kappa_tools {archetypal}

Compute kappa tools for data dimensionality analysis

Description

For a given data set and a given Archetypal Analysis (AA) solution, it finds a set of useful proxies for the dimensionality.

Usage

kappa_tools(aa, df = NULL, numBins = 100, chvertices = NULL, verbose = FALSE, ...)

Arguments

aa

An object of the class 'archetypal'

df

The data frame that was used for AA

numBins

The number of bins to be used for computing entropy

chvertices

The Convex Hull vertices, if they are given

verbose

Logical, set to TRUE if details must be printed

...

Other areguments, not used.

Details

The ECDF for the Squared Errors (SE) is computed and then the relevant curve is classified as 'convex' or 'concave' and its UIK & inflcetion point is found. Then the number of used rows for cfreating archetypes is found. A procedure for creating BIC and andjusted BIC is used. Finally the pecentage of used points that lie on the exact Convex Hull is given.

Value

A list with next arguments:

ecdf

The ECDF of SE

Convexity

The convex or concave classification for ECDF curve

UIK

The UIK points of ECDF curve by using [1]

INFLECTION

The inflection points of ECDF curve by using [2]

NumberRowsUsed

The number of rows used for creating archetypes

RowsUsed

The exact rows used for creating archetypes

SSE

The Sum of SE

BIC

The computed BIC by using [3], [4]

adjBIC

The computed adjusted BIC by using [3], [4]

CXHE

The percentage of used points that lie on the exact Convex Hull

Author(s)

Demetris T. Christopoulos, David F. Midgley (creator of BIC and adjBIC procedures)

References

[1] Demetris T. Christopoulos, Introducing Unit Invariant Knee (UIK) As an Objective Choice for Elbow Point in Multivariate Data Analysis Techniques (March 1, 2016). Available at SSRN: https://ssrn.com/abstract=3043076 or http://dx.doi.org/10.2139/ssrn.3043076

[2] Demetris T. Christopoulos, On the efficient identification of an inflection point,International Journal of Mathematics and Scientific Computing,(ISSN: 2231-5330), vol. 6(1), 2016.

[3] Felix Abramovich, Yoav Benjamini, David L. Donoho, Iain M. Johnstone. "Adapting to unknown sparsity by controlling the false discovery rate." The Annals of Statistics, 34(2) 584-653 April 2006. https://doi.org/10.1214/009053606000000074

[4] Murari, Andrea, Emmanuele Peluso, Francesco Cianfrani, Pasquale Gaudio, and Michele Lungaroni. 2019. "On the Use of Entropy to Improve Model Selection Criteria" Entropy 21, no. 4: 394. https://doi.org/10.3390/e21040394

Examples

{
	## Use the sample data "wd2"
	data(wd2)
	require("geometry")
	ch=convhulln(as.matrix(wd2),'Fx')
	chlist=as.list(ch)
	chvertices = unique(do.call(c,chlist))
	aa=archetypal(wd2, 3)
	out=kappa_tools(aa ,  df = wd2, numBins = 100, chvertices, verbose = T )
	out

}