zbMATH — the first resource for mathematics

Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design. (English) Zbl 0581.93017
The authors suggest a common approach based on a least squares technique to investigate model identifiability, estimability and optimal sampling design.
As a starting point the two levels of parameters encountered in the identifiability analysis are emphasized: the ”structural” and ”experimental” parameters, related with system and experiment; the ”observational” parameters, that is some functions of the previous ones which can be uniquely identified by means of the observations.
A least squares procedure is employed, based on a set of observations generated at initial parameter estimates, then the Jacobian matrix of the equations involved is constructed such that the parameters appear structured into three groups: those that are observable and identifiable, those that are observable but not identifiable, those that are not observable and not identifiable.
It is pointed out, and numerous examples are provided as an illustration, that the correlation matrix provides some information to separate the previous groups and highlights the difficulties that can arise in estimating some parameters, because of the high correlations between them.
Finally, to optimise sampling times, a strategy, which maximises the sensitivity of the sums of square deviations, is chosen to change the parameters to be estimated; this leads to equations very similar to those encountered in the identifiability study.
Reviewer: L.d’Angio

93B30 System identification
92Cxx Physiological, cellular and medical topics
93C57 Sampled-data control/observation systems
93B40 Computational methods in systems theory (MSC2010)
65D99 Numerical approximation and computational geometry (primarily algorithms)
Full Text: DOI
[1] Koopmans, T.C.; Reiersol, O., Identification of structural characteristics, Ann. math. statist., 21, 165-181, (1950) · Zbl 0038.29303
[2] Fisher, F.M., Generalization of the rank and order conditions for identifiability, Econometrica, 27, 431-477, (1959) · Zbl 0090.11703
[3] Astrom, K.J.; Cykhoff, P., System identificatio—a survey, Automatica, 7, 123-162, (1971) · Zbl 0219.93004
[4] Mehra, R.K., Topics in stochastic control theory—identification in control and econometrics: similarities and differences, Ann. econ. soc. meas., 3, 24-47, (1974)
[5] Bellman, R.; Astrom, K.J., On structural identifiability, Math. biosci., 7, 329-339, (1970)
[6] Nguyen, V.V.; Wood, E.F., Review and unification of linear identifiability concepts, SIAM rev., 24, 34-51, (1982)
[7] Walter, E., Identifiability of state space models, ()
[8] Anderson, D.H., Compartmental modeling and tracer kinetics, () · Zbl 0509.92001
[9] Jacquez, J.A., The inverse problem for compartmental systems, Math. comput. simulation, 24, 452-459, (1982) · Zbl 0562.93018
[10] Glover, K.; Willems, J.C., Parametrizations of linear dynamical systems; canonical forms and identifiability, IEEE trans. automat. control, 19, 640-646, (1974) · Zbl 0296.93008
[11] Glover, K.; Willems, J.C., On the identifiability of linear dynamical systems, Proceedings of the 3rd IFAC symposium on identification and system parameter estimation, 867-870, (1973), The Hague
[12] Grewal, M.S.; Glover, K., Identifiability of linear and nonlinear dynamical systems, IEEE trans. automat. control, 21, 833-837, (1976) · Zbl 0344.93022
[13] Cobelli, C.; DiStefano, J.J., Parameter and structural identifiability concepts and ambiguities: A critical review and analysis, Amer. J. physiol., 239, R7-R24, (1980)
[14] Cobelli, C.; Lepschy, A.; Romanin-Jacur, G., Identifiability of compartmental systems and related structural properties, Math. biosci., 44, 1-18, (1979) · Zbl 0431.93019
