Kernel density based global two-sample comparison test
Kernel density based global two-sample comparison test for 1- to 6-dimensional data.
kde.test(x1, x2, H1, H2, h1, h2, psi1, psi2, var.fhat1, var.fhat2,
binned=FALSE, bgridsize, verbose=FALSE)x1,x2 |
vector/matrix of data values |
H1,H2,h1,h2 |
bandwidth matrices/scalar bandwidths. If these are
missing, |
psi1,psi2 |
zero-th order kernel functional estimates |
var.fhat1,var.fhat2 |
sample variance of KDE estimates evaluated at x1, x2 |
binned |
flag for binned estimation. Default is FALSE. |
bgridsize |
vector of binning grid sizes |
verbose |
flag to print out progress information. Default is FALSE. |
The null hypothesis is H_0: f_1 = f_2 where f_1, f_2 are the respective density functions. The measure of discrepancy is the integrated squared error (ISE) int [ f_1(x) - f_2(x)]^2 dx. If we rewrite this as T = psi_0,1 - psi_0,12 - psi_0,21 + psi_0,2 where psi_0,uv = int f_u(x) f_v(x) dx, then we can use kernel functional estimators. This test statistic has a null distribution which is asymptotically normal, so no bootstrap resampling is required to compute an approximate p-value.
If H1,H2 are missing then the plug-in selector Hpi.kfe
is automatically called by kde.test to estimate the
functionals with kfe(, deriv.order=0). Likewise for missing
h1,h2.
For ks >= 1.8.8, kde.test(,binned=TRUE) invokes binned
estimation for the computation of the bandwidth selectors, and not the
test statistic and p-value.
A kernel two-sample global significance test is a list with fields:
Tstat |
T statistic |
zstat |
z statistic - normalised version of Tstat |
pvalue |
p-value of the double sided test |
mean,var |
mean and variance of null distribution |
var.fhat1,var.fhat2 |
sample variances of KDE values evaluated at data points |
n1,n2 |
sample sizes |
H1,H2 |
bandwidth matrices |
psi1,psi12,psi21,psi2 |
kernel functional estimates |
Duong, T., Goud, B. & Schauer, K. (2012) Closed-form density-based framework for automatic detection of cellular morphology changes. PNAS, 109, 8382-8387.
set.seed(8192) samp <- 1000 x <- rnorm.mixt(n=samp, mus=0, sigmas=1, props=1) y <- rnorm.mixt(n=samp, mus=0, sigmas=1, props=1) kde.test(x1=x, x2=y)$pvalue ## accept H0: f1=f2 library(MASS) data(crabs) x1 <- crabs[crabs$sp=="B", c(4,6)] x2 <- crabs[crabs$sp=="O", c(4,6)] kde.test(x1=x1, x2=x2)$pvalue ## reject H0: f1=f2
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