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Kernel Estimator and Bandwidth Selection for Density

and its Derivatives

The kedd Package

Version 1.0.3

by Arsalane Chouaib Guidoum

∗

Revised October 30, 2015

1 Introduction

In statistics, the univariate kernel density estimation (KDE) is a non-parametric way to estimate

the probability density function f(x) of a random variable X, is a fundamental data smoothing

problem where inferences about the population are made, based on a ﬁnite data sample. This

techniques are widely used in various inference procedures such as signal processing, data mining

and econometrics, see e.g., Silverman [1986], Wand and Jones [1995], Jeﬀrey [1996], Wolfgang et

all [2004], Alexandre [2009]. The kernel estimator are standard in many books with applications

and computer vision, see Wolfgang [1991], Scott [1992], Bowman and Azzalini [1997], Venables

and Ripley [2002], for computational complexity and with implementation in S, for an overview.

Estimation of the density derivatives also comes up in various other applications like estimation of

modes and inﬂexion points of densities, a good list of applications which require the estimation of

density derivatives can be found in Singh [1977].

There already exist a number of packages that can perform kernel density estimation in R

(density in R base); see for example KernSmooth [Wand and Ripley, 2013], sm [Bowman and Az-

zalini, 2013], np [Tristen and Jeﬀrey, 2008] and feature [Duong and Matt, 2013], they exist also of

functions for kernel density derivative estimation (KDDE), e.g., kdde in ks package [Duong, 2007].

We introduce in this vignette a new R package kedd [Guidoum, 2015] for use with the statistical

programming environment R Development Core Team [2015], which implements smoothing tech-

niques and computing bandwidth selectors of the r

derivative of a probability density f(x) for

univariate data, using several kernels functions.

2 Convolutions and derivatives in kernels

In non-parametric statistics, a kernel is a weighting function used in non-parametric estimation

techniques. Kernels are used in kernel density estimation to estimate random variables density

functions f(x), or in kernel regression to estimate the conditional expectation of a random variable,

see e.g., Silverman [1986], Wand and Jones [1995]. In general any functions having the following

assumptions can be used as a kernel:

(A1) K(x) ≥ 0 and

K(x)dx = 1.

(A2) Symmetric about the origin, e.g.,

xK(x)dx = 0.

∗

Department of Probabilities & Statistics.

Faculty of Mathematics.

University of Science and Technology Houari Boumediene.

BP 32 El-Alia, U.S.T.H.B, Algeria.

acguidoum@usthb.dz

(A3) Has ﬁnite second moment, e.g., µ

(K) =

K(x)dx < ∞. We denote R(K) =

(K(x))

dx.

If K(x) is a kernel, then so is the function

K(x) deﬁned by

K(x) = λK(λx), where λ > 0, this can

be used to select a scale that is appropriate for the data. The kernel function is very important to

spreading a probability mass of 1/n, the most widely used kernel is the Gaussian of zero mean and

unit variance. Some classical of kernel function K(x; r) (r is the maximum derivative of kernel) in

kedd package are the following:

Kernel K(x; r) R(K) µ

(K)

Gaussian K(x; ∞) =

√

2π

exp



−



]−∞,+∞[

1/ (2

√

π) 1

Epanechnikov K(x; 2) =



1 − x



(|x|≤1)

3/5 1/5

Uniform K(x; 0) =

(|x|≤1)

1/2 1/3

Triangular K(x; 1) = (1 − |x|)1

(|x|≤1)

2/3 1/6

Triweight K(x; 6) =



1 − x



(|x|≤1)

350/429 1/9

Tricube K(x; 9) =



1 − |x|



(|x|≤1)

175/247 35/243

Biweight K(x; 4) =



1 − x



(|x|≤1)

5/7 1/7

Cosine K(x; ∞) =

cos





(|x|≤1)

/16



−8 + π



/π

Table 1: Kernel functions in kedd pakage.

The r

derivative of the kernel function K(x) is written as:

(r)

(x) =

K(x) (1)

and convolution of K

(r)

(x) is:

(r)

∗ K

(r)

(x) =

(r)

(x)K

(r)

(x − y)dy (2)

for example the r

derivative of the Gaussian kernel is given by:

(r)

(x) = (−1)

(x)K(x)

and the r

convolution can be written as:

(r)

∗ K

(r)

(x) = (−1)

(x)H

(x − y)K(x)K(x − y)dy

where H

(x) is the r

Hermite polynomial, see e.g., Olver et all [2010]. We use kernel.fun for

kernel derivative deﬁned by (1), and kernel.conv for kernel convolution deﬁned by (2).

For example the ﬁrst derivative of the Gaussian kernel displayed on the left in Figure 1. On the

right is the ﬁrst convolution of the Gaussian kernel.

