blockCor: Correlation Matrix without or with Derivatives

In gek: Gradient-Enhanced Kriging

Correlation Matrix without or with Derivatives

Description

Calculation of a correlation matrix with or without derivatives according to the specification of a correlation structure.

Usage

blockCor(x, ...)

## Default S3 method:
blockCor(x, theta, covtype = c("matern5_2", "matern3_2", "gaussian"), 
	derivatives = FALSE, ...)
## S3 method for class 'gekm'
blockCor(x, ...)

Arguments

x	a numeric matrix or an object of class gekm.
theta	numeric vector of length d for the hyperparameters.
covtype	character specifying the correlation function to be used. Must be one of "matern5_2", "matern3_2" or "gaussian". See ‘Details’.
derivatives	logical, if TRUE the first and second partial derivatives of the correlation matrix are calculated, otherwise not.
...	further arguments passed to or from other methods.

Details

The correlation matrix with derivatives is defined as a block matrix of the form

\begin{pmatrix} K & R^{\rm T} \\ R & S \end{pmatrix}.

Three correlation functions are currently implemented in blockCor:

• Matérn 5/2 kernel with covtype = "matern5_2":

  \phi(x, x'; \theta) = \prod_{k = 1}^d \left(1 + \frac{\sqrt{5}|x_k - x_k'|}{\theta_k} + \frac{5 |x_k - x_k'|^2}{3 \theta_k^2}\right) \exp\left( -\frac{\sqrt{5}|x_k - x_k'|}{\theta_k}\right)

• Matérn 3/2 kernel with covtype = "matern3_2":

  \phi(x, x'; \theta) = \prod_{k = 1}^d \left(1 + \frac{\sqrt{3} |x_k - x_k'|}{\theta_k} \right) \exp\left( -\frac{\sqrt{3} |x_k - x_k'|}{\theta_k}\right)

• Gaussian kernel with covtype = "gaussian":

  \phi(x, x'; \theta) = \prod_{k = 1}^d \exp\left( -\frac{(x_k - x_k')^2}{2\theta_k^2}\right)

Value

blockCor returns a list with the following components:

K	The correlation matrix without derivatives.
R	If derivatives = TRUE, the correlation matrix with first partial derivatives, otherwise NULL.
S	If derivatives = TRUE, the correlation matrix with second partial derivatives, otherwise NULL.

The components of the list can be combined in the following form to constructed the complete correlation matrix with derivatives: cbind(rbind(K, R), rbind(t(R), S)).

Author(s)

Carmen van Meegen

References

Koehler, J. and Owen, A. (1996). Computer Experiments. In Ghosh, S. and Rao, C. (eds.), Design and Analysis of Experiments, volume 13 of Handbook of Statistics, pp. 261–308. Elsevier Science. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0169-7161(96)13011-X")}.

Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. The MIT Press. https://gaussianprocess.org/gpml/.

Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics. Springer-Verlag. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-1-4612-1494-6")}.

See Also

blockChol for block Cholesky decomposition.

tangents for drawing tangent lines.

Examples

# Some examples of correlation matrices:

x <- matrix(1:4, ncol = 1)
blockCor(x, theta = 1)
blockCor(x, theta = 1, covtype = "gaussian")
blockCor(x, theta = 1, covtype = "gaussian", derivatives = TRUE)
blockCor(x, theta = 1, covtype = "matern3_2", derivatives = TRUE)

res <- blockCor(x, theta = 2, covtype = "gaussian", derivatives = TRUE)
cbind(rbind(res$K, res$R), rbind(t(res$R), res$S))


# Illustration of correlation functions and their derivatives:

x <- seq(0, 1, length = 100)
X <- matrix(x, ncol = 1)
gaussian <- blockCor(X, theta = 0.25, covtype = "gaussian", derivatives = TRUE)
matern5_2 <- blockCor(X, theta = 0.25, covtype = "matern5_2", derivatives = TRUE)
matern3_2 <- blockCor(X, theta = 0.25, covtype = "matern3_2", derivatives = TRUE)

# Correlation functions and first partial derivatives:
index <- c(10, 20, 40, 80)
par(mar = c(5.1, 5.1, 4.1, 2.1))

# Matern 3/2
plot(x, matern3_2$K[1, ], type = "l", xlab = expression(group("|", x - x*minute, "|")),
	ylab = expression(phi(x, x*minute, theta == 0.25)), lwd = 2)
tangents(x[index], matern3_2$K[index, 1], matern3_2$R[index, 1], 
	length = 0.15, lwd = 2, col = 2)
points(x[index], matern3_2$K[index, 1], pch = 16)

# Matern 5/2
lines(x, matern5_2$K[1, ], lwd = 2, col = 3)
tangents(x[index], matern5_2$K[index, 1], matern5_2$R[index, 1], 
	length = 0.15, lwd = 2, col = 2)
points(x[index], matern5_2$K[index, 1], pch = 16)

# Gaussian
lines(x, gaussian$K[1, ], lwd = 2, col = 4)
tangents(x[index], gaussian$K[index, 1], gaussian$R[index, 1], 
	length = 0.15, lwd = 2, col = 2)
points(x[index], gaussian$K[index, 1], pch = 16)

legend("topright", lty = 1, lwd = 2, col = c(1, 3, 4), bty = "n",
	legend = c("Matern 3/2", "Matern 5/2", "Gaussian"))


# First and second partial derivatives of correlation functions:
index <- c(5, 10, 20, 40, 80)
par(mar = c(5.1, 6.1, 4.1, 2.1))

# Gaussian
plot(x, matern3_2$R[1, ], type = "l", xlab = expression(group("|", x - x*minute, "|")),
	ylab = expression(frac(partialdiff * phi(x, x*minute, theta == 0.25), 
		partialdiff * x * minute)), lwd = 2)
tangents(x[index], matern3_2$R[1, index], matern3_2$S[index, 1], 
	length = 0.15, lwd = 2, col = 2)
points(x[index], matern3_2$R[1, index], pch = 16)

# Matern 5/2
lines(x, matern5_2$R[1, ], lwd = 2, col = 3)
tangents(x[index], matern5_2$R[1, index], matern5_2$S[index, 1], 
	length = 0.15, lwd = 2, col = 2)
points(x[index], matern5_2$R[1, index], pch = 16)

# Matern 3/2
lines(x, gaussian$R[1, ], lwd = 2, col = 4)
tangents(x[index], gaussian$R[1, index], gaussian$S[index, 1], 
	length = 0.15, lwd = 2, col = 2)
points(x[index], gaussian$R[1, index], pch = 16)

legend("topright", lty = 1, lwd = 2, col = c(1, 3, 4), bty = "n",
	legend = c("Matern 3/2", "Matern 5/2", "Gaussian"))

gek documentation built on April 4, 2025, 12:35 a.m.