Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density estimation with improved statistical properties. Apart from histograms, other types of density estimators include parametric, spline, wavelet and Fourier series. Kernel density estimators were first introduced in the scientific literature for univariate data in the 1950s and 1960s^[1][2] and subsequently have been widely adopted. It was soon recognised that analogous estimators for multivariate data would be an important addition to multivariate statistics. Based on research carried out in the 1990s and 2000s, multivariate kernel density estimation has reached a level of maturity comparable to its univariate counterparts.^[3]

Motivation

We take an illustrative synthetic bivariate data set of 50 points to illustrate the construction of histograms. This requires the choice of an anchor point (the lower left corner of the histogram grid). For the histogram on the left, we choose (−1.5, −1.5): for the one on the right, we shift the anchor point by 0.125 in both directions to (−1.625, −1.625). Both histograms have a binwidth of 0.5, so any differences are due to the change in the anchor point only. The colour-coding indicates the number of data points which fall into a bin: 0=white, 1=pale yellow, 2=bright yellow, 3=orange, 4=red. The left histogram appears to indicate that the upper half has a higher density than the lower half, whereas the reverse is the case for the right-hand histogram, confirming that histograms are highly sensitive to the placement of the anchor point.^[4]

One possible solution to this anchor point placement problem is to remove the histogram binning grid completely. In the left figure below, a kernel (represented by the grey lines) is centred at each of the 50 data points above. The result of summing these kernels is given on the right figure, which is a kernel density estimate. The most striking difference between kernel density estimates and histograms is that the former are easier to interpret since they do not contain artifices induced by a binning grid. The coloured contours correspond to the smallest region which contains the respective probability mass: red = 25%, orange + red = 50%, yellow + orange + red = 75%, thus indicating that a single central region contains the highest density.

The goal of density estimation is to take a finite sample of data and to make inferences about the underlying probability density function everywhere, including where no data are observed. In kernel density estimation, the contribution of each data point is smoothed out from a single point into a region of space surrounding it. Aggregating the individually smoothed contributions gives an overall picture of the structure of the data and its density function. In the details to follow, we show that this approach leads to a reasonable estimate of the underlying density function.

Definition

The previous figure is a graphical representation of kernel density estimate, which we now define in an exact manner. Let x₁, x₂, ..., x_n be a sample of d-variate random vectors drawn from a common distribution described by the density function ƒ. The kernel density estimate is defined to be

${\displaystyle {\hat {f}}_{\mathbf {H} }(\mathbf {x} )={\frac {1}{n}}\sum _{i=1}^{n}K_{\mathbf {H} }(\mathbf {x} -\mathbf {x} _{i})}$

where

• x = (x₁, x₂, …, x_d)^T, x_i = (x_i1, x_i2, …, x_id)^T, i = 1, 2, …, n are d-vectors;
• H is the bandwidth (or smoothing) d×d matrix which is symmetric and positive definite;
• K is the kernel function which is a symmetric multivariate density;
• ${\displaystyle K_{\mathbf {H} }(\mathbf {x} )=|\mathbf {H} |^{-1/2}K(\mathbf {H} ^{-1/2}\mathbf {x} )}$.

The choice of the kernel function K is not crucial to the accuracy of kernel density estimators, so we use the standard multivariate normal kernel throughout: ${\textstyle K_{\mathbf {H} }(\mathbf {x} )={(2\pi )^{-d/2}}\mathbf {|H|} ^{-1/2}e^{-{\frac {1}{2}}\mathbf {x^{T}} \mathbf {H^{-1}} \mathbf {x} }}$, where H plays the role of the covariance matrix. On the other hand, the choice of the bandwidth matrix H is the single most important factor affecting its accuracy since it controls the amount and orientation of smoothing induced.^{[5]: 36–39} That the bandwidth matrix also induces an orientation is a basic difference between multivariate kernel density estimation from its univariate analogue since orientation is not defined for 1D kernels. This leads to the choice of the parametrisation of this bandwidth matrix. The three main parametrisation classes (in increasing order of complexity) are S, the class of positive scalars times the identity matrix; D, diagonal matrices with positive entries on the main diagonal; and F, symmetric positive definite matrices. The S class kernels have the same amount of smoothing applied in all coordinate directions, D kernels allow different amounts of smoothing in each of the coordinates, and F kernels allow arbitrary amounts and orientation of the smoothing. Historically S and D kernels are the most widespread due to computational reasons, but research indicates that important gains in accuracy can be obtained using the more general F class kernels.^[6][7]

