Simulate UMI counts

PROSSTT simulates UMI counts using a negative binomial distribution where the variance of the expression of each gene \(g\) depends on its average expression \(\mu_g\):

\(\sigma_g^2 = \alpha_g \mu^2 + \beta_g \mu_g\)

This relationship is preserved through pseudotime; as average expression changes with time, so does the variance, always obeying the same relationship.

\(\sigma_g^2(t) = \alpha_g \mu^2(t) + \beta_g \mu_g(t)\)

A negative binomial is the distribution of the number of successes in a sequence of i.i.d. Bernoulli trials before a specified number of failures occurs. The negative binomial can be parametrized by its mean and variance or by a pair \(p \in (0, 1), r > 0\), where \(p\) is the success probability in each Bernoulli trial and \(r\) the number of failures. While the negative binomial is originally a discrete probability distribution, it can easily be extended into a continuous one, preserving most of its attributes.

Here we use the implementation of the negative binomial distribution by the scipy package, after translating the mean and variance of each distribution to the \(p, r\) equivalents.

The gene-specific parameters \(\alpha_g, \beta_g\) are sampled from ranges found in real data. Users can set \(\alpha_g\) to 0 and \(\beta_g\) to 1 to have genes with Poisson distributions, or only set \(\alpha_g\) to 0 to have genes with scaled Poissonian noise.