8000 DOC Add Dropdown to Module 1.7. Gaussian Processes by anamfatima1304 · Pull Request #27414 · scikit-learn/scikit-learn · GitHub
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Nov 21, 2023
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DOC Add Dropdown to Module 1.7. Gaussian Processes #27414

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Add Dropdown
anamfatima1304 Sep 18, 2023
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Merge branch 'main' into dropdown_GaussianProcesses
glemaitre Nov 2, 2023
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Merge branch 'main' into dropdown_GaussianProcesses
anamfatima1304 Nov 3, 2023
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ArturoAmorQ Nov 21, 2023
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18 changes: 15 additions & 3 deletions doc/modules/gaussian_process.rst
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Expand Up @@ -239,8 +239,10 @@ also invariant to rotations in the input space. For more details, we refer to
Chapter 4 of [RW2006]_. For guidance on how to best combine different kernels,
we refer to [Duv2014]_.

Gaussian Process Kernel API
---------------------------
|details-start|
**Gaussian Process Kernel API**
|details-split|

The main usage of a :class:`Kernel` is to compute the GP's covariance between
datapoints. For this, the method ``__call__`` of the kernel can be called. This
method can either be used to compute the "auto-covariance" of all pairs of
Expand Down Expand Up @@ -323,6 +325,7 @@ only isotropic distances. The parameter ``gamma`` is considered to be a
hyperparameter and may be optimized. The other kernel parameters are set
directly at initialization and are kept fixed.

|details-end|

Basic kernels
-------------
Expand Down Expand Up @@ -383,7 +386,13 @@ Matérn kernel
-------------
The :class:`Matern` kernel is a stationary kernel and a generalization of the
:class:`RBF` kernel. It has an additional parameter :math:`\nu` which controls
the smoothness of the resulting function. It is parameterized by a length-scale parameter :math:`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 :math:`x` (anisotropic variant of the kernel). The kernel is given by:
the smoothness of the resulting function. It is parameterized by a length-scale parameter :math:`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 :math:`x` (anisotropic variant of the kernel).

|details-start|
**Mathematical implementation of Matérn kernel**
|details-split|

The kernel is given by:

.. math::

Expand Down Expand Up @@ -413,6 +422,9 @@ differentiable (as assumed by the RBF kernel) but at least once (:math:`\nu =

The flexibility of controlling the smoothness of the learned function via :math:`\nu`
allows adapting to the properties of the true underlying functional relation.

|details-end|

The prior and posterior of a GP resulting from a Matérn kernel are shown in
the following figure:

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