Defining kernels

To select another kernel, this just needs to be passed to the GP class during initialisation:

>>> gp = GeePea.GP(x,y,p,kf=GeePea.Matern32)

or alternatively added using the GP.set_pars method:

>>> gp.set_pars(kf=GeePea.Matern32)

This is important to do via the set_pars method as the number of parameters accepted by the kernel function and the kernel type is automatically set for built-in kernels. This would be ignored if the kernel function is set directly:

>>> gp.kf = some_kernel

For a list of built in kernel functions see kernels and toeplitz kernels.

It is also very easy to design your own kernels:

def MySqExponential(X,Y,theta,white_noise=False):
  # This is just a copy of GeePea.SqExponential

  #Calculate distance matrix with scaling - multiply each coord by sqrt(eta)
  D2 = GeePea.EuclideanDist2(X,Y,v=1./(np.array(theta[1:-1])))#

  #Calculate covariance matrix
  K = theta[0]**2 * np.exp( - 0.5 * D2 )

  #Add white noise
  if white_noise is True: K += np.identity(X[:,0].size) * (theta[-1]**2)

  return np.matrix(K)

# optional - add some attributes
MySqExponential.n_par = lambda D: D+2
MySqExponential.kernel_type = "Full"

Then the kernel can be added to the GP initialisation:

>>> gp = GeePea.GP(x,y,p,kf=MySqExponential)
# or later via:
>>> gp = GeePea.GP(x,y,p)
>>> gp.set_pars(kf=MySqExponential)

It is important that the function follows the same format, and accepts three input arguments X, Y and theta, plus one additional boolean argument, white_noise. X and Y are input matrices containing the arguments to the kernel, theta contains the parameters of the mean function, and if white_noise is True, then the white noise term is added along the diagonal of the covariance matrix. This format is required so that it can return arbitrary blocks of the covariance matrix for GP regression, however, usually it is called as:

>>> kf(X,X,white_noise=True)

in order to calculate the full covariance matrix of the input data and calculate the likelihood.

The attributes n_par and kernel_type are optional, and are used for the GP class to automatically set the number of parameters the kernel takes and the type of kernel. These can however be set directly via the GP class initialisation, via:

>>> gp = GeePea.GP(x,y,p,kf=GeePea.Matern32,n_hp=3,kernel_type='F')

The kernel_type is ‘Full’ by default, so it only needs to be set if the kernel is reset from a different type, or if Toeplitz/White kernels are defined by the user without a kernel_type attribute.