Gaussian process

In probability theory and statistics, a Gaussian process (also known as kriging) is a statistical model where observations occur in a continuous domain, e.g. time or space. In a Gaussian process, every point in some continuous input space is associated with a normally distributed random variable. Moreover, every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

Viewed as a machine-learning algorithm, a Gaussian process uses lazy learning and a measure of the similarity between points (this is the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information -- it is a one-dimensional Gaussian distribution (which is the marginal distribution at that point).

For some kernel functions, matrix algebra can be used to calculate the predictions, as described in the kriging article. When a parameterised kernel is used, optimisation software is typically used to fit a Gaussian process model.

The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions.

Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times.

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