> library(kedd)

> kernel.fun(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx

[1] 0.007977250 0.003989223 0.000000000 -0.003989223 -0.007977250

> kernel.conv(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx

[1] -0.1410051 -0.1410368 -0.1410474 -0.1410368 -0.1410051

> plot(kernel.fun(deriv.order = 1, kernel = "gaussian"))

> plot(kernel.conv(deriv.order = 1, kernel = "gaussian"))

−4 −2 0 2 4

−0.2

−0.1

0.0

0.1

0.2

−5 0 5

−0.10

−0.05

0.00

0.05

Figure 1: (Left) First derivative of the Gaussian kernel. (Right) Convolution of the ﬁrst derivative

Gaussian kernel.

3 Kernel density derivative estimator

Let (X

, X

, . . . , X

) be a data sample, independent and identically distributed of a continuous

random variable X, with density function f(x). If the kernel K is diﬀerentiable r times then a

natural estimator of the r

derivative of f(x) the r

derivative of the kernel estimate [Bhattacharya,

1967, Schuster, 1969, Alekseev, 1972]:

(r)

(x) =

i=1



x − X



r+1

i=1

(r)



x − X



(3)

where K

(r)

is r

derivative of the kernel function K, which we take to be a symmetric probability

density with at least r non zero derivatives when estimating f

(r)

(x), and h is the bandwidth, this

parameter is very important that controls the degree of smoothing applied to the data.

The following assumptions on the density f

(r)

(x), the bandwidth h, and the kernel K:

(A4) The (r + 2) derivatives f

(r+2)

(x) is continuous, square integrable and ultimately monotone.

(A5) In the asymptotic framework, as lim

n→∞

= 0 and lim

n→∞

2r+1

= ∞, i.e., as the number

of sample n is increased h approaches zero at a rate slower than 1/n

2r+1

(A6) Assumptions about K are introduced in the previous section.

As seen in Equation (3), when working with a kernel estimator of the r

derivative function two

choices must be made: the kernel function K and the smoothing parameter or bandwidth h. The

choice of K is a problem of less importance, because K is not very sensitive to the shape of estimator,

and diﬀerent functions that produce good results can be used. In practice, the choice of an eﬃcient

method for the computation of h, for an observed data sample is a crucial problem, because of the

eﬀect of the bandwidth on the shape of the corresponding estimator. If the bandwidth is small, we

will obtain an under smoothed estimator, with high variability. On the contrary, if the value of h

is big, the resulting estimator will be over smooth and farther from the function that we are trying

to estimate.

An example is drawn in Figure 2 where we show in left four diﬀerent kernel (Gaussian, biweight,

triweight and tricube) estimators of the ﬁrst derivative of a bimodal (separated) Gaussian density

(Equation 5), and a given value of h = 0.6. On the right, using the Gaussian kernel and four

diﬀerent values for the bandwidth.

−4 −2 0 2 4

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

density derivative function

TRUE

gaussian

biweight

triweight

tricube

−3 −2 −1 0 1 2 3

−0.5

0.0

0.5

1.0

density derivative function

TRUE

h = 0.14

h = 0.3

h = 0.6

h = 1.2

Figure 2: (Left) Diﬀerent kernels for estimation, with h = 0.6. (Right) Eﬀect of the bandwidth on

the kernel estimator.

We have implemented in R the function dkde corresponds to the derivative of kernel density

estimator (Equation 3). Eight possibilities are allowed for the kernel functions that are summarized

in Table 1. We enumerate the arguments and results of this function in Table 2.

Arguments Description

x The data sample.

y The points of the grid at which the density derivative is to be estimated.

The default are 4h outside of range(x).

deriv.order Derivative order (scalar).

h The smoothing bandwidth to be used. The default, ”ucv” unbiased cross-

validation.

kernel The kernel function (see Table 1), by default "gaussian".

Results Description

eval.points The coordinates of the points where the density derivative is estimated.

est.fx The estimated density derivative values (Equation 3).

Table 2: Summary of arguments and results of dkde.

Working with the dataset ’bimodal’ correspond to data sample of 200 random numbers of a bi-

modality (separated) of a two-component Gaussian mixture density (Equation 4), with the following

parameters: −µ

= µ

= 3/2 and σ

= σ

= 1/2. The dkde function enables to compute the r

derivative of kernel density estimator over a grid of points, with a bandwidth selected by the user,

but it also allows to estimate directly this parameter by the unbiased cross-validation method h.ucv

(see following Section). We have chosen this method as the automatic one because it is the fastest

in computation time terms. Now we estimate the ﬁrst three derivatives of f(x), can be written as:

f(x) = 0.5φ(µ

, σ

) + 0.5φ(µ

, σ

) (4)

(1)

(x) = 0.5(−4x − 6)φ(µ

, σ

) + 0.5(−4x + 6)φ(µ

, σ

) (5)

(2)

(x) = 0.5



(−4x − 6)

− 4



φ(µ

, σ

) + 0.5



(−4x + 6)

− 4



φ(µ

, σ

) (6)

(3)

(x) = 0.5(−4x − 6)



(−4x − 6)

− 12



φ(µ

, σ

) + 0.5(−4x + 6)



(−4x + 6)

− 12



φ(µ

, σ

) (7)

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