Multivariate kernel density estimation

Statistics is a field of quantitative analysis concerned with quantifying uncertainty. The main building block of statistical analysis is a random variable. A random variable is a mathematics function which assigns a numerical value to each possible value of the variable of interest. The complete behaviour of a random variable is contained in its distribution function. For continuous random variables, the partial derivative of the distribution function is known as probability density function or pdf. So density estimation is a fundamental question in statistics.

Kernel density estimation is one of the most popular techniques for density estimation. It can be viewed as a generalisation of histogram density estimation with improved statistical properties. 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 analagous 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 their univariate counterparts.

Motivation

To motivate the definition of multivariate kernel density estimators, we take as an illustrative bivariate data set drawn from ....

Problems with bivariate histograms.

Definition

Let ${\displaystyle {\mathbf {X}}_{1},{\mathbf {X}}_{2},\dots ,{\mathbf {X}}_{n}}$  be a d-variate random sample drawn from a common density function f. The kernel density estimate is defined to be

${\displaystyle {\hat {f}}_{\mathbf {H}}({\mathbf {x}})=n^{-1}|{\mathbf {H}}|^{-1/2}\sum _{i=1}^{n}K_{\mathbf {H}}({\mathbf {x}}-{\mathbf {X}}_{i})}$ 

where

• ${\displaystyle {\mathbf {x}}=(x_{1},x_{2},\dots ,x_{d})^{T}}$ , ${\displaystyle {\mathbf {X}}_{i}=(X_{i1},X_{i2},\dots ,X_{id})^{T},i=1,2,\dots ,n}$  are d-vectors
• K is the kernel function which is a symmetric density function, with ${\displaystyle K_{\mathbf {H}}({\mathbf {x}})=|{\mathbf {H}}|^{-1/2}K({\mathbf {H}}^{-1/2}{\mathbf {x}})}$ 
• H is the bandwidth (or smoothing) matrix which is a symmetric, positive definite d x d matrix.

The choice of the kernel function K is not crucial to the accuracy of kernel density estimators, whereas the choice of the bandwidth matrix H is the single most important factor affecting its accuracy ^[3](pp. 36-39). So we use the standard multivariate normal or Gaussian density function as our kernel K

${\displaystyle K({\mathbf {x}})=(2\pi )^{-d/2}\exp(-{\tfrac {1}{2}}\,{\mathbf {x}}^{T}{\mathbf {x}}).}$ 

Optimal bandwidth matrix selection

The most commonly used optimality criterion for selecting a bandwidth matrix is the MISE or Mean Integrated Squared Error

${\displaystyle \operatorname {MISE} ({\mathbf {H}})=E\int [{\hat {f}}_{\mathbf {H}}({\mathbf {x}})-f({\mathbf {x}})]^{2}\,d{\mathbf {x}}.}$ 

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

${\displaystyle \operatorname {AMISE} ({\mathbf {H}})=n^{-1}|{\mathbf {H}}|^{-1/2}R(K)+{\tfrac {1}{4}}m_{2}(K)^{2}\int \operatorname {tr} ^{2}({\mathbf {H}}\operatorname {D} ^{2}f({\mathbf {x}}))\,d{\mathbf {x}}}$ 

where

• ${\displaystyle R(K)=\int K({\mathbf {x} })^{2}\,d{\mathbf {x} }}$ . For the normal kernel ${\displaystyle K}$ , ${\displaystyle R(K)=(4\pi )^{-d/2}}$ 
• ${\displaystyle \int {\mathbf {x}}{\mathbf {x}}^{T}K({\mathbf {x}})^{2}\,d{\mathbf {x}}}$  = m_2(K) \bold{I}_d</math>, with ${\displaystyle {\mathbf {I}}_{d}}$  is the d x d identity matrix
• ${\displaystyle \operatorname {D} ^{2}f}$  is the d x d Hessian matrix of second order partial derivatives of ${\displaystyle f}$ 

This formula of the AMISE is due to ^[3](p. 97). The quality of the AMISE approximation to the MISE is given by

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

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

The ideal optimal bandwidth selector is ${\displaystyle {\mathbf {H}}_{\operatorname {AMISE} }=\operatorname {argmin} _{{\mathbf {H}}\in F}\,\operatorname {AMISE} ({\mathbf {H}})}$  where F is the space of all symmetric, positive definite matrices. Since this ideal selector contains the unknown density function >em>f, 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 practise: smoothed cross validation and plug-in selectors.

