Local polynomial estimation

Estimates a multidimensional regression function (and its first derivatives) using local polynomial kernel smoothing (and linear binning).

locpol(x, ...)

# S3 method for default
locpol(
  x,
  y,
  h = NULL,
  nbin = NULL,
  degree = 1 + as.numeric(drv),
  drv = FALSE,
  hat.bin = FALSE,
  ncv = 0,
  set.NA = FALSE,
  ...
)

# S3 method for bin.data
locpol(
  x,
  h = NULL,
  degree = 1 + as.numeric(drv),
  drv = FALSE,
  hat.bin = FALSE,
  ncv = 0,
  ...
)

# S3 method for svar.bin
locpol(x, h = NULL, degree = 1, drv = FALSE, hat.bin = TRUE, ncv = 0, ...)

# S3 method for bin.den
locpol(x, h = NULL, degree = 1 + as.numeric(drv), drv = FALSE, ncv = 0, ...)

locpolhcv(
  x,
  y,
  nbin = NULL,
  objective = c("CV", "GCV", "MASE"),
  degree = 1 + as.numeric(drv),
  drv = FALSE,
  hat.bin = FALSE,
  set.NA = FALSE,
  ncv = ifelse(objective == "CV", 2, 0),
  cov.dat = NULL,
  ...
)

Arguments

x	a (data) object used to select a method.
...	further arguments passed to or from other methods (e.g. to hcv.data).
y	vector of data (response variable).
h	(full) bandwidth matrix (controls the degree of smoothing; only the upper triangular part of h is used).
nbin	vector with the number of bins on each dimension.
degree	degree of the local polynomial used. Defaults to 1 (local linear estimation).
drv	logical; if TRUE, the matrix of estimated first derivatives is returned.
hat.bin	logical; if TRUE, the hat matrix of the binned data is returned.
ncv	integer; determines the number of cells leaved out in each dimension. Defaults to 0 (the full data is used) and it is not normally changed by the user in this setting. See "Details" below.
set.NA	logical. If TRUE, sets the bin averages corresponding to cells without data to NA.
objective	character; optimal criterion to be used ("CV", "GCV" or "MASE").
cov.dat	covariance matrix of the data or semivariogram model (of class extending svarmod). Defaults to the identity matrix (uncorrelated data).

Value

Returns an S3 object of class locpol.bin (locpol + bin data + grid par.). A bin.data object with the additional (some optional) 3 components:

est

vector or array (dimension nbin) with the local polynomial estimates.

locpol

a list with 7 components:

• degree degree of the polinomial.

• h bandwidth matrix.

• rm residual mean.

• rss sum of squared residuals.

• ncv number of cells ignored in each direction.

• hat (if requested) hat matrix of the binned data.

• nrl0 (if appropriate) number of cells with data (binw > 0) and missing estimate (est == NA).

deriv

(if requested) matrix of first derivatives.

locpol.svar.bin returns an S3 object of class np.svar (locpol semivar + bin semivar + grid par.). locpol.bin.den returns an S3 object of class np.den (locpol den + bin den + grid par.).

Details

Standard generic function with a default method (interface to the fortran routine lp_raw), in which argument x is a vector or matrix of covariates (e.g. spatial coordinates).

If parameter nbin is not specified is set to pmax(25, rule.binning(x)).

A multiplicative triweight kernel is used to compute the weights.

If ncv > 0, estimates are computed by leaving out 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 position. \(ncv = 1\) corresponds with traditional cross-validation and \(ncv > 1\) with modified CV (see e.g. Chu and Marron, 1991, for the one dimensional case).

Setting set.NA = TRUE (equivalent to biny[binw == 0] <- NA) may be useful for plotting the binned averages $biny (the hat matrix should be handled with care).

locpolhcv calls hcv.data to obtain an "optimal" bandwith (additional arguments ... are passed to this function). Argument ncv is only used here at the bandwith selection stage (estimation is done with all the data).

References

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

Rupert D. and Wand M.P. (1994) Multivariate locally weighted least squares regression. The Annals of Statistics, 22, 1346-1370.

See also

binning, data.grid, np.svariso, svar.bin, np.den, bin.den, hcv.data, rule.binning.

Examples

lp <- locpol(earthquakes[, c("lon", "lat")], earthquakes$mag, h = diag(2, 2), nbin = c(41,41))
simage(lp, main = "Smoothed magnitude")
contour(lp, add = TRUE)

bin <- binning(earthquakes[, c("lon", "lat")], earthquakes$mag, nbin = c(41,41))
lp2 <- locpol(bin, h = diag(2, 2))
all.equal(lp, lp2)
#> [1] TRUE

## Alternatively:
## lp <- locpolhcv(earthquakes[, c("lon", "lat")], earthquakes$mag, ncv = 4)

den <- locpol(as.bin.den(bin), h = diag(1, 2))
plot(den, log = FALSE, main = 'Estimated density')