Kernel smoothing for data from 1- to 6-dimensions.
There are three main types of functions in this package:
computing kernel estimators - these function names begin with `k'
computing bandwidth selectors - these begin with `h' (1-d) or `H' (>1-d)
displaying kernel estimators - these begin with `plot'.
The kernel used throughout is the normal (Gaussian) kernel \(K\). For 1-d data, the bandwidth \(h\) is the standard deviation of the normal kernel, whereas for multivariate data, the bandwidth matrix \(\bold{{\rm H}}\) is the variance matrix.
--For kernel density estimation, kde computes
$$\hat{f}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i).$$
The bandwidth matrix \(\bold{{\rm H}}\) is a matrix of smoothing
parameters and its choice is crucial for the performance of kernel
estimators. For display, its plot method calls plot.kde.
--For kernel density estimation, there are several varieties of bandwidth selectors
least squares (or unbiased) cross validation (LSCV or UCV) hlscv (1-d);
Hlscv, Hlscv.diag (2- to 6-d)
smoothed cross validation (SCV) hscv (1-d);
Hscv, Hscv.diag (2- to 6-d)
--For kernel density derivative estimation, the main function is kdde
$${\sf D}^{\otimes r}\hat{f}(\bold{x}) = n^{-1} \sum_{i=1}^n {\sf
D}^{\otimes r}K_{\bold{{\rm H}}} (\bold{x} -
\bold{X}_i).$$
The bandwidth selectors are a modified subset of those for
kde, i.e. Hlscv, Hns, Hpi, Hscv with deriv.order>0.
Its plot method is plot.kdde for plotting each
partial derivative singly.
--For kernel discriminant analysis, the main function is
kda which computes density estimates for each the
groups in the training data, and the discriminant surface.
Its plot method is plot.kda. The wrapper function
hkda, Hkda computes
bandwidths for each group in the training data for kde,
e.g. hpi, Hpi.
--For kernel functional estimation, the main function is
kfe which computes the \(r\)-th order integrated density functional
$$\hat{{\bold \psi}}_r = n^{-2} \sum_{i=1}^n \sum_{j=1}^n {\sf D}^{\otimes r}K_{\bold{{\rm H}}}(\bold{X}_i-\bold{X}_j).$$ The plug-in selectors are hpi.kfe (1-d), Hpi.kfe (2- to 6-d).
Kernel functional estimates are usually not required to computed
directly by the user, but only within other functions in the package.
--For kernel-based 2-sample testing, the main function is
kde.test which computes the integrated
\(L_2\) distance between the two density estimates as the test
statistic, comprising a linear combination of 0-th order kernel
functional estimates:
$$\hat{T} = \hat{\psi}_{0,1} + \hat{\psi}_{0,2} - (\hat{\psi}_{0,12} +
\hat{\psi}_{0,21}),$$ and the corresponding p-value. The \(\psi\) are
zero order kernel functional estimates with the subscripts indicating
that 1 = sample 1 only, 2 = sample 2 only, and 12, 21 =
samples 1 and 2. The bandwidth selectors are hpi.kfe,
Hpi.kfe with deriv.order=0.
--For kernel-based local 2-sample testing, the main function is
kde.local.test which computes the squared distance
between the two density estimates as the test
statistic $$\hat{U}(\bold{x}) = [\hat{f}_1(\bold{x}) - \hat{f}_2(\bold{x})]^2$$ and the corresponding local
p-values. The bandwidth selectors are those used with kde,
e.g. hpi, Hpi.
--For kernel cumulative distribution function estimation, the main
function is kcde
$$\hat{F}(\bold{x}) = n^{-1} \sum_{i=1}^n
\mathcal{K}_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i)$$
where \(\mathcal{K}\) is the integrated kernel.
The bandwidth selectors are hpi.kcde,
Hpi.kcde. Its plot method is
plot.kcde.
There exist analogous functions for the survival function \(\hat{\bar{F}}\).
--For kernel estimation of a ROC (receiver operating characteristic)
curve to compare two samples from \(\hat{F}_1,
\hat{F}_2\), the main function is kroc
$$(\hat{F}_{\hat{Y}_1}(z),
\hat{F}_{\hat{Y}_2}(z))$$ based on the cumulative distribution functions of
\(\hat{Y}_j = \hat{\bar{F}}_1(\bold{X}_j), j=1,2\).
The bandwidth selectors are those used with kcde,
e.g. hpi.kcde, Hpi.kcde for
\(\hat{F}_{\hat{Y}_j}, \hat{\bar{F}}_1\). Its plot method
is plot.kroc.
--For kernel estimation of a copula, the
main function is kcopula
$$\hat{C}(\bold{z}) = \hat{F}(\hat{F}_1^{-1}(z_1), \dots,
\hat{F}_d^{-1}(z_d))$$
where \(\hat{F}_j^{-1}(z_j)\) is
the \(z_j\)-th quantile of of the \(j\)-th marginal
distribution \(\hat{F}_j\).
The bandwidth selectors are those used with kcde for
\(\hat{F}, \hat{F}_j\).
Its plot method is plot.kcde.
--For kernel mean shift clustering, the main function is kms. The mean shift recurrence relation of the
candidate point \({\bold x}\)
$${\bold x}_{j+1} = {\bold x}_j + \bold{{\rm H}} {\sf D} \hat{f}({\bold
x}_j)/\hat{f}({\bold x}_j),$$
where \(j\geq 0\) and \({\bold x}_0 = {\bold x}\),
is iterated until \({\bold x}\) converges to its
local mode in the density estimate \(\hat{f}\) by following
the density gradient ascent paths. This mode determines the cluster
label for \(\bold{x}\).
The bandwidth selectors are those used with kdde(,deriv.order=1).
-- For kernel feature significance, the main function
kfs. The hypothesis test at a point \(\bold{x}\) is
\(H_0(\bold{x}): \mathsf{H} f(\bold{x}) < 0\),
i.e. the density Hessian matrix \(\mathsf{H} f(\bold{x})\) is negative definite.
The test statistic is
$$W(\bold{x}) = \Vert
\mathbf{S}(\bold{x})^{-1/2} \mathrm{vech} \ \mathsf{H} \hat{f} (\bold{x})\Vert ^2$$
where \({\sf H}\hat{f}\) is the
Hessian estimate, vech is the vector-half operator, and
\(\mathbf{S}\) is an estimate of the null variance.
\(W(\bold{x})\) is
approximately \(\chi^2\) distributed with
\(d(d+1)/2\) degrees of freedom.
If \(H_0(\bold{x})\) is rejected, then \(\bold{x}\)
belongs to a significant modal region.
The bandwidth selectors are those used with
kdde(,deriv.order=2). Its plot method is
plot.kfs.
--Binned kernel estimation is an approximation to the exact kernel estimation and is available for d=1, 2, 3, 4. This makes kernel estimators feasible for large samples.
--For an overview of this package with 2-d density estimation, see
vignette("kde").
Bowman, A. & Azzalini, A. (1997) Applied Smoothing Techniques for Data Analysis. Oxford University Press, Oxford.
Duong, T. (2004) Bandwidth Matrices for Multivariate Kernel Density Estimation. Ph.D. Thesis, University of Western Australia.
Scott, D.W. (1992) Multivariate Density Estimation: Theory, Practice, and Visualization. John Wiley & Sons, New York.
Silverman, B. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, London.
Simonoff, J. S. (1996) Smoothing Methods in Statistics. Springer-Verlag, New York.
Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing. Chapman & Hall/CRC, London.
feature, sm, KernSmooth