kernel.conv: Convolutions of r'th Derivative for Kernel Function

In kedd: Kernel Estimator and Bandwidth Selection for Density and Its Derivatives

Convolutions of r'th Derivative for Kernel Function

Description

The (S3) generic function kernel.conv computes the convolution of r'th derivative for kernel function.

Usage

kernel.conv(x, ...)
## Default S3 method:
kernel.conv(x = NULL, deriv.order = 0,kernel = c("gaussian","epanechnikov", 
             "uniform", "triangular", "triweight", "tricube", 
             "biweight", "cosine", "silverman"), ...)

Arguments

x	points at which the convolution of kernel derivative is to be evaluated.
deriv.order	derivative order (scalar).
kernel	a character string giving the smoothing kernel to be used, with default "gaussian".
...	further arguments for (non-default) methods.

Details

The convolution of r'th derivative for kernel function is written K^{(r)}\ast K^{(r)}. It is defined as the integral of the product of the derivative for kernel. As such, it is a particular kind of integral transform:

K^{(r)} \ast K^{(r)}(x) = \int_{-\infty}^{+\infty} K^{(r)}(y)K^{(r)}(x-y)dy

where:

K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)

for r = 0, 1, 2, \dots

Value

kernel	name of kernel to use.
deriv.order	the derivative order to use.
x	the n coordinates of the points where the convolution of kernel derivative is evaluated.
kx	the convolution of kernel derivative values.

Author(s)

Arsalane Chouaib Guidoum [e-mail]

References

Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press, New York, USA.

Scott, D. W. (1992). Multivariate Density Estimation. Theory, Practice and Visualization. New York: Wiley.

Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC. London.

Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.

Wolfgang, H. (1991). Smoothing Techniques, With Implementation in S. Springer-Verlag, New York.

See Also

plot.kernel.conv, kernapply in package "stats" for computes the convolution between an input sequence, and convolve use the Fast Fourier Transform (fft) to compute the several kinds of convolutions of two sequences.

Examples

kernels <- eval(formals(kernel.conv.default)$kernel)
kernels

## gaussian
kernel.conv(x = 0,kernel=kernels[1],deriv.order=0)
kernel.conv(x = 0,kernel=kernels[1],deriv.order=1)

## silverman
kernel.conv(x = 0,kernel=kernels[9],deriv.order=0)
kernel.conv(x = 0,kernel=kernels[9],deriv.order=1)

kedd documentation built on May 29, 2024, 1:45 a.m.