Kernel feature significance

Kernel feature significance for 1- to 6-dimensional data.

Usage

kfs(x, H, h, deriv.order=2, gridsize, gridtype, xmin, xmax, supp=3.7,
    eval.points, binned, bgridsize, positive=FALSE, adj.positive, w, 
    verbose=FALSE, signif.level=0.05)

Arguments

x: matrix of data values
H,h: bandwidth matrix/scalar bandwidth. If these are missing, Hpi or hpi is called by default.
deriv.order: derivative order (scalar)
gridsize: vector of number of grid points
gridtype: not yet implemented
xmin,xmax: vector of minimum/maximum values for grid
supp: effective support for standard normal
eval.points: vector or matrix of points at which estimate is evaluated
binned: flag for binned estimation
bgridsize: vector of binning grid sizes
positive: flag if 1-d data are positive. Default is FALSE.
adj.positive: adjustment applied to positive 1-d data
w: vector of weights. Default is a vector of all ones.
verbose: flag to print out progress information. Default is FALSE.
signif.level: overall level of significance for hypothesis tests. Default is 0.05.

Value

A kernel feature significance estimate is an object of class kfs which is a list with fields

x: data points - same as input
eval.points: vector or list of points at which the estimate is evaluated
estimate: binary matrix for significant feature at eval.points: 0 = not signif., 1 = signif.
h: scalar bandwidth (1-d only)
H: bandwidth matrix
gridtype: "linear"
gridded: flag for estimation on a grid
binned: flag for binned estimation
names: variable names
w: vector of weights
deriv.order: derivative order (scalar)
deriv.ind: martix where each row is a vector of partial derivative indices.

This is the same structure as a kdde object, except that estimate is a binary matrix rather than real-valued.

Details

Feature significance is based on significance testing of the gradient (first derivative) and curvature (second derivative) of a kernel density estimate. Only the latter is currently implemented, and is also known as significant modal regions.

The hypothesis test at a grid 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 \(p\)-values are computed for each \(\bold{x}\) using that the test statistic is approximately chi-squared distributed with \(d(d+1)/2\) d.f. We then use a Hochberg-type simultaneous testing procedure, based on the ordered \(p\)-values, to control the overall level of significance to be signif.level. If \(H_0(\bold{x})\) is rejected then \(\bold{x}\) belongs to a significant modal region.

The computations are based on kdde(x, deriv.order=2) so kfs inherits its behaviour from kdde. If the bandwidth H is missing, then the default bandwidth is the plug-in selector Hpi(deriv.order=2). Likewise for missing h. The effective support, binning, grid size, grid range, positive parameters are the same as kde.

This function is similar to the featureSignif function in the feature package, except that it accepts unconstrained bandwidth matrices.

References

Chaudhuri, P. & Marron, J.S. (1999) SiZer for exploration of structures in curves. Journal of the American Statistical Association 94, 807–823.

Duong, T., Cowling, A., Koch, I. & Wand, M.P. (2008) Feature significance for multivariate kernel density estimation. Computational Statistics and Data Analysis 52, 4225–4242.

Godtliebsen, F., Marron, J.S. & Chaudhuri, P. (2002) Significance in scale space for bivariate density estimation. Journal of Computational and Graphical Statistics 11, 1–22.

Examples

data(geyser, package="MASS")
geyser.fs <- kfs(geyser$duration, binned=TRUE)
plot(geyser.fs, xlab="duration")


## see example in ? plot.kfs