Cross-validation methods for bandwidth selection

Selects the bandwidth of a local polynomial kernel (regression, density or variogram) estimator using (standart or modified) CV, GCV or MASE criteria.

h.cv(bin, ...)

# S3 method for bin.data
h.cv(
  bin,
  objective = c("CV", "GCV", "MASE"),
  h.start = NULL,
  h.lower = NULL,
  h.upper = NULL,
  degree = 1,
  ncv = ifelse(objective == "CV", 2, 0),
  cov.bin = NULL,
  DEalgorithm = FALSE,
  warn = TRUE,
  tol.mask = npsp.tolerance(2),
  ...
)

# S3 method for bin.den
h.cv(
  bin,
  h.start = NULL,
  h.lower = NULL,
  h.upper = NULL,
  degree = 1,
  ncv = 2,
  DEalgorithm = FALSE,
  ...
)

# S3 method for svar.bin
h.cv(
  bin,
  loss = c("MRSE", "MRAE", "MSE", "MAE"),
  h.start = NULL,
  h.lower = NULL,
  h.upper = NULL,
  degree = 1,
  ncv = 1,
  DEalgorithm = FALSE,
  warn = FALSE,
  ...
)

hcv.data(
  bin,
  objective = c("CV", "GCV", "MASE"),
  h.start = NULL,
  h.lower = NULL,
  h.upper = NULL,
  degree = 1,
  ncv = ifelse(objective == "CV", 1, 0),
  cov.dat = NULL,
  DEalgorithm = FALSE,
  warn = TRUE,
  ...
)

Arguments

bin	object used to select a method (binned data, binned density or binned semivariogram).
...	further arguments passed to or from other methods (e.g. parameters of the optimization routine).
objective	character; optimal criterion to be used ("CV", "GCV" or "MASE").
h.start	vector; initial values for the parameters (diagonal elements) to be optimized over. If DEalgorithm == FALSE (otherwise not used), defaults to (3 + ncv) * lag, where lag = bin$grid$lag.
h.lower	vector; lower bounds on each parameter (diagonal elements) to be optimized. Defaults to (1.5 + ncv) * bin$grid$lag.
h.upper	vector; upper bounds on each parameter (diagonal elements) to be optimized. Defaults to 1.5 * dim(bin) * bin$grid$lag.
degree	degree of the local polynomial used. Defaults to 1 (local linear estimation).
ncv	integer; determines the number of cells leaved out in each dimension. (0 to GCV considering all the data, \(>0\) to traditional or modified cross-validation). See "Details" bellow.
cov.bin	(optional) covariance matrix of the binned data or semivariogram model (svarmod-class) of the (unbinned) data. Defaults to the identity matrix.
DEalgorithm	logical; if TRUE, the differential evolution optimization algorithm in package DEoptim is used.
warn	logical; sets the handling of warning messages (normally due to the lack of data in some neighborhoods). If FALSE all warnings are ignored.
tol.mask	tolerance used in the aproximations. Defaults to npsp.tolerance(2).
loss	character; CV error. See "Details" bellow.
cov.dat	covariance matrix of the data or semivariogram model (of class extending svarmod). Defaults to the identity matrix (uncorrelated data).

Value

Returns a list containing the following 3 components:

h

the best (diagonal) bandwidth matrix found.

value

the value of the objective function corresponding to h.

objective

the criterion used.

Details

Currently, only diagonal bandwidths are supported.

h.cv methods use binning approximations to the objective function values (in almost all cases, an averaged squared error). If ncv > 0, estimates are computed by leaving out binning cells with indexes within the intervals \([x_i - ncv + 1, x_i + ncv - 1]\), at each dimension i, where \(x\) denotes the index of the estimation location. \(ncv = 1\) corresponds with traditional cross-validation and \(ncv > 1\) with modified CV (it may be appropriate for dependent data; see e.g. Chu and Marron, 1991, for the one dimensional case). Setting ncv >= 2 would be recommended for sparse data (as linear binning is used). For standard GCV, set ncv = 0 (the whole data would be used). For theoretical MASE, set bin = binning(x, y = trend.teor), cov = cov.teor and ncv = 0.

If DEalgorithm == FALSE, the "L-BFGS-B" method in optim is used.

The different options for the argument loss in h.cv.svar.bin() define the CV error considered in semivariogram estimation:

"MSE"

Mean squared error

"MRSE"

Mean relative squared error

"MAE"

Mean absolute error

"MRAE"

Mean relative absolute error

hcv.data evaluates the objective function at the original data (combining a binning approximation to the nonparametric estimates with a linear interpolation), this can be very slow (and memory demanding; consider using h.cv instead). If ncv > 1 (modified CV), a similar algorithm to that in h.cv is used, estimates are computed by leaving out binning cells with indexes within the intervals \([x_i - ncv + 1, x_i + ncv - 1]\).

References

Chu, C.K. and Marron, J.S. (1991) Comparison of Two Bandwidth Selectors with Dependent Errors. The Annals of Statistics, 19, 1906-1918.

Francisco-Fernandez M. and Opsomer J.D. (2005) Smoothing parameter selection methods for nonparametric regression with spatially correlated errors. Canadian Journal of Statistics, 33, 539-558.

See also

locpol, locpolhcv, binning, np.svar.

Examples

bin <- binning(earthquakes[, c("lon", "lat")], earthquakes$mag)
hcv <- h.cv(bin, ncv = 2)
lp <- locpol(bin, h = hcv$h)
## Alternatively:
## lp <- locpolhcv(earthquakes[, c("lon", "lat")], earthquakes$mag, ncv = 2)

simage(lp, main = 'Smoothed magnitude')
contour(lp, add = TRUE)
with(earthquakes, points(lon, lat, pch = 20))

## Density estimation
hden <- h.cv(as.bin.den(bin))
den <- np.den(bin, h = hden$h)

plot(den, main = 'Estimated log(density)')