An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Gröbner bases. (English) Zbl 1179.93066
Summary: The parameter identifiability problem for dynamic system ODE models has been extensively studied. Nevertheless, except for linear ODE models, the question of establishing identifiable combinations of parameters when the model is unidentifiable has not received as much attention and the problem is not fully resolved for nonlinear ODEs. Identifiable combinations are useful, for example, for the reparameterization of an unidentifiable ODE model into an identifiable one. We extend an existing algorithm for finding globally identifiable parameters of nonlinear ODE models to generate the ‘simplest’ globally identifiable parameter combinations using Gröbner bases. We also provide sufficient conditions for the method to work, demonstrate our algorithm and find associated identifiable reparameterizations for several linear and nonlinear unidentifiable biomodels.
MSC:
| 93B30 | System identification |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93B27 | Geometric methods |
Keywords:
identifiability; differential algebra; reparameterization; dynamic systems; systems biologyReferences:
| [1] | Bellman, R.; Astrom, K. J., On structural identifiability, Math. Biosci., 7, 329-339 (1970) |
| [2] | Cobelli, C.; DiStefano, J. J., Parameter and structural identifiability concepts and ambiguities: a critical review and analysis, Am. J. Physiol.: Regul., Integr. Compar. Physiol., 239, R7-R24 (1980) |
| [3] | Godfrey, K. R.; DiStefano, J. J., Identifiability of model parameters, (Walter, E., Identifiability of Parametric Models (1987), Pergamon: Pergamon Oxford), 1-20 · Zbl 0648.93009 |
| [4] | Chappell, M. J.; Godfrey, K. R.; Vajda, S., Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods, Math Biosci., 102, 41-73 (1990) · Zbl 0789.93039 |
| [5] | Evans, N. D.; Chappell, M. J., Extensions to a procedure for generating locally identifiable reparameterisations of unidentifiable systems, Math. Biosci., 168, 137-159 (2000) · Zbl 0989.93021 |
| [6] | Gunn, R. N.; Chappell, M. J.; Cunningham, V. J., Reparameterisation of unidentifiable systems using the Taylor series approach, (Linkens, D. A.; Carson, E., Proceedings of the Third IFAC Symposium on Modelling and Control in Biomedical Systems (1997), Pergamon: Pergamon Oxford), 247 · Zbl 0958.93509 |
| [7] | Pohjanpalo, H., System identifiability based on the power series expansion of the solution, Math. Biosci., 41, 21-33 (1978) · Zbl 0393.92008 |
| [8] | Chappell, M. J.; Gunn, R. N., A procedure for generating locally identifiable reparameterisations of unidentifiable non-linear systems by the similarity transformation approach, Math. Biosci., 148, 21-41 (1998) · Zbl 0910.93027 |
| [9] | Denis-Vidal, L.; Joly-Blanchard, G., Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems, Automatica, 40, 287-292 (2004) · Zbl 1034.93008 |
| [10] | L. Denis-Vidal, G. Joly-Blanchard, C. Noiret, M. Petitot, An algorithm to test identifiability of non-linear systems, in: Proceedings of the Fifth IFAC Conference Nonlinear Control Systems, NOLCOS, St. Petersburg, Russia, 2001, pp. 174-178.; L. Denis-Vidal, G. Joly-Blanchard, C. Noiret, M. Petitot, An algorithm to test identifiability of non-linear systems, in: Proceedings of the Fifth IFAC Conference Nonlinear Control Systems, NOLCOS, St. Petersburg, Russia, 2001, pp. 174-178. |
| [11] | Denis-Vidal, L.; Joly-Blanchard, G.; Noiret, C., Some effective approaches to check the identifiability of uncontrolled nonlinear systems, Math. Comput. Simul., 57, 35-44 (2001) · Zbl 0984.93011 |
| [12] | Verdière, N.; Denis-Vidal, L.; Joly-Blanchard, G.; Domurado, D., Identifiability and estimation of pharmacokinetic parameters for the ligands of the macrophage mannose receptor, Int. J. Appl. Math Comput. Sci., 15-4, 517-526 (2005) · Zbl 1104.92027 |
| [13] | Boulier, F., Differential elimination and biological modelling, Radon Series Comp. Appl. Math., 2, 111-139 (2007) |
| [14] | M.P. Saccomani, S. Audoly, G. Bellu, L. D’Angiò, A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions, in: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2001, pp. 3108-3113.; M.P. Saccomani, S. Audoly, G. Bellu, L. D’Angiò, A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions, in: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2001, pp. 3108-3113. |
| [15] | Bellu, G.; Saccomani, M. P.; Audoly, S.; D’Angiò, L., DAISY: a new software tool to test global identifiability of biological and physiological systems, Comput. Meth. Programs Biomed., 88, 52-61 (2007) |
| [16] | DiStefano, J. J.; Cobelli, C., On parameter and structural identifiability: nonunique observability/reconstructibility for identifiable systems, other ambiguities, and new definitions, IEEE Trans. Autom. Control, 25-4, 830-833 (1980) · Zbl 0454.93004 |
| [17] | Ljung, L.; Glad, T., On global identifiability for arbitrary model parameterization, Automatica, 30-2, 265-276 (1994) · Zbl 0795.93026 |
| [18] | F. Ollivier, Le problème de l’identifiabilité structurelle globale: étude théoritique, méthodes effectives et bornes de complexité, PhD Thesis, Ecole Polytechnique, 1990.; F. Ollivier, Le problème de l’identifiabilité structurelle globale: étude théoritique, méthodes effectives et bornes de complexité, PhD Thesis, Ecole Polytechnique, 1990. |
| [19] | F. Ollivier, Identifiabilité des systèmes, Technical Report 97-04, GAGE, Ecole Polytechnique, 1997, pp. 43-54.; F. Ollivier, Identifiabilité des systèmes, Technical Report 97-04, GAGE, Ecole Polytechnique, 1997, pp. 43-54. |
| [20] | Boulier, F.; Lazard, D.; Ollivier, F.; Petitot, M., Computing representations for radicals of finitely generated differential ideals, Appl. Algebra Eng., Commun. Comput., 20-1, 73-121 (2009) · Zbl 1185.12003 |
| [21] | M.P. Saccomani, G. Bellu, DAISY: an efficient tool to test global identifiability. Some case studies, in :Proceedings of the 16th Mediterranean Conference on Control and Automation, Congress Centre, Ajaccio, France, 2008, pp. 1723-1728.; M.P. Saccomani, G. Bellu, DAISY: an efficient tool to test global identifiability. Some case studies, in :Proceedings of the 16th Mediterranean Conference on Control and Automation, Congress Centre, Ajaccio, France, 2008, pp. 1723-1728. |
| [22] | Shannon, D.; Sweedler, M., Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms and determine birational equivalence, J. Symb. Comput., 6, 2-3, 267-273 (1988) · Zbl 0681.68052 |
| [23] | Capasso, V., Mathematical Structures of Epidemic Systems (1993), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0798.92024 |
| [24] | Margaria, G.; Riccomagno, E.; Chappell, M. J.; Wynn, H. P., Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences, Math. Biosci., 174, 1-26 (2001) · Zbl 1006.92003 |
| [25] | Saccomani, M. P.; Audoly, S.; D’Angiò, L., Parameter identifiability of nonlinear systems: the role of initial conditions, Automatica, 39, 619-632 (2003) · Zbl 1034.93014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
