Notations¶

This page contains the definitions and notations of the variables used in Leaspy.

Notations, concepts, and names in DAG¶

The following table displays the relationships between a variable mathematical notation, the associated concept, and the string name used in the DAG if this variable is handled this way.

Notation	Concept	Name in the DAG
\( \tau_i \)	estimated reference time	`tau`
\( \xi_i \)	individual log speed factor	`xi`
\( \gamma_i(t) \)	individual trajectory	`model`
\( \psi_i(t) \)	latent disease age	`rt`
\( \mathbf{A} \)	mixing matrix	`mixing_matrix`
\( t_0 \)	population reference time	`tau_mean`
\( \mathbf{s}_i\)	sources	`sources`
\( w_{i,k} \)	space shift	`space_shifts`
\( u_{i,l} \)	survival shift	`survival_shifts`

Concept definitions¶

This section explains each mathematical concept and provides mathematical definitions when needed.

Estimated reference time¶

The estimated reference time for a given individual \( i \) is denoted as \( \tau_i \). It follows \(\tau_i \sim \mathcal{N}(t_0, \sigma^2_{\tau})\).

Individual log speed factor¶

The individual log speed factor for a given individual \( i \) is denoted as \( \xi_i \) . It follows \(\xi_i \sim \mathcal{N}(0, \sigma^2_{\xi})\).

Individual trajectory¶

The individual trajectory for a given individual \( i \) is denoted as \( \gamma_i(t) \) and represents the disease progression of the patient \(i\). It can be indexed by \(k\) when \(K\) outcomes are estimated.

Latent disease age¶

The latent disease age, for a given individual \( i \), is denoted as \( \psi_i(t) \).

It represents a transformation from chronological time \(t\) to latent disease age that is related to its stage in the disease and is defined as:

\[ \psi_i(t) = e^{\xi_i}(t - \tau_i) + t_0 \]

where :

\( e^{\xi_i} \) is the individual speed factor of individual \( i \).
\( \tau_i \) is the estimated reference time of individual \( i \).
\( t_0 \) is the population reference time.

Mixing matrix¶

The mixing matrix is denoted as \( \mathbf{A} \) and is defined as a matrix that describes the mixing of different sources in the model.

Population reference time¶

The population reference time is denoted as \( t_0 \) and is defined as the reference time for the entire population.

Sources¶

The sources, for a given individual \( i \), are denoted as \( \mathbf{s}_i\) and defined as the different origins of information or data that contribute to the individual’s disease progression model.

Space shift¶

The space shift, are more interpretable than sources \(\mathbf{s}_i\), as they encapsulate total spatial variability effects. for a given individual \( i \) and a given longitudinal outcome \( k \), is denoted as \( w_{i,k} \) and defined as:

\[ w_{i,k} = \sum_{j=1}^{N_w} \eta_{k,j} s_{i,j} \]

where:

\( N_w \) is the number of spatial sources.
\( \eta_{k,j} \) is the weight for spatial source \( j \) in outcome \( k \).
\( s_{i,j} \) is the contribution of spatial source \( j \) for individual \( i \).

Survival shift¶

The survival shift, for a given individual \( i \) and a given event \( l \) is denoted as \( u_{i,l} \) and defined as:

\[ u_{i,l} = \sum_{m=1}^{N_s} \zeta_{l,m} s_{i,m} \]

where:

\( N_s \) is the number of survival sources.
\( \zeta_{l,m} \) is the weight for survival source \( m \) in event \( l \).
\( s_{i,m} \) is the contribution of survival source \( m \) for individual \( i \).