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 \hat{p} has the following expression, where K is the univariate kernel, n the sample size and (X_1, \cdots, X_n) \in \Rset^n the univariate random sample with \forall i, \, \, X_i \in \Rset:

(1)\hat{p}(x) = \displaystyle \frac{1}{nh}\sum_{i=1}^{n} K\left(\frac{x-X_i}{h}\right)

The kernel K is a function satisfying \int K(x)\, dx=1. 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 d>1, the kernel may be defined as a product kernel K_d, as follows where \vect{x} = (x_1, \cdots, x_d)\in \Rset^d:

K_d(\vect{x}) = \displaystyle \prod_{j=1}^{d} K(x_j)

which leads to the kernel smoothed probability density function in dimension d, where (\vect{X}_1, \cdots, \vect{X}_n) is the d-variate random sample which components are denoted \vect{X}_i = (X_{i1}, \dots, X_{id}):

\hat{p}(\vect{x}) = \displaystyle \frac{1}{N \prod_{j=1}^{d}h_j} \sum_{i=1}^{N} K_d\left(\frac{x_1 - X_{i1} }{h_1}, \dots, \frac{x_d - X_{id}}{h_d}\right)

Let’s note that the bandwidth is the vector \vect{h} = (h_1, \cdots, h_d).

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

\left\{ \begin{array}{lcl} AMISE(\hat{p}) & = & \mbox{two first terms in the series expansion with respect to $n$ in } MISE(\hat{p}) \\ MISE(\hat{p}) & = & \mathbb{E}_\vect{X}\left[||\hat{p} - p||^2_{L_2}\right] = \int \, MSE(\hat{p}, \vect{x}) \, d\vect{x} \\ MSE(\hat{p}, \vect{x})& = & \left[ \mathbb{E}_\vect{X}\left[\hat{p}(\vect{x})\right] - p(\vect{x})\right]^2 + {\rm Var}_\vect{X}\left[\hat{p}(\vect{x})\right] \end{array} \right.

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)h_{AMISE}(K) = \displaystyle \left[ \frac{R(K)}{\mu_2(K)^2R(p^{(2)})}\right]^{\frac{1}{5}}n^{-\frac{1}{5}}

where R(K) = \int K(\vect{x})^2\, d\vect{x} and \mu_2(K) = \int \vect{x}^2K(\vect{x})\, d\vect{x} = \sigma_K^2.

If we note that R(p^{(r)}) = (-1)^r\Phi_{2r} with \Phi_r = \int p^{(r)}p(x)\, dx = \mathbb{E}_\vect{X}\left[p^{(r)}\right], then relation writes:

(3)h_{AMISE}(K) = \displaystyle \left[ \frac{R(K)}{\mu_2(K)^2\Phi_4}\right]^{\frac{1}{5}}n^{-\frac{1}{5}}

Several methods exist to evaluate the optimal bandwidth h_{AMISE}(K) based on different approximations of \Phi_4:

  • Silverman’s rule in dimension 1,

  • the plug-in bandwidth selection,

  • Scott’s rule in dimension d.

Silverman’s rule (dimension 1)

In the case where the density p is normal with standard deviation \sigma, then the term \Phi_4 can be exactly evaluated. In that particular case, the optimal bandwidth of relation (3) with respect to the AMISE criteria writes as follows:

(4)h^{p = normal}_{AMISE}(K) = \displaystyle \left[ \frac{8\sqrt{\pi} R(K)}{3\mu_2(K)^2}\right]^{\frac{1}{5}}\sigma n^{-\frac{1}{5}}

An estimator of h^{p= normal}_{AMISE}(K) is obtained by replacing \sigma by its estimator \hat{\sigma}^n, evaluated from the sample (X_1, \dots, X_n):

(5)\hat{h}^{p = normal}_{AMISE}(K) = \displaystyle \left[ \frac{8\sqrt{\pi} R(K)}{3\mu_2(K)^2}\right]^{\frac{1}{5}}\hat{\sigma}^n n^{-\frac{1}{5}}

The Silverman rule consists in considering \hat{h}^{p = normal}_{AMISE}(K) of relation (5) even if the density p is not normal:

(6)h^{Silver}(K) = \displaystyle \left[ \frac{8\sqrt{\pi} R(K)}{3\mu_2(K)^2}\right]^{\frac{1}{5}}\hat{\sigma}^n n^{-\frac{1}{5}}

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)

The plug-in bandwidth selection method improves the estimation of the rugosity of the second derivative of the density. Instead of making the gaussian assumption, the method uses a kernel smoothing method in order to make an approximation of higher derivatives of the density.

Relation (3) requires the evaluation of the quantity \Phi_4. As a general rule, we use the estimator \hat{\Phi}_r of \Phi_r defined by:

(7)\hat{\Phi}_r = \displaystyle \frac{1}{n}\sum_{i=1}^{n} \hat{p}^{(r)}(X_i)

Deriving relation (1) leads to:

(8)\hat{p}^{(r)}(x) = \displaystyle \frac{1}{nh^{r+1}}\sum_{i=1}^{n} K^{(r)}\left(\frac{x-X_i}{h}\right)

and then the estimator \hat{\Phi}_r(h) is defined as:

(9)\hat{\Phi}_r(h) = \displaystyle \frac{1}{n^2h^{r+1}}\sum_{i=1}^{n}\sum_{j=1}^{n} K^{(r)}\left(\frac{X_i-X_j}{h}\right)

