R: Confidence intervals and tests based on smoothing an...

sm.variogram {sm} | R Documentation

## Confidence intervals and tests based on smoothing an empirical variogram.

### Description

This function constructs an empirical variogram, using the robust form of construction based on square-root absolute value differences of the data. Flexible regression is used to assess a variety of questions about the structure of the data used to construct the variogram, including independence, isotropy and stationarity. Confidence bands for the underlying variogram, and reference bands for the independence, isotropy and stationarity models, can also be constructed under the assumption that the errors in the data are approximately normally distributed.

### Usage

    sm.variogram(x, y, h, df.se = "automatic", max.dist = NA, original.scale = TRUE,
                          varmat = FALSE, ...)

### Arguments

`x` | a vector or two-column matrix of spatial location values.
`y` | a vector of responses observed at the spatial locations.
`h` | a smoothing parameter to be used on the distance scale. A normal kernel function is used and `h` is its standard deviation. However, if this argument is omitted `h` will be selected by an approximate degrees of freedom criterion, controlled by the `df` parameter. See `sm.options` for details.
`df.se` | the degrees of freedom used when smoothing the empirical variogram to estimate standard errors. The default value of "automatic" selects the degrees of smoothing described in the Bowman and Crujeiras (2013) reference below.
`max.dist` | this can be used to constrain the distances used in constructing the variogram. The default is to use all distances.
`original.scale` | a logical value which determines whether the plots are constructed on the original variogram scale (the default) or on the square-root absolute value scale on which the calculations are performed.
`varmat` | a logical value which determines whether the variance matrix of the estimated variogram is returned.
`...` | other optional parameters are passed to the `sm.options` function, through a mechanism which limits their effect only to this call of the function. An important parameter here is `model` which, for `sm.variogram`, can be set to `"none"`, `"independent"`, `"isotropic"` or `"stationary"`. Other relevant parameters are `add`, `eval.points`, `ngrid`, `se`, `xlab`, `ylab`, `xlim`, `ylim`, `lty`; see the documentation of `sm.options` for their description. See the details section below for a discussion of the `display` and `se` parameters in this setting.

### Details

The reference below describes the statistical methods used in the function.

Note that, apart from the simple case of the indpendence model, the calculations required are extensive and so the function can be slow.

The `display` argument has a special meaning for this function. Its default value is `"binned"`, which plots the binned version of the empirical variogram. As usual, the value `"none"` will suppress the graphical display. Any other value will lead to a plot of the individual differences between all observations. This will lead to a very large number of plotted points, unless the dataset is small.

### Value

A list with the following components:

``sqrtdiff`, `distance`` | the raw differences and distances
``sqrtdiff.mean`, `distance.mean`` | the binned differences and distances
``weights`` | the frequencies of the bins
``estimate`` | the values of the estimate at the evaluation points
``eval.points`` | the evaluation points
``h`` | the value of the smoothing parameter used
``ibin`` | an indicator of the bin in which the distance between each pair of observations was placed
``ipair`` | the indices of the original observations used to construct each pair

The suitability of a particular model can be assessed by setting the `model` argument, in which case the following components may also be returned, determined by the arguments passed in ... or the settings in `sm.options`.

``p`` | the p-value of the test
``se`` | the standard errors of the binned values (if the argument `se` was set to `TRUE`)
``se.band`` | when an independence model is examined, this gives the standard error of the difference between the smooth estimate and the mean of all the data points, if a reference band has been requested
``V`` | the variance matrix of the binned variogram. When `model` is set to `"isotropic"` or `"stationary"`, the variance matrix is computed under those assumptions.
``sdiff`` | the standardised difference between the estiamte of the variogram and the reference model, evaluated at `eval.points`
``levels`` | the levels of standarised difference at which contours are drawn in the case of `model = "isotropy"`.

### Side Effects

a plot on the current graphical device is produced, unless the option `display="none"` is set.

### References

Diblasi, A. and Bowman, A.W. (2001). On the use of the variogram for checking independence in a Gaussian spatial process. Biometrics, 57, 211-218.

