ProbOnto is a knowledge base and ontology of probability distributions. ProbOnto 2.5 (released on January 16, 2017) contains over 150 uni- and multivariate distributions and alternative parameterizations, more than 220 relationships and re-parameterization formulas, supporting also the encoding of empirical and univariate mixture distributions.
Introduction ProbOnto was initially designed to facilitate the encoding of nonlinear-mixed effect models and their annotation in Pharmacometrics Markup Language (PharmML) developed by DDMoRe, an Innovative Medicines Initiative project. However, ProbOnto, due to its generic structure can be applied in other platforms and modeling tools for encoding and annotation of diverse models applicable to discrete (e.g. count, categorical and time-to-event) and continuous data.
Knowledge base
The knowledge base stores for each distribution:
Probability density or mass functions and where available cumulative distribution, hazard and survival functions. Related quantities such as mean, median, mode and variance. Parameter and support/range definitions and distribution type. LaTeX and R code for mathematical functions. Model definition and references.
Relationships ProbOnto stores in Version 2.5 over 220 relationships between univariate distributions with re-parameterizations as a special case, see figure. While this form of relationships is often neglected in literature, and the authors concentrate one a particular form for each distribution, they are crucial from the interoperability point of view. ProbOnto focuses on this aspect and features more than 15 distributions with alternative parameterizations.
Alternative parameterizations Many distributions are defined with mathematically equivalent but algebraically different formulas. This leads to issues when exchanging models between software tools. The following examples illustrate that.
Normal distribution Normal distribution can be defined in at least three ways
Normal1(μ,σ) with mean, μ, and standard deviation, σ
Normal2(μ,υ) with mean, μ, and variance, υ = σ^2 or
Normal3(μ,τ) with mean, μ, and precision, τ = 1/υ = 1/σ^2.
Re-parameterization formulas The following formulas can be used to re-calculate the three different forms of the normal distribution (we use abbreviations i.e. N 1 {\displaystyle N1} instead of N o r m a l 1 {\displaystyle Normal1} etc.)
N 1 ( μ , σ ) → N 2 ( μ , v ) : v = σ 2 and N 2 ( μ , v ) → N 1 ( μ , σ ) : σ = v ; {\displaystyle N1(\mu ,\sigma )\rightarrow N2(\mu ,v):v=\sigma ^{2}{\mbox{ and }}N2(\mu ,v)\rightarrow N1(\mu ,\sigma ):\sigma ={\sqrt {v}};}
N 1 ( μ , σ ) → N 3 ( μ , τ ) : τ = 1 / σ 2 and N 3 ( μ , τ ) → N 1 ( μ , σ ) : σ = 1 / τ ; {\displaystyle N1(\mu ,\sigma )\rightarrow N3(\mu ,\tau ):\tau =1/\sigma ^{2}{\mbox{ and }}N3(\mu ,\tau )\rightarrow N1(\mu ,\sigma ):\sigma =1/{\sqrt {\tau }};}
N 2 ( μ , v ) → N 3 ( μ , τ ) : τ = 1 / v and N 3 ( μ , τ ) → N 2 ( μ , v ) : v = 1 / τ . {\displaystyle N2(\mu ,v)\rightarrow N3(\mu ,\tau ):\tau =1/v{\mbox{ and }}N3(\mu ,\tau )\rightarrow N2(\mu ,v):v=1/\tau .}
Log-normal distribution In the case of the log-normal distribution there are more options. This is due to the fact that it can be parameterized in terms of parameters on the natural and log scale, see figure. The available forms in ProbOnto 2.0 are
LogNormal1(μ,σ) with mean, μ, and standard deviation, σ, both on the log-scale
LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale
LogNormal3(m,σ) with median, m, on the natural scale and standard deviation, σ, on the log-scale
LogNormal4(m,cv) with median, m, and coefficient of variation, cv, both on the natural scale
LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale
LogNormal6(m,σg) with median, m, and geometric standard deviation, σg, both on the natural scale
LogNormal7(μN,σN) with mean, μN, and standard deviation, σN, both on the natural scale
ProbOnto knowledge base stores such re-parameterization formulas to allow for a correct translation of models between tools.
Examples for re-parameterization Consider the situation when one would like to run a model using two different optimal design tools, e.g. PFIM and PopED. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results. For the transition L N 2 ( μ , v ) → L N 7 ( μ N , σ N ) {\displaystyle LN2(\mu ,v)\rightarrow LN7(\mu _{N},\sigma _{N})} following formulas hold
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