Contents
- 1 Which is the kernel function of the Gaussian process?
- 2 Which is a prior on the Gaussian process distribution?
- 3 How is a linear kernel different from a stationary covariance function?
- 4 How are kernel ridge regression and Gaussian process regression related?
- 5 How is the length of a kernel parameterized?
- 6 What kind of kernel is a radial basis function?
Which is the kernel function of the Gaussian process?
To summarize the kernel function k(x,x′) k ( x, x ′) models the covariance between each pair in x x. The kernel function together with the mean function m(x) m ( x) define the Gaussian process distribution:
Which is a prior on the Gaussian process distribution?
In the first post we introduced the concept of the kernel which defines a prior on the Gaussian process distribution. To summarize the kernel function k(x,x′) k ( x, x ′) models the covariance between each pair in x x.
How is a linear kernel different from a stationary covariance function?
The linear kernel is not like the others in that it’s non-stationary. A stationary covariance function is one that only depends on the relative position of its two inputs, and not on their absolute location. That means that the parameters of the linear kernel are about specifying the origin:
How to choose a covariance function for your problem?
Here you’ll find concrete advice on how to choose a covariance function for your problem, or better yet, make your own. If you’re looking for software to implement Gaussian process models, I recommend GPML for Matlab, or GPy for Python. These software packages deliberately do not provide a default kernel.
How are Gaussian processes interoperable with sklearn metrics?
All Gaussian process kernels are interoperable with sklearn.metrics.pairwise and vice versa: instances of subclasses of Kernel can be passed as metric to pairwise_kernels from sklearn.metrics.pairwise. Moreover, kernel functions from pairwise can be used as GP kernels by using the wrapper class PairwiseKernel.
It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression.
How is the length of a kernel parameterized?
It is parameterized by a length scale parameter l > 0, which can either be a scalar (isotropic variant of the kernel) or a vector with the same number of dimensions as the inputs X (anisotropic variant of the kernel). The kernel is given by: where l is the length scale of the kernel and d ( ⋅, ⋅) is the Euclidean distance.
What kind of kernel is a radial basis function?
Radial-basis function kernel (aka squared-exponential kernel). The RBF kernel is a stationary kernel. It is also known as the “squared exponential” kernel.