[15] Jacquez, J.A., Compartmental models of biological systems: linear and nonlinear, (), 185-205 · Zbl 0219.92005
[16] Cobelli, C.; Lepschy, A.; Romanin-Jacur, G., Identifiability results on some constrained compartmental systems, Math. biosci., 47, 173-195, (1979) · Zbl 0431.93019
[17] Delforge, J., New results on the problem of identifiability of a linear system, Math. biosci., 52, 73-96, (1980) · Zbl 0455.93019
[18] Delforge, J., Necessary and sufficient conditions for local identifiability of a system with linear compartments, Math. biosci., 54, 159-180, (1981) · Zbl 0459.92004
[19] DiStefano, J.J., Complete parameter bounds and quasidentifiable linear systems, Math. biosci., 65, 51-68, (1983) · Zbl 0516.93016
[20] Eisenfeld, J., On identifiability of impulse-response in compartmental systems, Math. biosci., 47, 15-34, (1979) · Zbl 0429.92006
[21] Milanese, M.; Molino, G.P., Structural identifiability of compartmental models and pathophysiological information from the kinetics of drugs, Math. biosci., 26, 175-190, (1975) · Zbl 0308.92008
[22] Norton, J.P., Normal-mode identifiability analysis of linear compartmental systems in linear stages, Math. biosci., 50, 95-115, (1980) · Zbl 0433.93064
[23] Norton, J.P., An investigation of the sources of nonuniqueness in deterministic identifiability, Math. biosci., 60, 89-108, (1982) · Zbl 0506.93015
[24] Travis, C.; Haddock, G., On structural identification, Math. biosci., 56, 157-173, (1981) · Zbl 0465.93027
[25] Pohjanpalo, H., System identifiability based on the power series expansion of the solution, Math. biosci., 4, 21-33, (1978) · Zbl 0393.92008
[26] Vajda, S., Structural equivalence of linear systems and compartmental models, Math. biosci., 55, 39-64, (1981) · Zbl 0472.93024
[27] Walter, E.; Lecourtier, Y., Unidentifiable compartmental models, Math. biosci., 56, 1-25, (1981), What to do? · Zbl 0465.93028
[28] Delforge, J., Sur l’identifiabilité et l’identification des modèles linéaires, ()
[29] Chen, C.T., Introduction to linear system theory, (1970), Holt, Rinehart and Winston New York
[30] Raksanyi, A.; Lecourtier, Y.; Walter, E.; Venot, A., Identifiability and distinguishability testing via computer algebra, Math. biosci., 77, 245-266, (1985) · Zbl 0574.93019
[31] Berman, M.; Schoenfeld, R., Invariants in experimental data on linear kenetics and the formulation of models, J. appl. phys., 27, 1361-1370, (1956)
[32] Jacquez, J.A., Compartmental analysis in biology and medicine, (1985), Univ. of Michigan Press, Chapter 14 · Zbl 0703.92001
[33] Aitken, A.C., Determinants and matrices, (1962), Oliver and Boyd Edinburgh · Zbl 0108.01501
[34] Milanese, M.; Sorrentino, N., Decomposition methods for the identifiability analysis of large systems, Internat. J. control, 28, 71-79, (1978) · Zbl 0391.93001
[35] Kalman, R.E., Mathematical description of linear dynamical systems, SIAM J. control, 1, 152-192, (1963) · Zbl 0145.34301
[36] Box, G.E.R.; Lucas, H.L., Design of experiments in non-linear situations, Biometrika, 46, 77-90, (1959) · Zbl 0086.34803
[37] Landaw, E.M., Optimal experimental design for biologic compartmental systems with applications to pharmacokinetics, ()
[38] Fedorov, V.V., Theory of optimal experiments, (1972), Academic New York
[39] Brown, R.F.; Godfrey, K.R., Problems of determinacy in compartmental modeling with application to bilirubin kinetics, Math. biosci., 40, 205-224, (1978) · Zbl 0393.92006
[40] Jacquez, J.A., Compartmental analysis in biology and medicine, (1972), Elsevier Amsterdam, Chapter 7 · Zbl 0703.92001
[41] DiStefano, J.J., Complete parameter bounds and quasiidentifiable linear systems, Math. biosci., 65, 51-658, (1983) · Zbl 0516.93016
[42] Hadaegh, F.Y.; Bekey, G.A., Near-identifiability of dynamical systems, Math biosci., 77, 325-340, (1985) · Zbl 0574.93018
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.