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

(10)h^{(r)}_{AMSE} = \displaystyle \left(\frac{-2K^{(r)}(0)}{\mu_2(K)\Phi_{r+2}}\right)^{\frac{1}{r+3}}n^{-\frac{1}{r+3}}

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

  • Relation (3) defines h_{AMISE}(K) as a function of \Phi_4 we denote here as:

    (11)h_{AMISE}(K) = t(\Phi_4)

  • The term \Phi_4 is approximated by its estimator defined in (9) evaluated with its optimal parameter h^{(4)}_{AMSE} defined in (10):

    (12)h^{(4)}_{AMSE} = \displaystyle \left(\frac{-2K^{(4)}(0)}{\mu_2(K)\Phi_{6}}\right)^{\frac{1}{7}}n^{-\frac{1}{7}}

    which leads to a relation of type:

    (13)\Phi_4 \simeq \hat{\Phi}_4(h^{(4)}_{AMSE})

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

    (14)h^{(4)}_{AMSE} = \displaystyle \left( \frac{-2K^{(4)}(0)\mu_2(K)\Phi_4}{R(K)\Phi_{6}}\right) ^{\frac{1}{7}}h_{AMISE}(K)^{\frac{5}{7}}

    which rewrites:

    (15)h^{(4)}_{AMSE} = l(h_{AMISE}(K))

  • Relation (14) depends on both terms \Phi_4 and \Phi_6 which are evaluated with their estimators defined in (9) respectively with their AMSE optimal parameters g_1 and g_2 (see relation (10)). It leads to the expressions:

    (16)\left\{ \begin{array}{lcl} g_1 & = & \displaystyle \left(\frac{-2K^{(4)}(0)}{\mu_2(K)\Phi_{6}}\right)^{\frac{1}{7}}n^{-\frac{1}{7}}\\ g_2 & = & \displaystyle \left(\frac{-2K^{(6)}(0)}{\mu_2(K)\Phi_{8}}\right)^{\frac{1}{7}}n^{-\frac{1}{9}} \end{array} \right.

  • In order to evaluate \Phi_6 and \Phi_8, we suppose that the density p is normal with a variance \sigma^2 which is approximated by the empirical variance of the sample, which leads to:

    (17)\left\{ \begin{array}{lcl} \hat{\Phi}_6 & = & \displaystyle \frac{-15}{16\sqrt{\pi}}\hat{\sigma}^{-7}\\ \hat{\Phi}_8 & = & \displaystyle \frac{105^{\strut}}{32\sqrt{\pi}}\hat{\sigma}^{-9} \end{array} \right.

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

(18)\boldsymbol{h_{AMISE}(K) = t \circ \hat{\Phi}_4 \circ l (h_{AMISE}(K))}

This method is due to (Sheather, Jones, 1991) who used ideas from (Park, Marron, 1990). The algorithm is presented in (Wand, Jones, 1994), page 74 under the “Solve the equation rule” name. The implementation uses ideas from (Raykar, Duraiswami, 2006), but the fast selection is not implemented.

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 N(0,1) to estimate it, then the optimal bandwidth vector \vect{h} with respect to the AMISE criteria writes as follows:

(19)\vect{h}^{Silver}(N) = \left(\left(\frac{4}{d+2}\right)^{1/(d+4)}\hat{\sigma}_i^n n^{-1/(d+4)}\right)_i

where \hat{\sigma}_i^n is the standard deviation of the i-th component of the sample (\vect{X}_1, \cdots, \vect{X}_n), and \sigma_K the standard deviation of the 1D kernel K.

Scott’ method is a simplification of Silverman’s rule, based on the fact that the coefficient \left(\frac{4}{d+2}\right)^{1/(d+4)} remains in [0.924, 1.059] when the dimension d varies. Thus, Scott fixed it to 1:

(20)\left(\frac{4}{d+2}\right)^{1/(d+4)} \simeq 1

which leads to the simplified expression:

(21)\vect{h}^{Silver}(N) \simeq \left( \hat{\sigma}_i^n n^{-1/(d+4)}\right)_i

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

(22)\frac{h_{AMISE}(K_1)}{h_{AMISE}(K_2)}=\frac{\sigma_{K_2}}{\sigma_{K_1}}\left[\frac{\sigma_{K_1}R(K_1)}{\sigma_{K_2}R(K_2)}\right]^{1/5}

Considering that \sigma_{K}R(K) \simeq 1 whatever the kernel K, relation (22) simplifies in:

(23)h_{AMISE}(K_1) \simeq h_{AMISE}(K_2)\frac{\sigma_{K_2}}{\sigma_{K_1}}

If we consider the normal kernel N(0,1) for K_2, then relation (23) writes in a more general notation:

(24)h_{AMISE}(K) \simeq h_{AMISE}(N)\frac{1}{\sigma_{K}}

If h_{AMISE}(N) is evaluated with the Silverman rule, (24) rewrites:

(25)h^{Silver}(K) \simeq h^{Silver}(N)\frac{1}{\sigma_{K}}

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

(26)\boldsymbol{\vect{h}^{Scott} = \left(\frac{\hat{\sigma}_i^n}{\sigma_K}n^{-1/(d+4)}\right)_i}

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 [min, min + h[ and ]max-h, max],

  • both sub-samples are transformed into their symmetric samples with respect their respective boundary: its results two samples within the range ]min-h, min] and [max, max+h[,

  • 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 [min, max] (conditional PDF).