# Extensions

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We have so far reviewed the most frequently used models for describing both the individual parameters $(\psi_i)$ and the observations $(y_i)$, but several extensions can be considered.

For instance, if we assume that a population consists of several homogeneous sub-populations, mixtures models can be very useful for describing different types of mixtures, such as mixtures of distributions, mixtures of structural models and mixtures of residual models (see Mixture models).

A stochastic component can also be introduced into the model by assuming some underlying stochastic dynamics, characterized either by a hidden Markov model (see Hidden Markov models) or a system of stochastic differential equations (see Stochastic differential equations based models).

Although we restrict ourselves to these extensions in this document, it should be noted that other extensions mentioned in the introduction (see What is a model? A joint probability distribution!) could also have been addressed:

• Population parameter models: introduce a priori information in an estimation context, or to model inter-population variability.
• Covariate models: mainly relevant in the context of wanting to simulate virtual individuals.
• Design models: measurement times, dose regimens, etc.