stochprofML (version 2.0.3)

d.sum.of.mixtures.EXPLN: Sums of mixtures of zero, one or more lognormal random variables and one exponential random variable

Description

Density and random generation of a sum of i.i.d. random variables, where each random variable is from the following mixture distribution: With probability p_i, it is of type i. For all but the largest i, it is lognormally distributed with log-mean mu_i and log-standard deviation sigma_i. Otherwise it is exponentially distributed with rate lambda.

Usage

d.sum.of.mixtures.EXPLN(y, n, p.vector, mu.vector, sigma.vector, lambda,
   logdens = T)
r.sum.of.mixtures.EXPLN(k, n, p.vector, mu.vector, sigma.vector, lambda, N.matrix)

Arguments

y

the argument at which the density is evaluated

k

number of i.i.d. random variables returned by this function (in the considered application: number of tissue samples)

n

the number of random variables entering each sum (in the considered application: number of cells per tissue sample). This can also be a vector stating how many cells are in each sample separatly.

p.vector

mu.vector

vector (mu1,mu2,...,mu(T-1)) containing the log-means for each lognormal type (types 1 to T-1)

sigma.vector

vector (sigma1,...,sigma(T-1)) containing the log-standard deviations sigma for each lognormal type (types 1 to T-1)

lambda

the rate for the exponential type (type T)

logdens

if TRUE, the log of the density is returned

N.matrix

optional. Matrix, that shows the decomposition of samples to be generated. Each row stands for a future sample and the columns show the decomposition of types for the specific sample such that the row sum should be n (either a value, i.e. all the same or a vector).

Value

'd.sum.of.mixtures.EXPLN' gives the density, and 'r.sum.of.mixtures.EXPLN' generates random variables.

Details

The lengths of mu.vector and sigma.vector have to be identical. p.vector has to have one component more. Its length automatically determines the number of different types. lambda has to be a scalar.

References

"Parameterizing cell-to-cell regulatory heterogeneities via stochastic transcriptional profiles" by Sameer S Bajikar*, Christiane Fuchs*, Andreas Roller, Fabian J Theis^ and Kevin A Janes^: PNAS 2014, 111(5), E626-635 (* joint first authors, ^ joint last authors) <doi:10.1073/pnas.1311647111>

"Pheno-seq - linking visual features and gene expression in 3D cell culture systems" by Stephan M. Tirier, Jeongbin Park, Friedrich Preusser, Lisa Amrhein, Zuguang Gu, Simon Steiger, Jan-Philipp Mallm, Teresa Krieger, Marcel Waschow, Bjoern Eismann, Marta Gut, Ivo G. Gut, Karsten Rippe, Matthias Schlesner, Fabian Theis, Christiane Fuchs, Claudia R. Ball, Hanno Glimm, Roland Eils & Christian Conrad: Sci Rep 9, 12367 (2019) <doi:10.1038/s41598-019-48771-4>

Examples

# NOT RUN {
# generate random variables
p <- c(0.25,0.75)
mu <- 2
sigma <- 0.3
lambda <- 5
set.model.functions("EXP-LN")

r <- r.sum.of.mixtures.EXPLN(10^4,10,p,mu,sigma,lambda)
hist(r,xlab="Sum of mixtures of lognormals",freq=FALSE,breaks=100,ylim=c(0,0.2))

# plot according theoretical density function
x <- seq(round(min(r)),round(max(r)),(round(max(r))-round(min(r)))/500)
y <- d.sum.of.mixtures.EXPLN(x,10,p,mu,sigma,lambda,logdens=FALSE)
lines(x,y,col="blue",lwd=3)
# }