Constant kernel:
Gabor kernel:
Linear kernel with ARD:
where is a diagnal matrix consist of length scale for each dimension .
Linear kernel:
Matern kernel:
with for , for and for .
where r is the distance and is a diagnal matrix consist of length scale for each dimension .
Independent noise kernel:
Periodic kernel:
Piecewise polynomial kernel:
with
where D is the dimension of input and is a diagnal matrix consist of length scale for each dimension and f is a function depending on v. See gpml matlab v3.4.
Polynomial kernel:
Squared exponential kernel:
Squared exponential kernel with ARD:
where is a diagnal matrix consist of length scale for each dimension .
Squared exponential kernel with unit magnitude:
Rational quadratic kernel:
Rational quadratic kernel with ARD:
where is a diagnal matrix consist of length scale for each dimension .
Constant kernel. hyp = [ log_sigma ]
Parameters:  log_sigma – signal deviation. 

Covariance function to be used together with the FITC approximation. The function allows for more than one output argument and does not respect the interface of a proper covariance function. Instead of outputing the full covariance, it returns crosscovariances between the inputs x, z and the inducing inputs xu as needed by infFITC
Gabor covariance function with length scale ell and period p. The covariance function is parameterized as:
k(x,z) = h( xz ) with h(t) = exp(t^2/(2*ell^2))*cos(2*pi*t/p).
The hyperparameters are:
hyp = [log(ell), log(p)]
Note that SM covariance implements a weighted sum of Gabor covariance functions, but using an alternative (spectral) parameterization.
Parameters: 


This is a base class of Kernel functions there is no computation in this class, it just defines rules about a kernel class should have each covariance function will inherit it and implement its own behaviour
Check validity of inputs for the method getCovMatrix()
Parameters: 


Check validity of inputs for the method getDerMatrix()
Parameters: 


Covariance function to be used together with the FITC approximation. Setting FITC gp model will implicitly call this method.
Returns:  an instance of FITCOfKernel 

Return the specific covariance matrix according to input mode
Parameters: 


Returns:  the corresponding covariance matrix 
Compute derivatives wrt. hyperparameters according to input mode
Parameters: 


Returns:  the corresponding derivative matrix 
Linear covariance function with Automatic Relevance Detemination. hyp = log_ell_list
Parameters: 


Linear kernel. hyp = [ log_sigma ].
Parameters:  log_sigma – signal deviation. 

Matern covariance function with nu = d/2 and isotropic distance measure. For d=1 the function is also known as the exponential covariance function or the OrnsteinUhlenbeck covariance in 1d. d will be rounded to 1, 3, 5 or 7 hyp = [ log_ell, log_sigma]
Parameters: 


Independent covariance function, i.e “white noise”, with specified variance. Normally NOT used anymore since noise is now added in liklihood. hyp = [ log_sigma ]
Parameters:  log_sigma – signal deviation. 

Stationary kernel for a smooth periodic function. hyp = [ log_ell, log_p, log_sigma]
Parameters: 


Piecewise polynomial kernel with compact support. hyp = [log_ell, log_sigma]
Parameters: 


Polynomial covariance function. hyp = [ log_c, log_sigma ]
Parameters: 


Precomputed kernel matrix. No hyperparameters and thus nothing will be optimised.
Parameters: 


Squared Exponential kernel with isotropic distance measure. hyp = [log_ell, log_sigma]
Parameters: 


Squared Exponential kernel with Automatic Relevance Determination. hyp = log_ell_list + [log_sigma]
Parameters: 


Squared Exponential kernel with isotropic distance measure with unit magnitude. i.e signal variance is always 1. hyp = [ log_ell ]
Parameters:  log_ell – characteristic length scale. 

Rational Quadratic covariance function with isotropic distance measure. hyp = [ log_ell, log_sigma, log_alpha ]
Parameters: 


Rational Quadratic covariance function with Automatic Relevance Detemination (ARD) distance measure. hyp = log_ell_list + [ log_sigma, log_alpha ]
Parameters: 


Gaussian Spectral Mixture covariance function. The covariance function is parameterized as:
k(x^p,x^q) = w’*prod( exp(2*pi^2*d^2*v)*cos(2*pi*d*m), 2), d = x^p,x^q
where m(DxQ), v(DxQ) are the means and variances of the spectral mixture components and w are the mixture weights. The hyperparameters are:
hyp = [ log(w), log(m(:)), log(sqrt(v(:))) ]
Copyright (c) by Andrew Gordon Wilson and Hannes Nickisch, 20131009.
For more details, see 1) Gaussian Process Kernels for Pattern Discovery and Extrapolation, ICML, 2013, by Andrew Gordon Wilson and Ryan Prescott Adams. 2) GPatt: Fast Multidimensional Pattern Extrapolation with Gaussian Processes, arXiv 1310.5288, 2013, by Andrew Gordon Wilson, Elad Gilboa, Arye Nehorai and John P. Cunningham, and http://mlg.eng.cam.ac.uk/andrew/pattern
Parameters: 


Constant mean function. hyp = [c]
Parameters:  c – constant value for mean 

Linear mean function. self.hyp = alpha_list
Parameters:  D – dimension of training data. Set if you want default alpha, which is 0.5 for each dimension. 

Alpha_list:  scalar alpha for each dimension 
Error function or cumulative Gaussian likelihood function for binary classification or probit regression.
Exact inference for a GP with Gaussian likelihood. Compute a parametrization of the posterior, the negative log marginal likelihood and its derivatives w.r.t. the hyperparameters.
Expectation Propagation approximation to the posterior Gaussian Process.
FITC approximation to the posterior Gaussian process. The function is equivalent to infExact with the covariance function: Kt = Q + G; G = diag(g); g = diag(KQ); Q = Ku’ * inv(Quu) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu, snu2 = sn2/1e6 is the noise of the inducing inputs and Quu = Kuu + snu2*eye(nu).
FITCEP approximation to the posterior Gaussian process. The function is equivalent to infEP with the covariance function: Kt = Q + G; G = diag(g); g = diag(KQ); Q = Ku’ * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise’s standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e6. For details, see The Generalized FITC Approximation, Andrew NaishGuzman and Sean Holden, NIPS, 2007.
FITCLaplace approximation to the posterior Gaussian process. The function is equivalent to Laplace with the covariance function: Kt = Q + G; G = diag(g); g = diag(KQ); Q = Ku’ * inv(Kuu + snu2 * eye(nu)) * Ku; where Ku and Kuu are covariances w.r.t. to inducing inputs xu and snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard deviation of the inducing inputs snu to be a one per mil of the measurement noise’s standard deviation sn. In case of a likelihood without noise parameter sn2, we simply use snu2 = 1e6.