Calculate Moments of Poisson Distribution

Computes moments of a Poisson distribution. The calculations are based on values of positive and total partitions or the theoretical lambda value.

moments(input)

Arguments

input

a vector with two elements (the first element is treated as a number of positive partitions and the second as a number of total partitions) or a matrix with two columns (first columns contains numbers of positive partitions and the second total numbers of total partitions) or an object of class dpcr .

Value

A data frame with four columns: name of the experiment, name of the replicate, method of computation (theoretical or empirical), name of the moment and the value of the moment. The theoretical moments are computed using the lambda value and the empirical using the sample values.

Note

Four first moments of a Poisson distribution.

Mean : \(\lambda\).

Variance: \(\lambda\).

Skewness: \(\sqrt{\lambda}\).

Kurtosis: \(\frac{1}{\lambda}\).

Examples

# moments for 100 positive partitions of 765 total partitions
moments(c(100, 765))
#>   experiment replicate      method   moment     value
#> 1    Unknown   Unknown theoretical     mean 0.1400888
#> 2    Unknown   Unknown theoretical variance 0.1400888
#> 3    Unknown   Unknown theoretical skewness 2.6717653
#> 4    Unknown   Unknown theoretical kurtosis 7.1383298
#> 5    Unknown   Unknown   empirical     mean 0.1307190
#> 6    Unknown   Unknown   empirical variance 0.1137802
#> 7    Unknown   Unknown   empirical skewness 2.1909760
#> 8    Unknown   Unknown   empirical kurtosis 2.8003759

# calculate moments for an array digital PCR
moments(six_panels)
#>     experiment replicate      method   moment       value
#> 1  Experiment1         1 theoretical     mean  0.01315808
#> 2  Experiment1         1 theoretical variance  0.01315808
#> 3  Experiment1         1 theoretical skewness  8.71773500
#> 4  Experiment1         1 theoretical kurtosis 75.99890350
#> 5  Experiment1         1   empirical     mean  0.01307190
#> 6  Experiment1         1   empirical variance  0.01291791
#> 7  Experiment1         1   empirical skewness  8.57398653
#> 8  Experiment1         1   empirical kurtosis 71.51324503
#> 9  Experiment1         2 theoretical     mean  0.01183446
#> 10 Experiment1         2 theoretical variance  0.01183446
#> 11 Experiment1         2 theoretical skewness  9.19233451
#> 12 Experiment1         2 theoretical kurtosis 84.49901380
#> 13 Experiment1         2   empirical     mean  0.01176471
#> 14 Experiment1         2   empirical variance  0.01164152
#> 15 Experiment1         2   empirical skewness  9.05604244
#> 16 Experiment1         2   empirical kurtosis 80.01190476
#> 17 Experiment2         1 theoretical     mean  0.05508444
#> 18 Experiment2         1 theoretical variance  0.05508444
#> 19 Experiment2         1 theoretical skewness  4.26074482
#> 20 Experiment2         1 theoretical kurtosis 18.15394645
#> 21 Experiment2         1   empirical     mean  0.05359477
#> 22 Experiment2         1   empirical variance  0.05078876
#> 23 Experiment2         1   empirical skewness  3.96423592
#> 24 Experiment2         1   empirical kurtosis 13.71516642
#> 25 Experiment2         2 theoretical     mean  0.05094936
#> 26 Experiment2         2 theoretical variance  0.05094936
#> 27 Experiment2         2 theoretical skewness  4.43027464
#> 28 Experiment2         2 theoretical kurtosis 19.62733335
#> 29 Experiment2         2   empirical     mean  0.04967320
#> 30 Experiment2         2   empirical variance  0.04726756
#> 31 Experiment2         2   empirical skewness  4.14534058
#> 32 Experiment2         2   empirical kurtosis 15.18384855
#> 33 Experiment3         1 theoretical     mean  0.14612199
#> 34 Experiment3         1 theoretical variance  0.14612199
#> 35 Experiment3         1 theoretical skewness  2.61602690
#> 36 Experiment3         1 theoretical kurtosis  6.84359673
#> 37 Experiment3         1   empirical     mean  0.13594771
#> 38 Experiment3         1   empirical variance  0.11761968
#> 39 Experiment3         1   empirical skewness  2.12440735
#> 40 Experiment3         1   empirical kurtosis  2.51310660
#> 41 Experiment3         2 theoretical     mean  0.13409177
#> 42 Experiment3         2 theoretical variance  0.13409177
#> 43 Experiment3         2 theoretical skewness  2.73085683
#> 44 Experiment3         2 theoretical kurtosis  7.45757903
#> 45 Experiment3         2   empirical     mean  0.12549020
#> 46 Experiment3         2   empirical variance  0.10988605
#> 47 Experiment3         2   empirical skewness  2.26102803
#> 48 Experiment3         2   empirical kurtosis  3.11224776