kedd (version 1.0.0)

dkde: Derivatives of Kernel Density Estimator

Description

The (S3) generic function dkde computes the r'th derivative of kernel density estimator for one-dimensional data. Its default method does so with the given kernel and bandwidth $h$ for one-dimensional observations.

Usage

dkde(x, ...)
## S3 method for class 'default':
dkde(x, y = NULL, deriv.order = 0, h, kernel = c("gaussian", 
         "epanechnikov", "uniform", "triangular", "triweight", 
         "tricube", "biweight", "cosine"), ...)

Arguments

x

the data from which the estimate is to be computed.

y

the points of the grid at which the density derivative is to be estimated; the defaults are $\tau * h$ outside of range($x$), where $\tau = 4$.

deriv.order

derivative order (scalar).

h

the smoothing bandwidth to be used, can also be a character string giving a rule to choose the bandwidth, see h.bcv. The default h.ucv.

kernel

a character string giving the smoothing kernel to be used, with default "gaussian".

...

further arguments for (non-default) methods.

Value

• x
  data points - same as input.
• data.name
  the deparsed name of the x argument.
• n
  the sample size after elimination of missing values.
• kernel
  name of kernel to use.
• deriv.order
  the derivative order to use.
• h
  the bandwidth value to use.
• eval.points
  the coordinates of the points where the density derivative is estimated.
• est.fx
  the estimated density derivative values.

newcommand

\CRANpkg

href

http://CRAN.R-project.org/package=#1

pkg

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eqn

$f(x)$

bold

• bias
• variance
• MSE

deqn

$$MSE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=\frac{f(x)R\left(K^{(r)}\right)}{nh^{2r+1}}+\frac{1}{4}h^{4}\mu_{2}^{2}(K) f^{(r+1)}(x)^{2}+o(h^{4}+1/nh^{2r+1})$$

cr

The MISE (Mean Integrated Squared Error) can be written as: $$MISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=AMISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)+o(h^{4}+1/nh^{2r+1})$$ where, $$AMISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=\frac{1}{nh^{2r+1}}R\left(K^{(r)}\right)+\frac{1}{4}h^{4}\mu_{2}^{2}(K)R\left(f^{(r+2)}\right)$$ with: $R\left(f^{(r)}(x)\right) = \int_{R} \left(f^{(r)}(x)\right)^{2}dx.$ The performance of kernel is measured by MISE or AMISE (Asymptotic MISE). If the bandwidth h is missing from dkde, then the default bandwidth is h.ucv(x,deriv.order,kernel) (Unbiased cross-validation, see h.ucv). For more details see references.

Details

A simple estimator for the density derivative can be obtained by taking the derivative of the kernel density estimate. If the kernel $K(x)$ is differentiable $r$ times then the r'th density derivative estimate can be written as: $$\hat{f}^{(r)}_{h}(x)=\frac{1}{nh^{r+1}}\sum_{i=1}^{n} K^{(r)}\left(\frac{x-X_{i}}{h}\right)$$ where, $$K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)$$ for $r = 0, 1, 2, \dots$ The following assumptions on the density $f^{(r)}(x)$, the bandwidth $h$, and the kernel $K(x)$:

{The $(r+2)$ derivative $f^{(r+2)}(x)$ is continuous, square integrable and ultimately monotone.} {$\lim_{n \to \infty} h = 0$ and $\lim_{n \to \infty}n h^{2r+1} = \infty$ i.e., as the number of samples $n$ is increased $h$ approaches zero at a rate slower than $1/n^{2r+1}$.} {$K(x) \geq 0$ and $\int_{R} K(x) dx = 1$. The kernel function is assumed to be symmetric about the origin i.e., $\int_{R} xK^{(r)}(x) dx = 0$ for even $r$ and has finite second moment i.e., $\mu_{2}(K)=\int_{R}x^{2} K(x) dx < \infty$.}

References

Alekseev, V. G. (1972). Estimation of a probability density function and its derivatives. Mathematical notes of the Academy of Sciences of the USSR. 12 (5), 808--811. Alexandre, B. T. (2009). Introduction to Nonparametric Estimation. Springer-Verlag, New York. Bowman, A. W. and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations. Oxford University Press, Oxford. Bhattacharya, P. K. (1967). Estimation of a probability density function and Its derivatives. Sankhya: The Indian Journal of Statistics, Series A, 29, 373--382. Jeffrey, S. S. (1996). Smoothing Methods in Statistics. Springer-Verlag, New York. Radhey, S. S. (1987). MISE of kernel estimates of a density and its derivatives. Statistics and Probability Letters, 5, 153--159. Scott, D. W. (1992). Multivariate Density Estimation. Theory, Practice and Visualization. New York: Wiley. Schuster, E. F. (1969) Estimation of a probability density function and its derivatives. The Annals of Mathematical Statistics, 40 (4), 1187--1195. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC. London. Stoker, T. M. (1993). Smoothing bias in density derivative estimation. Journal of the American Statistical Association, 88, 855--863. Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S. New York: Springer. 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.dkde, see density in package stats if deriv.order = 0, and kdde in package ks.

Examples

## EXAMPLE 1:  Simple example of a Gaussian density derivative

x <- rnorm(100)
dkde(x,deriv.order=0)  ## KDE of f
dkde(x,deriv.order=1)  ## KDDE of d/dx f
dkde(x,deriv.order=2)  ## KDDE of d^2/x^2 f
dkde(x,deriv.order=3)  ## KDDE of d^3/x^3 f
dev.new()
par(mfrow=c(2,2))
plot(dkde(x,deriv.order=0))
plot(dkde(x,deriv.order=1))
plot(dkde(x,deriv.order=2))
plot(dkde(x,deriv.order=3))

## EXAMPLE 2: Bimodal Gaussian density derivative
## show the kernels in the dkde parametrization

fx  <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5)
fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * 
                   dnorm(x,1.5,0.5)
				   
## 'h = 0.3' ; 'Derivative order = 0'

kernels <- eval(formals(dkde.default)$kernel)
dev.new()
plot(dkde(bimodal,h=0.3),sub=paste("Derivative order = 0",";",
     "Bandwidth =0.3 "),ylim=c(0,0.5), main = "Bimodal Gaussian Density")
for(i in 2:length(kernels))
   lines(dkde(bimodal, h = 0.3, kernel =  kernels[i]), col = i)
curve(fx,add=TRUE,lty=8)
legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),
          lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015)
	   
## 'h = 0.6' ; 'Derivative order = 1'

kernels <- eval(formals(dkde.default)$kernel)[-3]
dev.new()
plot(dkde(bimodal,deriv.order=1,h=0.6),main = "Bimodal Gaussian Density Derivative",sub=paste
         ("Derivative order = 1",";","Bandwidth =0.6"),ylim=c(-0.6,0.6))
for(i in 2:length(kernels))
   lines(dkde(bimodal,deriv.order=1, h = 0.6, kernel =  kernels[i]), col = i)
curve(fx1,add=TRUE,lty=8)
legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),
          lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015)