Kernel density derivative estimate

Kernel density derivative estimate for 1- to 6-dimensional data.

Usage

kdde(x, H, h, deriv.order=0, gridsize, gridtype, xmin, xmax, supp=3.7, 
    eval.points, binned, bgridsize, positive=FALSE, adj.positive, w,
    deriv.vec=TRUE, verbose=FALSE)
kcurv(fhat, compute.cont=TRUE)

# S3 method for class 'kdde'
predict(object, ..., x)

Arguments

x: matrix of data values
H,h: bandwidth matrix/scalar bandwidth. If these are missing, Hpi or hpi is called by default.
deriv.order: derivative order (scalar)
gridsize: vector of number of grid points
gridtype: not yet implemented
xmin,xmax: vector of minimum/maximum values for grid
supp: effective support for standard normal
eval.points: vector or matrix of points at which estimate is evaluated
binned: flag for binned estimation
bgridsize: vector of binning grid sizes
positive: flag if data are positive (1-d, 2-d). Default is FALSE.
adj.positive: adjustment applied to positive 1-d data
w: vector of weights. Default is a vector of all ones.
deriv.vec: flag to compute all derivatives in vectorised derivative. Default is TRUE. If FALSE then only the unique derivatives are computed.
verbose: flag to print out progress information. Default is FALSE.
compute.cont: flag for computing 1% to 99% probability contour levels. Default is TRUE.
fhat: object of class kdde with deriv.order=2
object: object of class kdde
...: other parameters

Value

A kernel density derivative estimate is an object of class kdde which is a list with fields:

x: data points - same as input
eval.points: vector or list of points at which the estimate is evaluated
estimate: density derivative estimate at eval.points
h: scalar bandwidth (1-d only)
H: bandwidth matrix
gridtype: "linear"
gridded: flag for estimation on a grid
binned: flag for binned estimation
names: variable names
w: vector of weights
deriv.order: derivative order (scalar)
deriv.ind: martix where each row is a vector of partial derivative indices

Details

For each partial derivative, for grid estimation, the estimate is a list whose elements correspond to the partial derivative indices in the rows of deriv.ind. For points estimation, the estimate is a matrix whose columns correspond to the rows of deriv.ind.

If the bandwidth H is missing from kdde, then the default bandwidth is the plug-in selector Hpi. Likewise for missing h.

The effective support, binning, grid size, grid range, positive parameters are the same as kde.

The summary curvature is computed by kcurv, i.e. $$\hat{s}(\bold{x})= - \bold{1}\{\mathsf{D}^2 \hat{f}(\bold{x}) < 0\} \mathrm{abs}(|\mathsf{D}^2 \hat{f}(\bold{x})|)$$ where $\mathsf{D}^2 \hat{f}(\bold{x})$ is the kernel Hessian matrix estimate. So $\hat{s}$ calculates the absolute value of the determinant of the Hessian matrix and whose sign is the opposite of the negative definiteness indicator.

Examples

set.seed(8192)
x <- rmvnorm.mixt(1000, mus=c(0,0), Sigmas=invvech(c(1,0.8,1)))
fhat <- kdde(x=x, deriv.order=1) ## gradient [df/dx, df/dy]
predict(fhat, x=x[1:5,])
#>             [,1]        [,2]
#> [1,]  0.08455865 -0.03043875
#> [2,]  0.06752571 -0.04804689
#> [3,] -0.05297611 -0.09152648
#> [4,]  0.18093462 -0.20108890
#> [5,] -0.19777717  0.20332981

## See other examples in ? plot.kdde