covmodel |
a list of two strings: family, form, where family indicates the family of covariance functions
including the Confluent Hypergeometric class, the Matérn class, the Cauchy class, the powered-exponential class. form indicates the
specific form of covariance structures including the isotropic form, tensor form, automatic relevance determination form.
- family
-
- CH
The Confluent Hypergeometric correlation function is given by
C(h) = \frac{\Gamma(\nu+\alpha)}{\Gamma(\nu)}
\mathcal{U}\left(\alpha, 1-\nu, \left(\frac{h}{\beta}\right)^2\right),
where \alpha is the tail decay parameter. \beta is the range parameter.
\nu is the smoothness parameter. \mathcal{U}(\cdot) is the confluent hypergeometric
function of the second kind. For details about this covariance,
see Ma and Bhadra (2023; doi:10.1080/01621459.2022.2027775).
- cauchy
The generalized Cauchy covariance is given by
C(h) = \left\{ 1 + \left( \frac{h}{\phi} \right)^{\nu}
\right\}^{-\alpha/\nu},
where \phi is the range parameter. \alpha is the tail decay parameter.
\nu is the smoothness parameter with default value at 2.
- matern
The Matérn correlation function is given by
C(h)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{h}{\phi} \right)^{\nu}
\mathcal{K}_{\nu}\left( \frac{h}{\phi} \right),
where \phi is the range parameter. \nu is the smoothness parameter.
\mathcal{K}_{\nu}(\cdot) is the modified Bessel function of the second kind of order \nu.
- exp
This is the Matérn correlation with \nu=0.5. This covariance should be specified as matern with smoothness parameter \nu=0.5.
- matern_3_2
This is the Matérn correlation with \nu=1.5.
This covariance should be specified as matern with smoothness parameter \nu=1.5.
- matern_5_2
This is the Matérn correlation with \nu=2.5.
This covariance should be specified as matern with smoothness parameter \nu=2.5.
- powexp
The powered-exponential correlation function is given by
C(h)=\exp\left\{-\left(\frac{h}{\phi}\right)^{\nu}\right\},
where \phi is the range parameter. \nu is the smoothness parameter.
- gauss
The Gaussian correlation function is given by
C(h)=\exp\left(-\frac{h^2}{\phi^2}\right),
where \phi is the range parameter.
- form
-
- isotropic
This indicates the isotropic form of covariance functions. That is,
C(\mathbf{h}) = C^0(\|\mathbf{h}\|; \boldsymbol \theta),
where \| \mathbf{h}\| denotes the
Euclidean distance or the great circle distance for data on sphere. C^0(\cdot) denotes
any isotropic covariance family specified in family.
- tensor
This indicates the tensor product of correlation functions. That is,
C(\mathbf{h}) = \prod_{i=1}^d C^0(|h_i|; \boldsymbol \theta_i),
where d is the dimension of input space. h_i is the distance along the ith input dimension. This type of covariance structure has been often used in Gaussian process emulation for computer experiments.
- ARD
This indicates the automatic relevance determination form. That is,
C(\mathbf{h}) = C^0\left(\sqrt{\sum_{i=1}^d\frac{h_i^2}{\phi^2_i}}; \boldsymbol \theta \right),
where \phi_i denotes the range parameter along the ith input dimension.
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