ProbOnto is a knowledge base and ontology of probability distributions. ProbOnto 2.5 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 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 and continuous data.
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.
Normal3 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
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. , MCSim, Monolix, PFIM, Phoenix NLME, PopED, R , Simcyp Simulator, Simulx and winBUGS The available forms in ProbOnto 2.0 are
LogNormal1 with mean, μ, and standard deviation, σ, both on the log-scale
LogNormal2 with mean, μ, and variance, υ, both on the log-scale
LogNormal3 with median, m, on the natural scale and standard deviation, σ, on the log-scale
LogNormal7 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 following formulas hold For the transition following formulas hold All remaining re-parameterisation formulas can be found in the specification document on the project website.
Ontology
The knowledge base is built from a simple ontological model. At its core, a probability distribution is an instance of the class thereof, a specialization of the class of mathematical objects. A distribution relates to a number of other individuals, which are instances of various categories in the ontology. For example, these are parameters and related functions associated with a given probability distribution. This strategy allows for the rich representation of attributes and relationships between domain objects. The ontology can be seen as a conceptual schema in the domain of mathematics and has been implemented as a PowerLoom knowledge base. An OWL version is generated programmatically using the Jena API. Output for ProbOnto are provided as supplementary materials and published on or linked from the probonto.org website. The OWL version of ProbOnto is available via Ontology Lookup Service to facilitate simple searching and visualization of the content. In addition the OLS API provides methods to programmatically access ProbOnto and to integrate it into applications. ProbOnto is also registered on the BioSharing portal.
This example shows how the zero-inflated Poisson distribution is encoded by using its codename and declaring that of its parameters. Model parameters Lambda and P0 are assigned to the parameter code names.
To specify any given distribution unambiguously using ProbOnto, it is sufficient to declare its code name and the code names of its parameters. More examples and a detailed specification can be found on the project website.
