spline.correlog function

Uni- and multivariate spline correlograms

spline.correlog is the function to estimate the spline (cross-)correlogram from spatial data. Either univariate or multivariate (time seres) for each site can be used.

spline.correlog(
  x,
  y,
  z,
  w = NULL,
  df = NULL,
  type = "boot",
  resamp = 1000,
  npoints = 300,
  save = FALSE,
  filter = FALSE,
  fw = 0,
  max.it = 25,
  xmax = FALSE,
  latlon = FALSE,
  na.rm = FALSE,
  quiet = FALSE
)

Arguments

• x: vector of length n representing the x coordinates (or longitude; see latlon).
• y: vector of length n representing the y coordinates (or latitude).
• z: vector of length n or matrix of dimension n x p representing p observation at each location.
• w: an optional second variable with identical dimension to z (to estimate cross-correlograms).
• df: degrees of freedom for the spline. Default is sqrt(n).
• type: takes the value "boot" (default) to generate a bootstrap distribution or "perm" to generate a null distribution
• resamp: the number of resamples for the bootstrap or the null distribution.
• npoints: the number of points at which to save the value for the spline function (and confidence envelope / null distribution).
• save: If TRUE, the whole matrix of output from the resampling is saved (a resamp x npoints dimensional matrix).
• filter: If TRUE, the Fourier filter method of Hall and coworkers is applied to ensure positive semidefiniteness of the estimator.
• fw: If filter is TRUE, it may be useful to truncate the function at some distance fw sets the truncation distance. When set to zero, no truncation is done.
• max.it: the maximum iteration for the Newton method used to estimate the intercepts.
• xmax: If FALSE, the max observed in the data is used. Otherwise all distances greater than xmax is omitted.
• latlon: If TRUE, coordinates are latitude and longitude.
• na.rm: If TRUE, NA's will be dealt with through pairwise deletion of missing values.
• quiet: If TRUE, the counter is suppressed during execution.

Returns

An object of class "spline.correlog" is returned, consisting of the following components: - real: the list of estimates from the data.

• $x.intercept: the lowest value at which the function is = 0. If correlation is initially negative, the distance is given as negative.

• $e.intercept: the lowest value at which the function 1/e.

• $y.intercept: the extrapolated value at x=0 (nugget).

• $predicted$x: the x-axes for the fitted covariance function.

• $predcited$y: the values for the covariance function.

• boot: a list with the analogous output from the bootstrap or null distribution.

• $summary: gives the full vector of output for the x.intercept, y.intercept, e.intercept, and a quantile summary for the resampling distribution.

• $boot: If save=TRUE, the full raw matrices from the resampling is saved.

• max.distance: the maximum spatial distance considered.

Details

If observations are univariate the spline (cross-)correlogram represents the generalization of the spatial (cross-)correlogram; if observations are multivariate the spline (cross-)correlogram represents the generalization of the Mantel (cross-)correlogram.

The spline (cross-)correlogram differs from the spatial correlogram (and Mantel correlogram) in that it estimated spatial dependence as a continuous functions of distance (rather than binning into distance classes). The spline correlogram differs from the nonparametric (cross-)correlation function in that the zero-correlation reference line in the former corresponds to the region-wide correlation reference line in the latter. The x-intercept in the spline correlogram is the distance at which object are no more similar than that expected by-chance-alone across the region.

Missing values are allowed -- values are assumed missing at random.

Examples

# first generate some sample data
x <- expand.grid(1:20, 1:5)[, 1]
y <- expand.grid(1:20, 1:5)[, 2]

# z data from an exponential random field
z <- cbind(
  rmvn.spa(x = x, y = y, p = 2, method = "exp"), 
  rmvn.spa(x = x, y = y, p = 2, method = "exp")
  )

# w data from a gaussian random field
w <- cbind(
  rmvn.spa(x = x, y = y, p = 2, method = "gaus"), 
  rmvn.spa(x = x, y = y, p = 2, method = "gaus")
  )

# univariate spline correlogram
fit1 <- spline.correlog(x = x, y = y, z = z[, 1], resamp = 100)
## Not run: plot.spline.correlog(fit1)
summary(fit1)

# multivariate spline correlogram
fit2 <- spline.correlog(x = x, y = y, z = z, resamp = 100)
## Not run: plot.spline.correlog(fit2)
summary(fit2)

# multivariate spline cross-correlogram
fit3 <- spline.correlog(x = x, y = y, z = z, w = w, resamp = 100)
## Not run: plot.spline.correlog(fit3)
summary(fit3)

References

Bjornstad, O.N. & Falck, W. (2001) Nonparametric spatial covariance functions: estimation and testing. Environmental and Ecological Statistics, 8:53-70. doi:10.1023/A:1009601932481

See Also

summary.spline.correlog, plot.spline.correlog, Sncf, spline.correlog2D, correlog

Author(s)

Ottar N. Bjornstad onb1@psu.edu