# Scalable mixed-domain Gaussian process modeling and model reduction for longitudinal data

arXiv: https://arxiv.org/abs/2111.02019 · HTML: https://mlsanity.com/paper/2111.02019/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 3 Nov 2021.

## Builds on (most cited first, 18 of 18)

- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Input Warping for Bayesian Optimization of Non-stationary Functions](https://mlsanity.com/paper/1402.0929/) (2014)
- [Dealing with Integer-valued Variables in Bayesian Optimization with Gaussian Processes](https://mlsanity.com/paper/1706.03673/) (2017)
- [A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors](https://mlsanity.com/paper/1806.07504/) (2018)
- [Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming](https://mlsanity.com/paper/2004.11408/) (2020)
- [Group kernels for Gaussian process metamodels with categorical inputs](https://mlsanity.com/paper/1802.02368/) (2018)
- [Scalable Gaussian Processes for Data-Driven Design using Big Data with Categorical Factors](https://mlsanity.com/paper/2106.15356/) (2021)
- [Comparison of Bayesian predictive methods for model selection](https://mlsanity.com/paper/1503.08650/) (2015)
- [Deep Mixed Effect Model using Gaussian Processes: A Personalized and Reliable Prediction for Healthcare](https://mlsanity.com/paper/1806.01551/) (2018)
- [Efficient Optimization for Sparse Gaussian Process Regression](https://mlsanity.com/paper/1310.6007/) (2013)
- [Projective Inference in High-dimensional Problems: Prediction and Feature Selection](https://mlsanity.com/paper/1810.02406/) (2018)
- [Scalable Gaussian Processes on Discrete Domains](https://mlsanity.com/paper/1810.10368/) (2018)
- [lgpr: An interpretable nonparametric method for inferring covariate effects from longitudinal data](https://mlsanity.com/paper/1912.03549/) (2019)
- [Using reference models in variable selection](https://mlsanity.com/paper/2004.13118/) (2020)
- [Projection Predictive Inference for Generalized Linear and Additive Multilevel Models](https://mlsanity.com/paper/2010.06994/) (2020)

## Abstract (arXiv)

Gaussian process (GP) models that combine both categorical and continuous input variables have found use in analysis of longitudinal data and computer experiments. However, standard inference for these models has the typical cubic scaling, and common scalable approximation schemes for GPs cannot be applied since the covariance function is non-continuous. In this work, we derive a basis function approximation scheme for mixed-domain covariance functions, which scales linearly with respect to the number of observations and total number of basis functions. The proposed approach is naturally applicable to also Bayesian GP regression with discrete observation models. We demonstrate the scalability of the approach and compare model reduction techniques for additive GP models in a longitudinal data context. We confirm that we can approximate the exact GP model accurately in a fraction of the runtime compared to fitting the corresponding exact model. In addition, we demonstrate a scalable model reduction workflow for obtaining smaller and more interpretable models when dealing with a large number of candidate predictors.