.. _MaternKernel1:


Matérn kernel
-------------


   

 

  

 

Gaussian processes model the correlation (or covariance) between  two function values :math:`f(x), f(x')` by means of a covariance function  :math:`k(x,x') = k(||x-x'||)`, also called kernel. The kernel is monotonically decreasing for increasing distance :math:`d = ||x-x'||` such that far apart function values are uncorrelated while close-by function values are strongly correlated.      

The distance :math:`d` between function values is defined as the  scaled Euclidean distance       

.. math::   d = ||x-x'||=\sqrt{\sum_{i=1}^{D}\frac{(x_i−x_i^{'})^{2}}{l_i^{2}}},      

where the hyperparameters :math:`l_1,\dots,l_D` determine the characteristic  length scales at which the covariance between separated function values  becomes negligible.      

The Matérn covariance function is defined as      

.. math::      

  k_{\nu}(x, x') = \frac{\sigma_0^2}{\Gamma(\nu)2^{\nu-1}}   \left[\sqrt{2\nu} d \right]^\nu    K_\nu\left( \sqrt{2\nu} d\right),      

where  :math:`\sigma_0` is a hyperparameter that determines the  standard deviation of possible function values,   :math:`K_\nu(\cdot)` is a modified Bessel function,  and :math:`\Gamma(\cdot)` is the gamma function.

.. _ActiveLearning.Interpolation.matern.matern_order:


matern_order (int)
""""""""""""""""""
    
   
   For half-integer :math:`\nu` the Matérn functions  simplify e.g. to  
   
   - :math:`k_{1/2}(x,x′)= \sigma^2 \exp(−d)` ,  
   
   - :math:`k_{3/2}(x,x′)= \sigma^2 (1+3-\sqrt{3}d)\exp(-\sqrt{3}d)` ,  
   
   - :math:`k_{5/2}(x,x′)= \sigma^2 (1+5-\sqrt{5}d + \frac{5}{3}d^2)\exp(-\sqrt{5}d)` .  
   
   A Gaussian process with Matérn covariance is  :math:`\lceil \nu \rceil` times differentiable in the  mean-square sense. The ``matern_order`` is defined as :math:`m = 2\nu`. Hence, :math:`m=5` is best suited for describing two-times differentiable  functions, :math:`m=3` for one-times differentiable functions, and :math:`m=1` for functions, that are not differentiable. 

   Default: ``5``
   Choices: ``1``, ``3``, ``5``.

   .. note:: For ``matern_order=1`` it is not possible to add  derivative information.