Sncf function

Nonparametric (cross-)correlation function for spatio-temporal data

Sncf is the function to estimate the nonparametric (cross-)correlation function using a smoothing spline as an equivalent kernel. The function requires multiple observations at each location (use spline.correlog otherwise).

Sncf(
  x,
  y,
  z,
  w = NULL,
  df = NULL,
  type = "boot",
  resamp = 1000,
  npoints = 300,
  save = FALSE,
  filter = FALSE,
  fw = 0,
  max.it = 25,
  xmax = FALSE,
  na.rm = FALSE,
  latlon = FALSE,
  circ = 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: matrix of dimension n x p representing p observation at each location.
• w: an optional second matrix of dimension n x p for species 2 (to estimate the spatial cross-correlation function).
• 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 for the estimator
• 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 semi-definiteness of the estimator. (more work may be needed on this.)
• fw: If filter is TRUE, it may be useful to truncate the function at some distance w 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.
• na.rm: If TRUE, NA's will be dealt with through pairwise deletion of missing values for each pair of time series -- it will dump if any one pair has less than two (temporally) overlapping observations.
• latlon: If TRUE, coordinates are latitude and longitude.
• circ: If TRUE, the observations are assumed to be angular (in radians), and circular correlation is used.
• quiet: If TRUE, the counter is suppressed during execution.

Returns

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

• $cbar: the regional average correlation.

• $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).

• $cbar.intercept: distance at which regional average correlation is reach.

• $predicted$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, cbar.intercept, cbar 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

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

The circ argument computes a circular version of the Pearson's product moment correlation (see cor2). This option is to calculate the 'nonparametric phase coherence function' (Grenfell et al. 2001)

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")
  )
# multivariate nonparametric covariance function
fit1 <- Sncf(x = x, y = y, z = z, resamp = 0)
## Not run: plot.Sncf(fit1)
summary(fit1)

# multivariate nonparametric cross-covariance function
fit2 <- Sncf(x = x, y = y, z = z, w = w, resamp = 0)
## Not run: plot(fit2)
summary(fit2)

References

Hall, P. and Patil, P. (1994) Properties of nonparametric estimators of autocovariance for stationary random fields. Probability Theory and Related Fields, 99:399-424. doi:10.1007/BF01199899

Hall, P., Fisher, N.I. and Hoffmann, B. (1994) On the nonparametric estimation of covariance functions. Annals of Statistics, 22:2115-2134 doi:10.1214/aos/1176325774.

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

Bjornstad, O.N., Ims, R.A. and Lambin, X. (1999) Spatial population dynamics: Analysing patterns and processes of population synchrony. Trends in Ecology and Evolution, 11:427-431 doi:10.1016/S0169-5347(99)01677-8.

Bjornstad, O. N., and J. Bascompte. (2001) Synchrony and second order spatial correlation in host-parasitoid systems. Journal of Animal Ecology 70:924-933 doi:10.1046/j.0021-8790.2001.00560.x.

Grenfell, B.T., Bjornstad, O.N., & Kappey, J. (2001) Travelling waves and spatial hierarchies in measles epidemics. Nature 414:716-723. doi:10.1038/414716a

See Also

Sncf2D, Sncf.srf

Author(s)

Ottar N. Bjornstad onb1@psu.edu