An Intuitive Tutorial to Gaussian Process Regression

TL;DR

This tutorial introduces Gaussian process regression by building from foundational concepts to practical implementation, emphasizing the GP prior over functions and uncertainty-aware predictions. It derives the standard predictive equations and demonstrates a vanilla GPR workflow, followed by hyperparameter optimization and a review of popular software packages. The work highlights kernel choice, especially the RBF kernel, and discusses computational limitations that motivate sparse GP approaches for large datasets. Overall, it equips readers with a rigorous, implementation-ready understanding of GPR and pointers to tools for real-world use.

Abstract

This tutorial aims to provide an intuitive introduction to Gaussian process regression (GPR). GPR models have been widely used in machine learning applications due to their representation flexibility and inherent capability to quantify uncertainty over predictions. The tutorial starts with explaining the basic concepts that a Gaussian process is built on, including multivariate normal distribution, kernels, non-parametric models, and joint and conditional probability. It then provides a concise description of GPR and an implementation of a standard GPR algorithm. In addition, the tutorial reviews packages for implementing state-of-the-art Gaussian process algorithms. This tutorial is accessible to a broad audience, including those new to machine learning, ensuring a clear understanding of GPR fundamentals.

Figures (12)

• Figure 1: A regression example: (a) Observed data points, (b) Five sample functions fitting the observed data points.
• Figure 2: Visualization of 1000 normally distributed data points as red vertical bars on the $X$-axis, alongside their PDF plotted as a two-dimensional bell curve.
• Figure 3: Two independent uni-variate Gaussian vector points plotted vertically within the $Y, x$ coordinates space.
• Figure 4: Connecting points of independent Gaussian vectors by lines: (a) Ten randomly selected points in vectors $x_1$ and $x_2$, (b) Ten randomly selected points in twenty vectors $x_1, x_2, \ldots, x_{20}$ .
• Figure 5: BVN PDF visualization: (a) a 3-D bell curve with the height representing the probability density, (b) 2-D ellipse contour projections showing the correlation between $x_1$ and $x_2$ points.