The first method relies on differential algebra tools to first generate the input-output equations while the second does not utilize these equations.įor unidentifiable models, we also introduce two novel approaches for finding identifiable functions of the parameters. The total number of such values over the complex numbers is called the identifiability degree which is computed in two ways.

For an identifiable model, we compute the finite number of values of the parameters given input-output data. In this paper, we use techniques from numerical algebraic geometry (e.g., see for a general overview) to investigate both identifiable and unidentifiable models. Since our goal is to reparametrize a model over identifiable functions of the parameters, simpler functions are preferred. It should be noted, however, that even in the linear case, the identifiable functions of parameters found using the transfer function method are not necessarily (and are usually not) the simplest identifiable functions of parameters. For example, the approach in requires the calculation of many Gröbner bases and can thus be computationally expensive. However, in nonlinear models, the problem has been more challenging with only ad hoc methods proposed, e.g. In linear models, this can be done using the transfer function method. Several methods have been proposed to find identifiable functions. These can then be used to reparameterize the model and also to give additional insight into which parameters should be experimentally measured. In the latter case, we attempt to find a set of identifiable functions of the parameters. However, if some subset of the parameters can take on an infinite number of values and yet yield the same input-output data, then the model and this subset of parameters are called unidentifiable. If the parameters of the model have a unique or finite number of values given input-output data, then the model and its parameters are said to be identifiable. There are two possible outcomes of the structural identifiability check of a mathematical model. Even if a model fails to be structurally identifiable, some useful information about the parameters can still be determined, which is the main motivation for this paper. For this reason, structural identifiability is often referred to as a priori identifiability. This is a necessary condition for the practical or numerical identifiability problem, which involves parameter estimation with real, and often noisy, data. In this paper, we address structural identifiability, which concerns whether the parameters of a model can be determined from perfect input-output data, i.e., noise-free and of any time duration required. Parameter identifiability analysis for dynamical system models consisting of ordinary differential equations (ODEs) addresses the question of which unknown parameters can be determined from given input-output data. Several examples are used to demonstrate the new techniques. For unidentifiable models, we present a novel numerical differential algebra technique aimed at computing a set of algebraically independent identifiable functions.

For identifiable models, we present a novel approach to compute the identifiability degree. In this work, we use numerical algebraic geometry to determine if a model given by polynomial or rational ordinary differential equations is identifiable or unidentifiable.

For unidentifiable models, a set of identifiable functions of the parameters are sought so that the model can be reparametrized in terms of these functions yielding an identifiable model. Unidentifiable models are models such that the unknown parameters can have an infinite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. A common problem when analyzing models, such as mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data.

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