TimeReparametrizedModel

Module: leaspy.models.time_reparametrized

This class introduces the concept of time reparametrization, which is central to Leaspy’s ability to model disease progression. It handles patient-specific time shifts (\(\tau\)) and acceleration factors (\(\xi\)).

Responsibilities

• Time Normalization: It maps real individual time (age) to a normalized “pathological” time used by the model.

• Individual Variability: It defines and manages the individual parameters responsible for temporal variability (\(\tau\), \(\xi\)) and optionally spatial variability (sources).

  • \(\tau\) (Time shift): Accounts for disease onset timing (subjects starting earlier or later).

  • \(\xi\) (Log-acceleration): Accounts for disease progression speed (subjects progressing faster or slower).

  • sources (Spatial components): Accounts for inter-subject variability in the pattern of progression (e.g. different symptoms worsening at different relative rates), if source_dimension > 0.

Key Methods

• get_variables_specs(): Defines the statistical nature of the parameters (e.g., priors, distributions). It introduces:

  • xi: Individual log-acceleration (Normal distribution).

  • tau: Individual time shift (Normal distribution).

• time_reparametrization(t, alpha, tau): The static method that implements the core math equation for time mapping: \(t_{reparam} = \alpha_i \cdot (t_{ij} - \tau_i)\).

  • sources: (If applicable) Independent component sources for spatial variability.

Key Attributes

• dimension (Inherited): The number of features (biomarkers) in the model. This determines the shape of the mixing matrix if sources are used.

• source_dimension: Number of independent sources modeling the spatial variability (optional).

Next Step

The RiemanianManifoldModel builds upon this time reparametrization by adding a geometric structure (Riemannian metric) to the space of observations, essential for consistent multivariate modeling.