Plug-in

The plug-in (PI) selector of the AMISE is formed by replacing the Hessian matrix by its estimator

${\displaystyle \operatorname {PI} ({\mathbf {H}})=n^{-1}|{\mathbf {H}}|^{-1/2}R(K)+{\tfrac {1}{4}}m_{2}(K)^{2}\int \operatorname {tr} ^{2}({\mathbf {H}}{\widehat {\operatorname {D} ^{2}f}}_{\mathbf {G}}({\mathbf {x}}))\,d{\mathbf {x}}}$ 

where ${\displaystyle {\widehat {\operatorname {D} ^{2}f}}_{\mathbf {G}}({\mathbf {x}})=n^{-1}\sum _{i=1}^{n}D^{2}K_{\mathbf {G}}({\mathbf {x}}-{\mathbf {X}}_{i})}$ . Thus ${\displaystyle {\hat {\mathbf {H} }}_{\operatorname {PI} }=\operatorname {argmin} _{{\mathbf {H} }\in F}\,\operatorname {PI} ({\mathbf {H} })}$  is the plug-in selector^[4][5]. These references also contain algorithms on optimal estimation of the pilot bandwidth matrix G.

Smoothed cross validation

Smoothed cross validation (SCV) is a subset of a larger class of cross validation techniques. The SCV estimator differs from the plug-in estimator in the second term

${\displaystyle \operatorname {SCV} ({\mathbf {H} })=n^{-1}|{\mathbf {H} }|^{-1/2}R(K)+n^{-2}\sum _{i=1}^{n}\sum _{j=1}^{n}(K_{2{\mathbf {H} }+2{\mathbf {G} }}-2K_{{\mathbf {H} }+2{\mathbf {G} }}+K_{2{\mathbf {G} }})({\mathbf {X} }_{i}-{\mathbf {X} }_{j})}$ 

Thus ${\displaystyle {\hat {\mathbf {H} }}_{\operatorname {SCV} }=\operatorname {argmin} _{{\mathbf {H} }\in F}\,\operatorname {SCV} ({\mathbf {H} })}$  is the SCV selector^[6][7]. These references also contain algorithms on optimal estimation of the pilot bandwidth matrix G.

Implementation in the R statistical software

The ks package in the R programming language implements the plug-in and smoothed cross validation selectors. This example is based on the Old Faithful Geyser in Yellowstone National Park, USA. This dataset contains 272 records with two measurements each: the eruption duration time (minutes) and the waiting time until the next eruption (minutes), contained in the base distribution of R.

 

This code snippet computes the kernel density estimate with the plug-in bandwidth matrix. The coloured contours correspond to the smallest region which contains that corresponding probability mass: red = 25%, orange + red = 50%, yellow + orange + red = 75%. To compute the SCV selector, replace Hpi with Hscv.

library(ks)
data(faithful)
H <- Hpi(x=faithful)
fhat <- kde(x=faithful, H=H)
plot(fhat, display="filled.contour2")
points(faithful)

References

1. Rosenblatt, M. (1956). "Remarks on some nonparametric estimates of a density function". Annals of Mathematical Statistics. 27: 832–837. doi:10.1214/aoms/1177728190.
2. Parzen, E. (1962). "On estimation of a probability density function and mode". Annals of Mathematical Statistics. 33: 1065–1076. doi:10.1214/aoms/1177704472.
3. Wand, M.P; Jones, M.C. (1995). Kernel Smoothing. London: Chapman & Hall. ISBN 0412552701. Cite error: The named reference "WJ1995" was defined multiple times with different content (see the help page).
4. Wand, M.P.; Jones, M.C. (1994). "Multivariate plug-in bandwidth selection". Computational Statistics. 9: 97–177.
5. Duong, T.; Hazelton, M.L. (2003). "Plug-in bandwidth matrices for bivariate kernel density estimation". Journal of Nonparametric Statistics. 15: 17–30. doi:10.1080/10485250306039.
6. Hall, P.; Marron, J.; Park, B. (1992). "Smoothed cross-validation". Probability Theory and Related Fields. 92: 1–20. doi:10.1007/BF01205233.
7. Duong, T.; Hazelton, M.L. (2005). "Cross validation bandwidth matrices for multivariate kernel density estimation". Scandinavian Journal of Statistics. 32: 485–506. doi:10.1111/j.1467-9469.2005.00445.x.

External links

• example.com

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