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Abstract
BackgroundGraphical models have long been used to describe biological networks for a variety of important tasks such as the determination of key biological parameters, and the structure of graphical model ultimately determines whether such unknown parameters can be unambiguously obtained from experimental observations (i.e., the identifiability problem). Limited by resources or technical capacities, complex biological networks are usually partially observed in experiment, which thus introduces latent variables into the corresponding graphical models. A number of previous studies have tackled the parameter identifiability problem for graphical models such as linear structural equation models (SEMs) with or without latent variables. However, the limited resolution and efficiency of existing approaches necessarily calls for further development of novel structural identifiability analysis algorithms.ResultsAn efficient structural identifiability analysis algorithm is developed in this study for a broad range of network structures. The proposed method adopts the Wright’s path coefficient method to generate identifiability equations in forms of symbolic polynomials, and then converts these symbolic equations to binary matrices (called identifiability matrix). Several matrix operations are introduced for identifiability matrix reduction with system equivalency maintained. Based on the reduced identifiability matrices, the structural identifiability of each parameter is determined. A number of benchmark models are used to verify the validity of the proposed approach. Finally, the network module for influenza A virus replication is employed as a real example to illustrate the application of the proposed approach in practice.ConclusionsThe proposed approach can deal with cyclic networks with latent variables. The key advantage is that it intentionally avoids symbolic computation and is thus highly efficient. Also, this method is capable of determining the identifiability of each single parameter and is thus of higher resolution in comparison with many existing approaches. Overall, this study provides a basis for systematic examination and refinement of graphical models of biological networks from the identifiability point of view, and it has a significant potential to be extended to more complex network structures or high-dimensional systems.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0287-y) contains supplementary material, which is available to authorized users.
Highlights
Graphical models have long been used to describe biological networks for a variety of important tasks such as the determination of key biological parameters, and the structure of graphical model determines whether such unknown parameters can be unambiguously obtained from experimental observations
The determination of unknown model parameter values from experimental data is of fundamental importance to many other critical tasks, and it should be stressed that parameter identifiability is one of the first questions that needs to be answered before any statistical method can be applied to obtain accurate and reliable estimates of unknown parameters [20]
In the structural equation model (SEM), each model variable Yi corresponds to a vertex Vi (i = 1, 2, ..., n), the structure of the coefficient matrix C = [cij] is specified by D, and the existence of disturbance correlation between two variables is given by U
Summary
Graphical models have long been used to describe biological networks for a variety of important tasks such as the determination of key biological parameters, and the structure of graphical model determines whether such unknown parameters can be unambiguously obtained from experimental observations (i.e., the identifiability problem). The determination of unknown model parameter values from experimental data is of fundamental importance to many other critical tasks (e.g., computer simulation or prediction, network structure refinement), and it should be stressed that parameter identifiability is one of the first questions that needs to be answered before any statistical method can be applied to obtain accurate and reliable estimates of unknown parameters [20]. Limited by resources or technical capabilities, it is not uncommon that only part of the nodes or interactions (i.e., edges) in a biological network can be experimentally observed such that the values of certain unknown parameters associated with those unobserved nodes or edges cannot be uniquely determined from experimental data due to the lack of information. It is necessary to develop identifiability analysis techniques for graphical models with or without latent variables
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