An eigen-analysis of the relationships between model structure, discrete data, measurement error and resulting parameter identification distributions
Practical rather than structural identifiability is often the determining factor whether effective parameter identification is possible in a physiological model. This paper presents analysis into relationships between the population outcomes, and the original model and data properties as part of ongoing research into a deterministic approach to evaluate a-priori identifiability. Data size, output noise variance and true parameter values were varied for a simple 2-parameter model with a linear regression equation Ax = b for discrete data points. Principal Component Analysis of a Monte Carlo simulation was compared to these varied properties and the eigendecomposition of ATA. Principal component vectors were found to be parallel with ATA eigenvectors and the eigenvalues were inversely related. Principal component eigenvalues decreased in inverse proportion to data size, were scaled by the sum of squared parameter values and noise variance. ATA eigenvalues on the other hand were unchanged by output noise and parameter value, but increased in linear, rather than inverse proportion, to data size. The ratio of principal component eigenvalues to each other was affected by data size and some parameter values, while the ATA eigenvalue ratio was affected by data size only. Deterministic relationships have been found between population parameter identification outcomes, model properties and data. If all of the factors determining principle components can be calculated then population variance can be estimated from a single set of data, facilitating confidence of individual outcomes and evaluation of practical identifiability.