Bowman, A.W. and Crujeiras, R.M. (2013). Inference for variograms. Computational Statistics and Data Analysis, 66, 19-31.

### See Also

`sm.regression`, `sm.options`

### Examples

    with(coalash, {
       Position <- cbind(East, North)
       sm.options(list(df = 6, se = TRUE))

       par(mfrow=c(2,2))
       sm.variogram(Position, Percent, original.scale = FALSE, se = FALSE)
       sm.variogram(Position, Percent, original.scale = FALSE)
       sm.variogram(Position, Percent, original.scale = FALSE, model = "independent")
       sm.variogram(East,     Percent, original.scale = FALSE, model = "independent")
       par(mfrow=c(1,1))
    })

    ## Not run:
    # Comparison of Co in March and September

    with(mosses, {

       nbins <- 12
       vgm.m <- sm.variogram(loc.m, Co.m, nbins = nbins, original.scale = TRUE,
                            ylim = c(0, 1.5))
       vgm.s <- sm.variogram(loc.s, Co.s, nbins = nbins, original.scale = TRUE,
                            add = TRUE, col.points = "blue")

       trns <- function(x) (x / 0.977741)^4
       del <- 1000
       plot(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean), type = "b",
             ylim = c(0, 1.5), xlab = "Distance", ylab = "Semi-variogram")
       points(vgm.s$distance.mean - del, trns(vgm.s$sqrtdiff.mean), type = "b",
             col = "blue", pch = 2, lty = 2)

       plot(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean), type = "b",
             ylim = c(0, 1.5), xlab = "Distance", ylab = "Semi-variogram")
       points(vgm.s$distance.mean - del, trns(vgm.s$sqrtdiff.mean), type = "b",
             col = "blue", pch = 2, lty = 2)
       segments(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean - 2 * vgm.m$se),
             vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean + 2 * vgm.m$se))
       segments(vgm.s$distance.mean - del, trns(vgm.s$sqrtdiff.mean - 2 * vgm.s$se),
             vgm.s$distance.mean - del, trns(vgm.s$sqrtdiff.mean + 2 * vgm.s$se),
             col = "blue", lty = 2)

       mn <- (vgm.m$sqrtdiff.mean + vgm.s$sqrtdiff.mean) / 2
       se <- sqrt(vgm.m$se^2 + vgm.s$se^2)
       plot(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean), type = "n",
            ylim = c(0, 1.5), xlab = "Distance", ylab = "Semi-variogram")
       polygon(c(vgm.m$distance.mean, rev(vgm.m$distance.mean)),
            c(trns(mn - se), rev(trns(mn + se))),
            border = NA, col = "lightblue")
       points(vgm.m$distance.mean, trns(vgm.m$sqrtdiff.mean))
       points(vgm.s$distance.mean, trns(vgm.s$sqrtdiff.mean), col = "blue", pch = 2)

       vgm1 <- sm.variogram(loc.m, Co.m, nbins = nbins, varmat = TRUE,
                            display = "none")
       vgm2 <- sm.variogram(loc.s, Co.s, nbins = nbins, varmat = TRUE,
                            display = "none")

       nbin  <- length(vgm1$distance.mean)
       vdiff <- vgm1$sqrtdiff.mean - vgm2$sqrtdiff.mean
       tstat <- c(vdiff %*% solve(vgm1$V + vgm2$V) %*% vdiff)
       pval  <- 1 - pchisq(tstat, nbin)
       print(pval)
    })

    # Assessing isotropy for Hg in March

    with(mosses, {
       sm.variogram(loc.m, Hg.m, model = "isotropic")
    })

    # Assessing stationarity for Hg in September

    with(mosses, {
       vgm.sty <- sm.variogram(loc.s, Hg.s, model = "stationary")
       i <- 1
       image(vgm.sty$eval.points[[1]], vgm.sty$eval.points[[2]], vgm.sty$estimate[ , , i],
             col = topo.colors(20))
       contour(vgm.sty$eval.points[[1]], vgm.sty$eval.points[[2]], vgm.sty$sdiff[ , , i],
             col = "red", add = TRUE)
    })

    ## End(Not run)

[Package sm version 2.2-5.4 Index]