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ProbOnto is a
knowledge base A knowledge base (KB) is a technology used to store complex structured and unstructured information used by a computer system. The initial use of the term was in connection with expert systems, which were the first knowledge-based systems. ...
and ontology of
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s.Main project website, URL: http://probonto.org ProbOnto 2.5 (released on January 16, 2017) contains over 150 uni- and
multivariate distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
s and alternative parameterizations, more than 220 relationships and re-parameterization formulas, supporting also the encoding of empirical and univariate
mixture distribution In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection ...
s.


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 The Innovative Medicines Initiative (IMI) is a European initiative to improve the competitive situation of the European Union in the field of pharmaceutical research. The IMI is a joint initiative ( public-private partnership) of the DG Resear ...
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 Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New Yor ...
, categorical and time-to-event) and continuous data.


Knowledge base

The knowledge base stores for each distribution: * Probability density or
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
functions and where available
cumulative distribution In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form. Types The cumu ...
,
hazard A hazard is a potential source of harm. Substances, events, or circumstances can constitute hazards when their nature would allow them, even just theoretically, to cause damage to health, life, property, or any other interest of value. The probab ...
and
survival Survival, or the act of surviving, is the propensity of something to continue existing, particularly when this is done despite conditions that might kill or destroy it. The concept can be applied to humans and other living things (or, hypotheti ...
functions. * Related quantities such as mean, median, mode and variance. *
Parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
and
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
/range definitions and distribution type. *
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosper ...
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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
can be defined in at least three ways * Normal1(μ,σ) with
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...
, μ, and standard deviation, σ Forbes et al. Probability Distributions (2011), John Wiley & Sons, Inc.
P(x;\boldsymbol\mu,\boldsymbol\sigma)= \frac\exp\Big \frac\Big/math>
* Normal2(μ,υ) with mean, μ, and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
, υ = σ^2 or
P(x;\boldsymbol\mu,\boldsymbol v)= \frac\exp\Big \frac\Big/math>
* Normal3(μ,τ) with mean, μ, and precision, τ = 1/υ = 1/σ^2.
P(x;\boldsymbol\mu,\boldsymbol\tau)= \sqrt \Big \frac(x-\mu)^2\Big/math>


= 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. N1 instead of Normal1 etc.) * N1(\mu,\sigma) \rightarrow N2(\mu,v): v=\sigma^2 \mbox N2(\mu,v) \rightarrow N1(\mu,\sigma): \sigma=\sqrt; * N1(\mu,\sigma) \rightarrow N3(\mu,\tau): \tau=1/\sigma^2 \mbox N3(\mu,\tau) \rightarrow N1(\mu,\sigma): \sigma=1/\sqrt; * N2(\mu,v) \rightarrow N3(\mu,\tau): \tau=1/v \mbox N3(\mu,\tau) \rightarrow N2(\mu,v): v=1/\tau.


Log-normal distribution

In the case of the
log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
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
P(x;\boldsymbol\mu,\boldsymbol\sigma)= \frac \exp\Big \frac\Big/math>
* LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale
P(x;\boldsymbol\mu,\boldsymbol )=\frac \exp\Big \frac\Big/math>
* LogNormal3(m,σ) with median, m, on the natural scale and standard deviation, σ, on the log-scale
P(x;\boldsymbol m,\boldsymbol \sigma) =\frac \exp\Big \frac\Big/math>
* LogNormal4(m,cv) with median, m, and
coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed ...
, cv, both on the natural scale
P(x;\boldsymbol m,\boldsymbol )= \frac \exp\Big \frac\Big/math>
* LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale
P(x;\boldsymbol\mu,\boldsymbol \tau)=\sqrt \frac \exp\Big \Big
* LogNormal6(m,σg) with median, m, and
geometric standard deviation In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. N ...
, σg, both on the natural scale
P(x;\boldsymbol m,\boldsymbol )=\frac \exp\Big \frac\Big/math>
* LogNormal7(μNN) with mean, μN, and standard deviation, σN, both on the natural scale
P(x;\boldsymbol ,\boldsymbol )= \frac \exp\Bigg( \frac\Bigg)
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 LN2(\mu, v) \rightarrow LN7(\mu_N, \sigma_N) following formulas hold \mu_N = \exp(\mu+v/2) \text \sigma_N = \exp(\mu+v/2)\sqrt. For the transition LN7(\mu_N, \sigma_N) \rightarrow LN2(\mu, v) following formulas hold \mu = \log\Big( \mu_N/\sqrt \Big) \text v = \log(1+\sigma_N^2/\mu_N^2). 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 (OLS) 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.ProbOnto on BioSharing, the database of biological databases, URL: https://biosharing.org/biodbcore-000772


ProbOnto in PharmML

A PharmML interface is provided in form of a generic XML schema for the definition of the distributions and their parameters. Defining functions, such as probability density function (PDF), probability mass function (PMF), hazard function (HF) and survival function (SF), can be accessed via methods provided in the PharmML schema.


Use example

This example shows how the zero-inflated Poisson distribution is encoded by using its ''codename'' and declaring that of its parameters (‘rate’ and ‘probabilityOfZero’). 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.


See also

*
List of probability distributions Many probability distributions that are important in theory or applications have been given specific names. Discrete distributions With finite support * The Bernoulli distribution, which takes value 1 with probability ''p'' and value 0 with pr ...
*
Ontology (computer science) In computer science and information science, an ontology encompasses a representation, formal naming, and definition of the categories, properties, and relations between the concepts, data, and entities that substantiate one, many, or all domains ...
*
Relationships among probability distributions In probability theory and statistics, there are several relationships among probability distributions. These relations can be categorized in the following groups: *One distribution is a special case of another with a broader parameter space * ...
*
Web Ontology Language The Web Ontology Language (OWL) is a family of knowledge representation languages for authoring ontologies. Ontologies are a formal way to describe taxonomies and classification networks, essentially defining the structure of knowledge for vario ...


References

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External links


ProbOnto website



Ultimate Univariate Probability Distribution Explorer
– most likely the largest, free collection of univariate distributions and their features.
UncertML
Probability distributions