logMINDEX {IDmining}

The Multipoint Morisita Index in 1, 2 or Higher Dimensions

Description

Computes the ln values of the multipoint Morisita index in 1, 2 or higher dimensional spaces.

Usage

logMINDEX(X, scaleQ=1:5, mMin=2, mMax=2)

Arguments

X

A N \times E matrix, data.frame or data.table where N is the number of data points and E is the number of variables (or features). Each variable is rescaled to the [0,1] interval by the function.

scaleQ

Either a single value or a vector. It contains the value(s) of \ell^{-1} chosen by the user (by default: scaleQ = 1:5).

mMin

The minimum value of m (by default: mMin = 2).

mMax

The maximum value of m (by default: mMax = 2).

Details

1. \ell is the edge length of the grid cells (or quadrats). Since the variables (and consenquently the grid) are rescaled to the [0,1] interval, \ell is equal to 1 for a grid consisting of only one cell.

2. \ell^{-1} is the number of grid cells (or quadrats) along each axis of the Euclidean space in which the data points are embedded.

3. \ell^{-1} is equal to Q^{(1/E)} where Q is the number of grid cells and E is the number of variables (or features).

4. \ell^{-1} is directly related to \delta (see References).

5. \delta is the diagonal length of the grid cells.

Value

A data.frame containing the \ln value of the m-Morisita index for each value of \ln (\delta) and m. Notice also that the values of \ln (\delta) are provided with regard to the [0,1] interval.

Author(s)

Jean Golay [e-mail]

References

J. Golay and M. Kanevski (2015). A new estimator of intrinsic dimension based on the multipoint Morisita index, Pattern Recognition 48 (12):4070–4081.

Examples

sim_dat <- SwissRoll(1000)

m <- 2
scaleQ <- 1:15 # It starts with a grid of 1^E cell (or quadrat).
               # It ends with a grid of 15^E cells (or quadrats).
lnmMI <- logMINDEX(sim_dat, scaleQ, m, m)

dev.new(width=5, height=4)
plot(exp(lnmMI[,1]),exp(lnmMI[,2]),pch=19,col="black",xlab="",ylab="")
title(xlab = expression(delta), cex.lab = 1.5,line = 2.5)
title(ylab = expression(I['2,'*delta]), cex.lab = 1.5,line = 2.5)

dev.new(width=5, height=4)
plot(lnmMI[,1],lnmMI[,2],pch=19,col="black",xlab="",ylab="")
title(xlab = expression(paste("log(",delta,")")), cex.lab = 1.5,line = 2.5)
title(ylab = expression(paste("log(",I['2,'*delta],")")), cex.lab = 1.5,line = 2.5)