par |
a numerical vector, with which numerical optimization routine such as optim can be
carried out directly. When the confluent Hypergeometric class is used, it is used to hold values
for range, tail, nugget, and nu if the smoothness parameter is estimated.
When the Matérn class or powered-exponential class is used, it is used to hold values
for range, nugget, and nu if the smoothness parameter is estimated.
The order of the parameter values in par cannot be changed. For tensor or ARD form correlation
functions, range and tail becomes a vector.
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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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