Optimal bandwidth matrix selection

The most commonly used optimality criterion for selecting a bandwidth matrix is the MISE or mean integrated squared error

${\displaystyle \operatorname {MISE} (\mathbf {H} )=\operatorname {E} \!\left.}$

This in general does not possess a closed-form expression, so it is usual to use its asymptotic approximation (AMISE) as a proxy

${\displaystyle \operatorname {AMISE} (\mathbf {H} )=n^{-1}|\mathbf {H} |^{-1/2}R(K)+{\tfrac {1}{4}}m_{2}(K)^{2}(\operatorname {vec} ^{T}\mathbf {H} )\mathbf {\Psi } _{4}(\operatorname {vec} \mathbf {H} )}$

where

• ${\displaystyle R(K)=\int K(\mathbf {x} )^{2}\,d\mathbf {x} }$, with R(K) = (4π)^−d/2 when K is a normal kernel
• ${\displaystyle \int \mathbf {x} \mathbf {x} ^{T}K(\mathbf {x} )\,d\mathbf {x} =m_{2}(K)\mathbf {I} _{d}}$,

with I_d being the d × d identity matrix, with m₂ = 1 for the normal kernel

• D²ƒ is the d × d Hessian matrix of second order partial derivatives of ƒ
• ${\displaystyle \mathbf {\Psi } _{4}=\int (\operatorname {vec} \,\operatorname {D} ^{2}f(\mathbf {x} ))(\operatorname {vec} ^{T}\operatorname {D} ^{2}f(\mathbf {x} ))\,d\mathbf {x} }$ is a d² × d² matrix of integrated fourth order partial derivatives of ƒ
• vec is the vector operator which stacks the columns of a matrix into a single vector e.g. ${\displaystyle \operatorname {vec} {\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}a&b&c&d\end{bmatrix}}^{T}.}$

The quality of the AMISE approximation to the MISE^[5]: 97 is given by

${\displaystyle \operatorname {MISE} (\mathbf {H} )=\operatorname {AMISE} (\mathbf {H} )+o(n^{-1}|\mathbf {H} |^{-1/2}+\operatorname {tr} \,\mathbf {H} ^{2})}$

where o indicates the usual small o notation. Heuristically this statement implies that the AMISE is a 'good' approximation of the MISE as the sample size n → ∞.

It can be shown that any reasonable bandwidth selector H has H = O(n^−2/(d+4)) where the big O notation is applied elementwise. Substituting this into the MISE formula yields that the optimal MISE is O(n^−4/(d+4)).^{[5]: 99–100} Thus as n → ∞, the MISE → 0, i.e. the kernel density estimate converges in mean square and thus also in probability to the true density f. These modes of convergence are confirmation of the statement in the motivation section that kernel methods lead to reasonable density estimators. An ideal optimal bandwidth selector is

${\displaystyle \mathbf {H} _{\operatorname {AMISE} }=\operatorname {argmin} _{\mathbf {H} \in F}\,\operatorname {AMISE} (\mathbf {H} ).}$

Since this ideal selector contains the unknown density function ƒ, it cannot be used directly. The many different varieties of data-based bandwidth selectors arise from the different estimators of the AMISE. We concentrate on two classes of selectors which have been shown to be the most widely applicable in practice: smoothed cross validation and plug-in selectors.

Plug-in

The plug-in (PI) estimate of the AMISE is formed by replacing Ψ₄ by its estimator ${\displaystyle {\hat {\mathbf {\Psi } }}_{4}}$

$$Zdroj:https://en.wikipedia.org?pojem=Multivariate_kernel_density_estimation
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