# Kernel smoothing¶

Kernel smoothing is a non parametric estimation method of the probability density function of a distribution.

In dimension 1, the kernel smoothed probability density function has the following expression, where K is the univariate kernel, n the sample size and the univariate random sample with :

(1)

The kernel K is a function satisfying . Usually K is chosen to be a unimodal probability density function that is symmetric about 0. The parameter h is called the bandwidth.

In dimension , the kernel may be defined as a product kernel , as follows where :

which leads to the kernel smoothed probability density function in dimension d, where is the d-variate random sample which components are denoted :

Let’s note that the bandwidth is the vector .

The quality of the approximation may be controlled by the AMISE (Asymptotic Mean Integrated Square error) criteria defined as:

The quality of the estimation essentially depends on the value of the bandwidth h. The bandwidth that minimizes the AMISE criteria has the expression (given in dimension 1):

(2)

where and .

If we note that with , then relation writes:

(3)

Several rules exist to evaluate the optimal bandwidth : all efforts are concentrated on the evaluation of the term . We give here the most usual rules:

• the Silverman rule in dimension 1,

• the plug-in bandwidth selection

• the Scott rule in dimension d.

## Silverman rule (dimension 1)¶

In the case where the density p is normal with standard deviation , then the term can be exactly evaluated. In that particular case, the optimal bandwidth of relation (3) with respect to the AMISE criteria writes as follows:

(4)

An estimator of is obtained by replacing by its estimator , evaluated from the sample :

(5)

The Silverman rule consists in considering of relation (5) even if the density p is not normal:

(6)

Relation (6) is empirical and gives good results when the density is not far from a normal one.

## Plug-in bandwidth selection method (dimension 1)¶

Relation (3) requires the evaluation of the quantity . As a general rule, we use the estimator of defined by:

(7)

(8)

and then the estimator is defined as:

(9)

We note that depends of the parameter h which can be taken in order to minimize the AMSE (Asymptotic Mean Square Error) criteria evaluated between and . The optimal parameter h is:

(10)

Given that preliminary results, the solve-the-equation plug-in method proceeds as follows:

• Relation (3) defines as a function of we denote here as:

(11)

• The term is approximated by its estimator defined in (9) evaluated with its optimal parameter defined in (10):

(12)

which leads to a relation of type:

(13)

• Relations (3) and (12) lead to the new one:

(14)

which rewrites:

(15)

• Relation (14) depends on both terms and which are evaluated with their estimators defined in (9) respectively with their AMSE optimal parameters and (see relation (10)). It leads to the expressions:

(16)

• In order to evaluate and , we suppose that the density p is normal with a variance which is approximated by the empirical variance of the sample, which leads to:

(17)

Then, to summarize, thanks to relations (11), (13), (15), (16) and (17), the optimal bandwidth is solution of the equation:

(18)

## Scott rule (dimension d)¶

The Scott rule is a simplification of the Silverman rule generalized to the dimension d which is optimal when the density p is normal with independent components. In all the other cases, it gives an empirical rule that gives good result when the density p is not far from the normal one. For examples, the Scott bandwidth may appear too large when p presents several maximum.

The Silverman rule given in dimension 1 in relation (6) can be generalized in dimension d as follows: if we suppose that the density p is normal with independent components, in dimension d and that we use the normal kernel to estimate it, then the optimal bandwidth vector with respect to the AMISE criteria writes as follows:

(19)

where is the standard deviation of the i-th component of the sample , and the standard deviation of the 1D kernel K.

The Scott proposition is a simplification of the Silverman rule, based on the fact that the coefficient remains in when the dimension d varies. Thus, Scott fixed it to 1:

(20)

which leads to the simplified expression:

(21)

Furthermore, in the general case, we have from relation (ref{AMISE1}) :

(22)

Considering that whatever the kernel K, relation (22) simplifies in:

(23)

If we consider the normal kernel for , then relation (23) writes in a more general notation:

(24)

If is evaluated with the Silverman rule, (24) rewrites:

(25)

At last, from relation (21) and (25) applied in each direction i, we deduce the Scott rule:

(26)

## Boundary treatment¶

In dimension 1, the boundary effects may be taken into account: the boundaries are automatically detected from the sample (with the min and max functions) and the kernel smoothed PDF is corrected in the boundary areas to remain within the boundaries, according to the mirroring technique:

• the Scott bandwidth is evaluated from the sample: h

• two sub-samples are extracted from the initial sample, containing all the points within the range and ,

• both sub-samples are transformed into their symmetric samples with respect their respective boundary: its results two samples within the range and ,

• a kernel smoothed PDF is built from the new sample composed with the initial one and the two new ones, with the previous bandwidth h,

• this last kernel smoothed PDF is truncated within the initial range (conditional PDF).

API:

Examples:

References:

• Kernel smoothing, M.P. Wand and M.C. Jones, Chapman & Hall/CRC edition, ISNB 0-412-55270-1.

• Multivariate Density Estimation, practice and Visualization, Theory, David W. Scott, Wiley edition.