Key Concepts of Gaussian Process Regression (GPR)

Gaussain Process

A non-parametric, probabilistic model called a Gaussian Process (GP) is utilized in statistics and machine learning for regression, classification, and uncertainty quantification. It depicts a group of random variables, each of which has a joint Gaussian distribution and can have a finite number. GPs are a versatile and effective technique for modeling intricate relationships in data and producing forecasts with related uncertainty.

Characteristics of Gaussian Processes:

• Non-Parametric Nature: GPs can adjust to the complexity of the data because they do not rely on a set number of model parameters
• Probabilistic Predictions: Predictions from GPs can be quantified because they deliver predictions as probability distributions.
• Interpolation and Smoothing: GPs are useful for noisy or irregularly sampled data because they are good at smoothing noisy data and interpolating between data points.
• Marginalization of Hyperparameters: By eliminating the requirement for explicit hyperparameter tweaking, they marginalize over hyperparameters, making the model simpler.

Mean Function

The predicted value of the function being modeled at each input point is represented by the mean function in Gaussian Processes (GPs). It functions as a foundational presumption regarding the underlying data structure. The mean function is frequently set to zero by default not necessarily and can be modified based on data properties or domain expertise. By influencing the central tendency of forecasts, it aids general practitioners in identifying patterns or trends in the data. GPs provide probabilistic predictions that contain uncertainty as well as point estimates by including the mean function

Covariance (Kernel) Function

The covariance function, also referred to as the kernel function, measures how similar the input data points are to one another in Gaussian Processes (GPs). It is essential in characterizing the behavior of the GP model, affecting the selection of functions from the previous distribution. The covariance function measures pairwise similarities to ascertain the correlation between function values. GPs can adjust to a broad range of data patterns, from smooth trends to complex structures, because different kernel functions capture different kinds of correlations. The model’s performance can be greatly impacted by the kernel selection.

Prior Distributions

The prior distribution, in Gaussian Processes (GPs), is our understanding of functions prior to the observation of any data. Usually, it is described by a covariance (kernel) function and a mean function. Whereas the covariance function describes the similarity or correlation between function values at various input points, the mean function encodes our previous expectations. This is used beforehand by GPs to create a distribution over functions. In GPs, priors can be selected to represent data uncertainty, integrate domain knowledge, or indicate smoothness.

Posterior Distributions

Gaussian Processes’ posterior distribution shows our revised assumptions about functions following data observation. It puts together the likelihood of the data given the function and the previous distribution. The posterior in GP regression offers a distribution over functions that most closely match the observed data. By allowing for probabilistic predictions and the quantification of uncertainty, the posterior distribution reflects the trade-off between the prior beliefs stored in the prior distribution and the information supplied by the data.

Regression and probabilistic classification issues can be resolved using the Gaussian process (GP), a supervised learning technique. Since each Gaussian process can be thought of as an infinite-dimensional generalization of multivariate Gaussian distributions, the term “Gaussian” appears in the name. We will discuss Gaussian processes for regression in this post, which is also referred to as Gaussian process regression (GPR). Numerous real-world issues in the fields of materials science, chemistry, physics, and biology have been resolved with the use of GPR.