kfs: Kernel feature significance

Description

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

Usage

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

Arguments

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. Default is FALSE.

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

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

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.

scalar bandwidth (1-d only)

bandwidth matrix

gridtype

"linear"

gridded

flag for estimation on a grid

binned

flag for binned estimation

names

variable names

weights

deriv.order

derivative order (scalar)

deriv.ind

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 from kfs, 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

Run this code

# NOT RUN {
## see example is ? plot.fks
# }

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