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. 2012;7(7):e39721.
doi: 10.1371/journal.pone.0039721. Epub 2012 Jul 3.

Declarative Representation of Uncertainty in Mathematical Models

Free PMC article

Declarative Representation of Uncertainty in Mathematical Models

Andrew K Miller et al. PLoS One. .
Free PMC article


An important aspect of multi-scale modelling is the ability to represent mathematical models in forms that can be exchanged between modellers and tools. While the development of languages like CellML and SBML have provided standardised declarative exchange formats for mathematical models, independent of the algorithm to be applied to the model, to date these standards have not provided a clear mechanism for describing parameter uncertainty. Parameter uncertainty is an inherent feature of many real systems. This uncertainty can result from a number of situations, such as: when measurements include inherent error; when parameters have unknown values and so are replaced by a probability distribution by the modeller; when a model is of an individual from a population, and parameters have unknown values for the individual, but the distribution for the population is known. We present and demonstrate an approach by which uncertainty can be described declaratively in CellML models, by utilising the extension mechanisms provided in CellML. Parameter uncertainty can be described declaratively in terms of either a univariate continuous probability density function or multiple realisations of one variable or several (typically non-independent) variables. We additionally present an extension to SED-ML (the Simulation Experiment Description Markup Language) to describe sampling sensitivity analysis simulation experiments. We demonstrate the usability of the approach by encoding a sample model in the uncertainty markup language, and by developing a software implementation of the uncertainty specification (including the SED-ML extension for sampling sensitivty analyses) in an existing CellML software library, the CellML API implementation. We used the software implementation to run sampling sensitivity analyses over the model to demonstrate that it is possible to run useful simulations on models with uncertainty encoded in this form.

Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.


Figure 1
Figure 1. The distribution of the initial position in x and y, and the initial x and y velocity components of the object, shown using both a density histogram and a kernel density plot.
Generated using Model S1.
Figure 2
Figure 2. The path of the object in the example model is plotted for ten runs of the model.
The path depends on the uncertain parameters. Generated using Model S1.
Figure 3
Figure 3. A sensitivity analysis of the example model, showing the position of the object at time 10 s.
Generated using Model S1.

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