Definition of probability distributions

Introduction

As defined previously, a large part of the modelling is defined using probability distributions (observation model, inter-individual variability, …). An explicit definition of a random variable can be provided in the block DEFINITION:.

Distribution definition

A random variable $X$ can be defined through its probability distribution. It is a list including a distribution name with specific parameters. Obviously, distribution is a keyword. The elements of this list are identified according to their names (the order does not matter).

Mlxtran only supports distributions based on the normal distribution. More precisely, we consider that there exists a monotonic transformation $h$ such that $h(X)$ is normally distributed:

normal distribution, used with keyword normal, and $h(X)=X$
log-normal distribution, used with keyword lognormal, and $h(X)=\log(X)$
logit-normal distribution, used with keyword logitnormal, and $h(X)=\frac{1}{1+e^{-X}}$
probit-normal distribution, used with keyword probitnormal, and $h(X)=\Psi^{-1}(X)$ , where $\Psi$ is the cumulative distribution function of the ${\cal N}(0, 1)$ distribution.

These distributions are defined by the mean and standard deviation of the normal distribution, i.e of the transformed variable $h(X)$ . Therefore, when we say that $X$ is log-normally distributed with a mean $\mu$ and a standard variation $\omega$ , it means that

$X \sim \log{\cal N}(\mu,\omega) \quad \Leftrightarrow \quad \log(X) \sim {\cal N}(\mu,\omega)$

Here, $\mu$ and $\omega$ are the mean and standard deviation of $\log(X)$ . Notice that mean and sd are keywords defining the the mean and standard deviation of the normal distribution.

Example

To define the previous example, X log-normally distributed with a mean $\mu$ and a standard variation $\omega$ , in Mlxtran as a individual parameter, one just has to write

input= {mu, sigma}
DEFINITION:
X = {distribution=lognormal, mean=mu, sd=sigma}

Then, from a syntax point of view :

$X = \{ \textrm{distribution}=\textrm{lognormal}, \textrm{mean}=\mu, \textrm{sd}=\sigma\} \Leftrightarrow X \sim \log{\cal N}(\mu,\omega) \quad \Leftrightarrow \quad \log(X) \sim {\cal N}(\mu,\omega)$

In that case, the mean definition in the distribution is the mean of the transformed value, i.e. $\mu=\textrm{mean}(\log(X))$ in the previous example. Therefore, we introduce a keyword typical which corresponds to the “typical” value of X around which you want to make the transformation. From a syntax point of view, we have

$X = \{ \textrm{distribution}=\textrm{lognormal}, \textrm{typical}=X_{pop}, \textrm{sd}=\sigma\} \Leftrightarrow X = \{ \textrm{distribution}=\textrm{lognormal}, \textrm{mean}=\log(X_{pop}), \textrm{sd}=\sigma\} \Leftrightarrow \quad \log(X) \sim {\cal N}(\mu,\omega)$

Therefore, the use of the keywords typical and mean depends if you are in the transformed referential or not.