We present a workflow to diagnose persistent noise from model fits and describe how to remodel accounting for correlated errors.ĭ x d t = h ( t, x, θ ), t ∈ ( 0, T ] and x ( 0 θ ) = x 0, } 1.1where x ( t θ ) ∈ R n is the state of the system θ ∈ R m are the parameters of the system t denotes time h( t, x, θ) can be a function of time, state and parameters and x 0 ∈ R n is the initial state. Considering the case when persistent noise arises owing only to measurement imperfections, we use the Fisher information matrix to quantify how uncertainty in parameter estimates is artificially reduced when erroneously assuming independent noise. Here, we use experimental data from dynamical systems drawn from cardiac physiology and electrochemistry to demonstrate that highly persistent differences between observations and modelled quantities can occur. However, like the deterministically modelled parts of a system, these latent factors can have persistent effects on observables. This noise model assumes ‘random’ latent factors affect the system in the ephemeral fashion resulting in unsystematic deviation of observables from their modelled counterparts. For this, independent Gaussian noise is commonly chosen, with its use so widespread that researchers typically provide no explicit justification for this choice. To perform likelihood-based parameter inference on these models, it is necessary to specify a statistical process representing the contribution of factors not explicitly included in the mathematical model.
Ordinary differential equation models are used to describe dynamic processes across biology.
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