These functions compute the `coverage coefficient' \(R_C\) for local principal curves, local principal points (i.e., kernel density estimates obtained through iterated mean shift), and other principal objects.
Rc(x,...)# S3 method for lpc
Rc(x,...)
# S3 method for lpc.spline
Rc(x,...)
# S3 method for ms
Rc(x,...)
base.Rc(data, closest.coords, type="curve")
an object used to select a method.
Further arguments passed to or from other methods (not needed yet).
A data matrix.
A matrix of coordinates of the projected data.
For principal curves, don't modify. For principal points, set "points".
J. Einbeck.
Rc computes the coverage coefficient \(R_C\), a quantity which
estimates the goodness-of-fit of a fitted principal object. This
quantity can be interpreted similar to the coefficient of determination in
regression analysis: Values close to 1 indicate a good fit, while values
close to 0 indicate a `bad' fit (corresponding to linear PCA).
For objects of type lpc, lpc.spline, and ms, S3 methods are available which use the generic function
Rc. This, in turn, calls the base function base.Rc, which
can also be used manually if the fitted object is of another class.
In principle, function base.Rc can be used for assessing
goodness-of-fit of any principal object provided that
the coordinates (closest.coords) of the projected data are
available. For instance, for HS principal curves fitted via
princurve, this information is contained in component $s,
and for a a k-means object, say fitk, this information can be
obtained via fitk$centers[fitk$cluster,]. Set type="points" in
the latter case.
The function Rc attempts to compute all missing information, so
computation will take the longer the less informative the given
object x is. Note also, Rc looks up the option scaled in the fitted
object, and accounts for the scaling automatically. Important: If the data
were scaled, then do NOT unscale the results by hand in order to feed
the unscaled version into base.Rc, this will give a wrong result.
In terms of methodology, these functions compute \(R_C\) directly through the mean reduction of absolute residual length, rather than through the area above the coverage curve.
These functions do currently not account for observation weights, i.e. \(R_C\) is computed through the unweighted mean reduction in absolute residual length (even if weights have been used for the curve fitting).
In the clustering context, a value of \(R_C=0.8\) means that, after the clustering, the mean absolute residual length has been reduced by \(80\%\) (compared to the distances to the overall mean).
Einbeck, Tutz, and Evers (2005). Local principal curves. Statistics and Computing 15, 301-313.
Einbeck (2011). Bandwidth selection for nonparametric unsupervised learning techniques -- a unified approach via self-coverage. Journal of Pattern Recognition Research 6, 175-192.
lpc.spline, ms, coverage.
data(calspeedflow)
lpc1 <- lpc.spline(lpc(calspeedflow[,3:4]), project=TRUE)
Rc(lpc1)
# \donttest{
# is the same as:
base.Rc(lpc1$lpcobject$data, lpc1$closest.coords)
# }
# \donttest{
ms1 <- ms(calspeedflow[,3:4], plot=FALSE)
Rc(ms1)
# is the same as:
base.Rc(ms1$data, ms1$cluster.center[ms1$closest.label,], type="points")
# }
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