cov.wendland {GeneralizedWendland}

Generalized Wendland Covariance Function

Description

A fully parametrized generalized Wendland covariance function for use in geostatistical modeling, as well as multiple methods of obtaining computationally inexpensive approximations.

\rho_{\beta,\kappa,\mu} = \begin{cases}\sigma + \theta \quad 0 \leq r < \epsilon \\ \frac{\sigma}{B(1+2\kappa,\mu)} \int_r^1 (u^2-r^2)^\kappa (1-u)^{\mu-1} du \quad \epsilon \leq r < 1 \\ 0 \quad 1 \leq r \end{cases}

where r=h/\beta

Usage

cov.wendland(h, theta, ..., cov.args = list())

Arguments

h

A numeric vector, matrix, or spam object storing distances.

theta

Numeric vector \vec{\theta}=(\beta, \sigma, \kappa, \mu, \theta) storing parameters.

...

Other arguments.

cov.args

Named list of arguments. See Details.

Details

Using the list cov.args, users can provide the following additional arguments:

numint.abstol (default: 1e-3)

Absolute tolerance for numerical integration.

numint.reltol (default: 1e-3)

Relative tolerance for numerical integration.

numint.qag_key (default: 0)

Method to use in QAG integration (Values 1 - 6)

numint.subintervals (default: 0)

Number of subintervals to use in QAG/QAGS integration.

interp.method (default: 'none')

Method to use for covariance interpolation. Valid methods are 'none', 'linear', 'polynomial', and 'cspline'.

interp.num_support (default: 0)

Number of support points to use for covariance interpolation.

cov.reparameterize (default: TRUE)

Whether to apply the reparameterization \mu=\frac{1+d}{2} + \kappa + \nu, where \nu takes the place of \mu in input vector \vec{\theta}. This allows users to use box constraints in maximum likelihood estimation, as the covariance function is valid for \nu \in [0,\infty) rather than \mu \in [\frac{1+d}{2} + \kappa,\infty).

cov.eps (default: .Machine$double.eps^0.5)

The threshold distance \epsilon below which the function will return \sigma + \theta.

cov.d_value (default: 2)

Dimensionality of space in which measurements were taken. This only takes effect if cov.reparameterize is TRUE.

Value

Returns an object of the same type as input object h which stores the computed covariance values, i.e. a spam object if input h was also a spam object.

Author(s)

Thomas Caspar Fischer

References

Moreno Bevilacqua and Tarik Faouzi and Reinhard Furrer and Emilio Porcu (2019) Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics, Annals of Statistics, 47(2), 828–856.

Examples

h <- seq(0, 1, 0.01)
plot(0, type = "n", xlab = "Distance", ylab = "Covariance",
     xlim = c(0, 1), ylim = c(0,1))

theta <- c(range=1, sill=1, kappa=1, mu=0, nugget=0)
cov.args <- list()
lines(x = h, y = cov.wendland(h, theta, cov.args = cov.args),
      lwd = 2)

theta <- c(range=1, sill=1, kappa=1, mu=0, nugget=0)
cov.args <- list(cov.reparametrize = FALSE, cov.d_value = 2)
theta[4] <- (1 + cov.args[[2]])/2 + theta[3] + theta[4]
lines(x = h, y = cov.wendland(h, theta, cov.args = cov.args),
      col = "red", lty = 3, lwd = 3.5)

theta <- c(range=0.5, sill=1, kappa=1, mu=0, nugget=0)
cov.args <- list(interp.method="cspline", interp.num_support=100)
lines(x = h, y = cov.wendland(h, theta, cov.args = cov.args),
      col = "green", lwd = 2)

legend("topright", legend = c("Default", "No reparameterization",
                              "Cubic spline interpolation"),
       col = c(1, 2, 3), lty = c(1,3,1), lwd = c(2, 3.5, 2))