Kernel discriminant analysis (kernel classification)

Kernel discriminant analysis (kernel classification) for 1- to d-dimensional data.

Usage

kda(x, x.group, Hs, hs, prior.prob=NULL, gridsize, xmin, xmax, supp=3.7,
  eval.points, binned, bgridsize, w, compute.cont=TRUE, approx.cont=TRUE,
  kde.flag=TRUE)
Hkda(x, x.group, Hstart, bw="plugin", ...)
Hkda.diag(x, x.group, bw="plugin", ...)
hkda(x, x.group, bw="plugin", ...)

# S3 method for class 'kda'
predict(object, ..., x)

compare(x.group, est.group, by.group=FALSE)
compare.kda.cv(x, x.group, bw="plugin", prior.prob=NULL, Hstart, by.group=FALSE,
   verbose=FALSE, recompute=FALSE, ...)
compare.kda.diag.cv(x, x.group, bw="plugin", prior.prob=NULL, by.group=FALSE, 
   verbose=FALSE, recompute=FALSE, ...)

Arguments

x: matrix of training data values
x.group: vector of group labels for training data
Hs,hs: (stacked) matrix of bandwidth matrices/vector of scalar bandwidths. If these are missing, Hkda or hkda is called by default.
prior.prob: vector of prior probabilities
gridsize: vector of grid sizes
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
w: vector of weights. Not yet implemented.
compute.cont: flag for computing 1% to 99% probability contour levels. Default is TRUE.
approx.cont: flag for computing approximate probability contour levels. Default is TRUE.
kde.flag: flag for computing KDE on grid. Default is TRUE.
object: object of class kda
bw: bandwidth: "plugin" = plug-in, "lscv" = LSCV, "scv" = SCV
Hstart: (stacked) matrix of initial bandwidth matrices, used in numerical optimisation
est.group: vector of estimated group labels
by.group: flag to give results also within each group
verbose: flag for printing progress information. Default is FALSE.
recompute: flag for recomputing the bandwidth matrix after excluding the i-th data item
...: other optional parameters for bandwidth selection, see Hpi, Hlscv, Hscv

Value

–For kde.flag=TRUE, a kernel discriminant analysis is an object of class kda which is a list with fields

x: list of data points, one for each group label
estimate: list of density estimates at eval.points, one for each group label
eval.points: vector or list of points that the estimate is evaluated at, one for each group label
h: vector of bandwidths (1-d only)
H: stacked matrix of bandwidth matrices or vector of bandwidths
gridded: flag for estimation on a grid
binned: flag for binned estimation
w: vector of weights
prior.prob: vector of prior probabilities
x.group: vector of group labels - same as input
x.group.estimate: vector of estimated group labels. If the test data eval.points are given then these are classified. Otherwise the training data x are classified.

For kde.flag=FALSE, which is always the case for d > 3, then only the vector of estimated group labels is returned.

–The result from Hkda and Hkda.diag is a stacked matrix of bandwidth matrices, one for each training data group. The result from hkda is a vector of bandwidths, one for each training group.

–The compare functions create a comparison between the true group labels x.group and the estimated ones. It returns a list with fields

cross: cross-classification table with the rows indicating the true group and the columns the estimated group
error: misclassification rate (MR)

In the case where the test data are independent of the training data, compare computes MR = (number of points wrongly classified)/(total number of points). In the case where the test data are not independent e.g. we are classifying the training data set itself, then the cross validated estimate of MR is more appropriate. These are implemented as compare.kda.cv (unconstrained bandwidth selectors) and compare.kda.diag.cv (for diagonal bandwidth selectors). These functions are only available for d > 1.

If by.group=FALSE then only the total MR rate is given. If it is set to TRUE, then the MR rates for each class are also given (estimated number in group divided by true number).

References

Simonoff, J. S. (1996) Smoothing Methods in Statistics. Springer-Verlag. New York

Details

If the bandwidths Hs are missing from kda, then the default bandwidths are the plug-in selectors Hkda(bw="plugin"). Likewise for missing hs. Valid options for bw are "plugin", "lscv" and "scv" which in turn call Hpi, Hlscv and Hscv.

The effective support, binning, grid size, grid range, positive parameters are the same as kde.

If prior probabilities are known then set prior.prob to these. Otherwise prior.prob=NULL uses the sample proportions as estimates of the prior probabilities.

For ks \(\geq\) 1.8.11, kda.kde has been subsumed into kda, so all prior calls to kda.kde can be replaced by kda. To reproduce the previous behaviour of kda, the command is kda(kde.flag=FALSE).

Examples

set.seed(8192)
x <- c(rnorm.mixt(n=100, mus=1), rnorm.mixt(n=100, mus=-1))
x.gr <- rep(c(1,2), times=c(100,100))
y <- c(rnorm.mixt(n=100, mus=1), rnorm.mixt(n=100, mus=-1))
y.gr <- rep(c(1,2), times=c(100,100))
kda.gr <- kda(x, x.gr)
y.gr.est <- predict(kda.gr, x=y)
compare(y.gr, y.gr.est)
#> $cross
#>          1 (est.) 2 (est.) Total
#> 1 (true)       85       15   100
#> 2 (true)       14       86   100
#> Total          99      101   200
#> 
#> $error
#> [1] 0.145
#> 

## See other examples in ? plot.kda