Motivation: disease progression modelling Covariate-GPLVM - PowerPoint PPT Presentation

Motivation: disease progression modelling Covariate-GPLVM Motivation: disease progression modelling Covariate-GPLVM Feature-level decomposition Feature-level decomposition Motivation: disease progression modelling Covariate-GPLVM Decomposing feature-level variation with Decomposing feature-level variation with Poster #261 Poster #261 Covariate Gaussian Process Latent Variable Models Covariate Gaussian Process Latent Variable Models Decomposing feature-level variation with Decomposing feature-level variation with Covariate Gaussian Process Latent Variable Models Covariate Gaussian Process Latent Variable Models y ( j ) y ( j ) = μ ( j ) + f ( j ) = μ ( j ) + f ( j ) z ( z ) + f ( j ) z ( z ) + f ( j ) x ( x ) + f ( j ) x ( x ) + f ( j ) GPLVM maps latent GPLVM maps latent GPLVM maps latent to observed data to observed data to observed data zx ( z , x ) + ε ij zx ( z , x ) + ε ij i i z i ∼ N (0, 1) z i ∼ N (0, 1) z i ∼ N (0, 1) using GP mappings using GP mappings using GP mappings Readily available for linear models , otherwise challenging: Readily available for linear models , otherwise challenging: Y Y Y Naive decompositions (with standard GP priors) can lead to misleading conclusions y ( j ) y ( j ) y ( j ) = f ( j ) ( z i ) + ε ij = f ( j ) ( z i ) + ε ij = f ( j ) ( z i ) + ε ij i i i With appropriate functional constraints we learn an identifiable non-linear decomposition Covariate-GPLVM extends GPLVM by: Covariate-GPLVM extends GPLVM by: 1. Incorporating covariates 1. Incorporating covariates x x y ( j ) y ( j ) = f ( j ) ( x i , z i ) + ε ij = f ( j ) ( x i , z i ) + ε ij i i 2. Providing a feature-level decomposition Kaspar Märtens, Kieran Campbell, Christopher Yau Kaspar Märtens, Kieran Campbell, Christopher Yau ( j ) = μ ( j ) + f ( j ) z ( z ) + f ( j ) x ( x ) + f ( j ) y zx ( z , x ) + ε ij i @kasparmartens @kasparmartens @kasparmartens @kasparmartens kasparmartens.rbind.io kasparmartens.rbind.io kasparmartens.rbind.io kasparmartens.rbind.io

Motivation: disease progression modelling

Motivation: disease progression modelling

Motivation: disease progression modelling

Covariate-GPLVM GPLVM maps latent to observed data z i ∼ N (0, 1) using GP mappings Y y ( j ) = f ( j ) ( z i ) + ε ij i

Covariate-GPLVM GPLVM maps latent to observed data z i ∼ N (0, 1) using GP mappings Y y ( j ) = f ( j ) ( z i ) + ε ij i Covariate-GPLVM extends GPLVM by: 1. Incorporating covariates x y ( j ) = f ( j ) ( x i , z i ) + ε ij i

Covariate-GPLVM GPLVM maps latent to observed data z i ∼ N (0, 1) using GP mappings Y y ( j ) = f ( j ) ( z i ) + ε ij i Covariate-GPLVM extends GPLVM by: 1. Incorporating covariates x y ( j ) = f ( j ) ( x i , z i ) + ε ij i 2. Providing a feature-level decomposition ( j ) = μ ( j ) + f ( j ) z ( z ) + f ( j ) x ( x ) + f ( j ) y zx ( z , x ) + ε ij i

Feature-level decomposition y ( j ) = μ ( j ) + f ( j ) z ( z ) + f ( j ) x ( x ) + f ( j ) zx ( z , x ) + ε ij i Readily available for linear models , otherwise challenging:

Feature-level decomposition y ( j ) = μ ( j ) + f ( j ) z ( z ) + f ( j ) x ( x ) + f ( j ) zx ( z , x ) + ε ij i Readily available for linear models , otherwise challenging: Naive decompositions (with standard GP priors) can lead to misleading conclusions With appropriate functional constraints we learn an identifiable non-linear decomposition

Decomposing feature-level variation with Decomposing feature-level variation with Covariate Gaussian Process Latent Variable Models Covariate Gaussian Process Latent Variable Models @kasparmartens @kasparmartens kasparmartens.rbind.io kasparmartens.rbind.io

Poster #261 Poster #261