the upper limit?

table(data>poisson_upper_tail(20))

 FALSE   TRUE 
986239  13761

What percentage of values were outside of the 95% CI?

(sum(data<poisson_lower_tail(20)) + sum(data>poisson_upper_tail(20))) * 100 / n
[1] 5.2548

Plot.

hist(data)
abline(v=poisson_lower_tail(20))
abline(v=poisson_upper_tail(20))

Version Author Date
e49a794 Dave Tang 2024-07-30

Webtool

Using the Poisson Confidence Interval Calculator and lambda = 20 returns:

  • 99% confidence interval: 10.35327 - 34.66800
  • 95% confidence interval: 12.21652 - 30.88838
  • 90% confidence interval: 13.25465 - 29.06202

which matches our 95% CI values.


sessionInfo()
R version 4.4.0 (2024-04-24)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 22.04.4 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so;  LAPACK version 3.10.0

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

time zone: Etc/UTC
tzcode source: system (glibc)

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

other attached packages:
 [1] lubridate_1.9.3 forcats_1.0.0   stringr_1.5.1   dplyr_1.1.4    
 [5] purrr_1.0.2     readr_2.1.5     tidyr_1.3.1     tibble_3.2.1   
 [9] ggplot2_3.5.1   tidyverse_2.0.0 workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] sass_0.4.9        utf8_1.2.4        generics_0.1.3    stringi_1.8.4    
 [5] hms_1.1.3         digest_0.6.35     magrittr_2.0.3    timechange_0.3.0 
 [9] evaluate_0.24.0   grid_4.4.0        fastmap_1.2.0     rprojroot_2.0.4  
[13] jsonlite_1.8.8    processx_3.8.4    whisker_0.4.1     ps_1.7.6         
[17] promises_1.3.0    httr_1.4.7        fansi_1.0.6       scales_1.3.0     
[21] jquerylib_0.1.4   cli_3.6.2         rlang_1.1.4       munsell_0.5.1    
[25] withr_3.0.0       cachem_1.1.0      yaml_2.3.8        tools_4.4.0      
[29] tzdb_0.4.0        colorspace_2.1-0  httpuv_1.6.15     vctrs_0.6.5      
[33] R6_2.5.1          lifecycle_1.0.4   git2r_0.33.0      fs_1.6.4         
[37] pkgconfig_2.0.3   callr_3.7.6       pillar_1.9.0      bslib_0.7.0      
[41] later_1.3.2       gtable_0.3.5      glue_1.7.0        Rcpp_1.0.12      
[45] highr_0.11        xfun_0.44         tidyselect_1.2.1  rstudioapi_0.16.0
[49] knitr_1.47        htmltools_0.5.8.1 rmarkdown_2.27    compiler_4.4.0   
[53] getPass_0.2-4