deriv_kernel: A wraper to construct the derivative of correlation matrix with respect to correlation parameters

Description

This function wraps existing built-in routines to construct the derivative of correlation matrix with respect to correlation parameters.

Usage

deriv_kernel(d, range, tail, nu, covmodel)

Value

a list of matrices

Arguments

d

a matrix or a list of distances returned from distance.

range

a vector of range parameters

tail

a vector of tail decay parameters

nu

a vector of smoothness parameters

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 (2019) at https://arxiv.org/abs/1911.05865.

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 $\nu$.

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 $i$th 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 $i$th input dimension.

Author

Pulong Ma mpulong@gmail.com

Examples

Run this code

input = seq(0,1,length=10)
d = distance(input,input,type="isotropic",dtype="Euclidean")
dR = deriv_kernel(d,range=0.5,tail=0.2,nu=2.5,
         covmodel=list(family="CH",form="isotropic"))