# Cluster-Specific Predictions with Multi-Task Gaussian Processes

### 40th International Conference on Machine Learning

Longitudinal data observed from multiple irregular sources

Gaussian processes offers an elegant and well-suited framework for modelling longitudinal data.

Until then, most multi-task approaches focus on the covariance structure. Let us present a novel multi-task GP paradigm sharing information through a common mean process.

Leroy et al. - Magma: Inference and Prediction using Multi-Task Gaussian Processes with Common Mean - Machine Learning - 2022

$y_i = \mu_0 + f_i + \epsilon_i$

with: $$\ \ \ \mu_0 \sim \mathcal{GP}(m_0, K_{\theta_0}), \ \ \ f_i \sim \mathcal{GP}(0, \Sigma_{\theta_i}), \ \ \ \epsilon_i \sim \mathcal{GP}(0, \sigma_i^2)$$  Leroy et al. - Cluster-Specific Predictions with Multi-Task Gaussian Processes - JMLR - 2023

$y_i \mid \{\color{orange}{Z_{ik}} = 1 \} = \mu_{\color{orange}{k}} + f_i + \epsilon_i$

with: $$\ \ \ \color{orange}{Z_{i}} \sim \mathcal{M}(1,\color{orange}{\boldsymbol{\pi}}), \ \ \ \mu_{\color{orange}{k}} \sim \mathcal{GP}(m_{\color{orange}{k}}, \color{orange}{C_{\gamma_{k}}}), \ \ \ f_i \sim \mathcal{GP}(0, \Sigma_{\theta_i}), \ \ \ \epsilon_i \sim \mathcal{GP}(0, \sigma_i^2).$$

$$\rightarrow$$ Variational EM algorithm to learn hyper-parameters and hyper-posterior distributions

$$\rightarrow$$ Cluster-specific predictions combined into a mixture of GPs

Performances and applications
Implemented as an R package MagmaClustR: https://github.com/ArthurLeroy/MagmaClustR