R: Defines and computes covariance functions

covariance {SpatialExtremes} | R Documentation

## Defines and computes covariance functions

### Description

This function defines and computes several covariance function either from a fitted “max-stable” model; either by specifying directly the covariance parameters.

### Usage

    covariance(fitted, nugget, sill, range, smooth, smooth2 = NULL, cov.mod =
    "whitmat", plot = TRUE, dist, xlab, ylab, col = 1, ...)

### Arguments

`fitted` | An object of class “maxstab”. Most often this will be the output of the `fitmaxstab` function. May be missing if `sill`, `range`, `smooth` and `cov.mod` are given.
`nugget,sill,range,smooth,smooth2` | The nugget, sill, scale and smooth parameters for the covariance function. May be missing if `fitted` is given.
`cov.mod` | Character string. The name of the covariance model. Must be one of "whitmat", "cauchy", "powexp", "bessel" or "caugen" for the Whittle-Matern, Cauchy, Powered Exponential, Bessel and Generalized Cauchy models. May be missing if `fitted` is given.
`plot` | Logical. If `TRUE` (default) the covariance function is plotted.
`dist` | A numeric vector corresponding to the distance at which the covariance function should be evaluated. May be missing.
`xlab,ylab` | The x-axis and y-axis labels. May be missing.
`col` | The color to be used for the plot.
`...` | Several option to be passed to the `plot` function.

### Details

Currently, four covariance functions are defined: the Whittle-Matern, powered exponential (also known as "stable"), Cauchy and Bessel models. These covariance functions are defined as follows for h > 0

Whittle-Matern

  γ(h) = σ 2^(1-κ) / Γ(κ) (h/λ)^κ K_κ(h / λ)

Powered Exponential

  γ(h) = σ exp[-(h/λ)^κ]

Cauchy

  γ(h) = σ [1 + (h/λ)^2]^(-κ)

Bessel

  γ(h) = σ (2 λ / h)^(κ) Gamma(κ + 1) J_κ(h / λ)

Generalized Cauchy

  γ(h) = σ [1 + (h / λ)^κ_2]^(-κ / κ_2)

where σ, λ and κ are the sill, the range and shape parameters, Γ is the gamma function, K_κ and J_κ are both modified Bessel functions of order κ. In addition a nugget effect can be set that is there is a jump at the origin, i.e., γ(o) = ν + σ, where ν is the nugget effect.

### Value

This function returns the covariance function. Eventually, if `dist` is given, the covariance function is computed for each distance given by `dist`. If `plot = TRUE`, the covariance function is plotted.

### Author(s)

Mathieu Ribatet

### Examples

    ## 1- Calling covariance using fixed covariance parameters
    covariance(nugget = 0, sill = 1, range = 1, smooth = 0.5, cov.mod = "whitmat")
    covariance(nugget = 0, sill = 0.5, range = 1, smooth = 0.5, cov.mod = "whitmat",
      dist = seq(0,5, 0.2), plot = FALSE)

    ## 2- Calling covariance from a fitted model
    ##Define the coordinate of each location
    n.site <- 30
    locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
    colnames(locations) <- c("lon", "lat")

    ##Simulate a max-stable process - with unit Frechet margins
    data <- rmaxstab(30, locations, cov.mod = "whitmat", nugget = 0, range =
    3, smooth = 1)

    ##Fit a max-stable model
    fitted <- fitmaxstab(data, locations, "whitmat", nugget = 0)
    covariance(fitted, ylim = c(0, 1))
    covariance(nugget = 0, sill = 1, range = 3, smooth = 1, cov.mod = "whitmat", add =
    TRUE, col = 3)
    title("Whittle-Matern covariance function")
    legend("topright", c("Theo.", "Fitted"), lty = 1, col = c(3,1), inset =
    .05)

[Package SpatialExtremes version 2.0-5 Index]