5) Model regularization:

In cases of low signal-to-noise ratio¹, regularization of the patient-specific mixing weights π can be applied to control variability of these estimates. Once the number and parameters of the population components have been determined, a penalized nonlinear least squares approach using the historical linear parameters as a Bayesian penalty for individual fit [34] is used for this approach. Regularization on π = π_k parameters for the kth patient is achieved using the following individual objective function:

where z={zik}i=1mk, A_Δ = [A_ijk]_i,j is a matrix of dimension m_k × J, and W = W_k are defined in Section II-B.1. All components in the vectorized mixing weights π are constrained to be nonnegative, and π₀ and Ω respectively denote the mean and covariance matrix of linear parameters obtained from population-based estimation. regularization parameter λ controls the trade-off between individual and population data contributions, larger values of λ constraining individual estimates to be closer to the population profile π₀, which suits higher-noise scenarios. The optimization is a quadratic problem for which we used the Goldfarb-Idnami method [33].

Parameter λ is set by minimizing the generalized cross-validation (GCV) score which is defined as the mean squared error adjusted by the effective degree of freedom:

where z^ik(λ) is C^P(tik∣Δk,πk,θ) based on minimization of the objective function defined in Eqn. (2), for a given λ.