Open Access Research article

Multi-way metamodelling facilitates insight into the complex input-output maps of nonlinear dynamic models

Kristin Tøndel1*, Ulf G Indahl1, Arne B Gjuvsland1, Stig W Omholt2 and Harald Martens13

Author Affiliations

1 Centre for Integrative Genetics (CIGENE), Dept. of Mathematical Sciences and Technology, Norwegian University of Life Sciences, P. O. Box 5003, N-1432, Ås, Norway

2 CIGENE, Dept. of Animal and Aquacultural Sciences, Norwegian University of Life Sciences, P. O. Box 5003, N-1432, Ås, Norway

3 Nofima, P. O. Box 210, N-1431, Ås, Norway

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BMC Systems Biology 2012, 6:88  doi:10.1186/1752-0509-6-88

Published: 20 July 2012



Statistical approaches to describing the behaviour, including the complex relationships between input parameters and model outputs, of nonlinear dynamic models (referred to as metamodelling) are gaining more and more acceptance as a means for sensitivity analysis and to reduce computational demand. Understanding such input-output maps is necessary for efficient model construction and validation. Multi-way metamodelling provides the opportunity to retain the block-wise structure of the temporal data typically generated by dynamic models throughout the analysis. Furthermore, a cluster-based approach to regional metamodelling allows description of highly nonlinear input-output relationships, revealing additional patterns of covariation.


By presenting the N-way Hierarchical Cluster-based Partial Least Squares Regression (N-way HC-PLSR) method, we here combine multi-way analysis with regional cluster-based metamodelling, together making a powerful methodology for extensive exploration of the input-output maps of complex dynamic models. We illustrate the potential of the N-way HC-PLSR by applying it both to predict model outputs as functions of the input parameters, and in the inverse direction (predicting input parameters from the model outputs), to analyse the behaviour of a dynamic model of the mammalian circadian clock. Our results display a more complete cartography of how variation in input parameters is reflected in the temporal behaviour of multiple model outputs than has been previously reported.


Our results indicated that the N-way HC-PLSR metamodelling provides a gain in insight into which parameters that are related to a specific model output behaviour, as well as variations in the model sensitivity to certain input parameters across the model output space. Moreover, the N-way approach allows a more transparent and detailed exploration of the temporal dimension of complex dynamic models, compared to alternative 2-way methods.

Parameter-phenotype map; Dynamic models; Metamodelling; N-way Partial Least Squares Regression; Hierarchical analysis; HC-PLSR; Cluster-analysis; Input-output relationships; Circadian clock