Kovarianssifunktion

The covariance function, often referred to as the kernel function in machine learning contexts, describes the similarity between points in a dataset. It quantifies how two variables change together. If the covariance function's value is high for two points, it suggests they are similar and likely to have related output values. Conversely, a low value indicates dissimilarity.

In statistical modeling, the covariance function is central to Gaussian Processes. It defines the prior distribution over functions. By specifying a covariance function, we implicitly define the properties of the functions that the model can learn, such as their smoothness and periodicity. Different choices of covariance functions lead to models with different inductive biases.

Commonly used covariance functions include the Radial Basis Function (RBF) kernel, also known as the squared exponential kernel, which assumes that points closer together are more similar. Another is the Matern kernel, which offers more flexibility in controlling the smoothness of the functions. Polynomial kernels and linear kernels are also employed, each capturing different types of relationships between data points.

The parameters within a covariance function are typically learned from the data by maximizing the marginal likelihood of the observations. This process allows the model to adapt the covariance structure to the specific dataset. The output of the covariance function is a covariance matrix, where each element represents the covariance between a pair of data points. This matrix is crucial for predicting new data points and quantifying uncertainty in those predictions.