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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; ) = $

which is also written as:

$ Pr(X = k) = e^{-}   k = 0, 1, 2, $

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) = $

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))

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