IDmining: Intrinsic Dimension for Data Mining

Description

Contains techniques for mining large and high-dimensional data sets by using the concept of Intrinsic Dimension (ID). Here the ID is not necessarily an integer. It is extended to fractal dimensions. And the Morisita estimator is used for the ID estimation, but other tools are included as well.

Author(s)

Jean Golay jeangolay@gmail.com and Mohamed Laib laib.med@gmail.com,

Maintainer: Jean Golay jeangolay@gmail.com

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.

J. Golay, M. Leuenberger and M. Kanevski (2017). Feature selection for regression problems based on the Morisita estimator of intrinsic dimension, Pattern Recognition 70:126–138.

J. Golay and M. Kanevski (2017). Unsupervised feature selection based on the Morisita estimator of intrinsic dimension, Knowledge-Based Systems 135:125-134.

J. Golay, M. Leuenberger and M. Kanevski (2015). Morisita-based feature selection for regression problems. Proceedings of the 23rd European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN), Bruges (Belgium).

See Also

Useful links:

• https://www.sites.google.com/site/jeangolayresearch/

Butterfly Data Set Generator

Description

Generates a random simulation of the butterfly data set with a given number of points.

Usage

Butterfly(N=10000)

Arguments

Value

A N \times 9 data.frame. The first eight columns are the input variables, and the last one is the output (or target) variable Y.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay, M. Leuenberger and M. Kanevski (2016). Feature selection for regression problems based on the Morisita estimator of intrinsic dimension, Pattern Recognition 70:126–138.

Examples

bf <- Butterfly(1000)

## Not run: 
require(colorRamps)
require(rgl)

c <- cut(bf$Y,breaks=64)
cols <- matlab.like(64)[as.numeric(c)]

plot3d(bf$X1,bf$X2,bf$Y,col=cols,radius=0.10,type="s",
       xlab="",ylab="",zlab="",box=F)
axes3d(lwd=3,cex.axis=3)
grid3d(c("x+","y-","z"),col="black",lwd=1)

## End(Not run)

Morisita-Based Filter for Regression Problems

Description

Executes the MBFR algorithm for supervised feature selection.

Usage

MBFR(XY, scaleQ, m=2, C=NULL)

Arguments

Details

1. \ell is the edge length of the grid cells (or quadrats). Since the data (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.

6. The values of \ell^{-1} in scaleQ must be chosen according to the linear part of the \log-\log plot relating the \log values of the multipoint Morisita index to the \log values of \delta (or, equivalently, to the \log values of \ell^{-1}) (see logMINDEX).

Value

A list of five elements:

1. a vector containing the identifier numbers of the original features in the order they are selected through the Sequential Forward Selection (SFS) search procedure.

2. the names of the corresponding features.

3. the corresponding values of Diss.

4. the ID estimate of the output variable.

5. a C \times 3 matrix containing: (column 1) the ID estimates of the subsets retained by the SFS procedure with the target variable; (column 2) the ID estimates of the subsets retained by the SFS procedure without the output variable; (column 3) the values of Diss of the subsets retained by the SFS procedure.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay, M. Leuenberger and M. Kanevski (2017). Feature selection for regression problems based on the Morisita estimator of intrinsic dimension, Pattern Recognition 70:126–138.

J. Golay, M. Leuenberger and M. Kanevski (2015). Morisita-based feature selection for regression problems.Proceedings of the 23rd European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN), Bruges (Belgium).

Examples

## Not run: 
bf <- Butterfly(10000)

fly_select <- MBFR(bf, 5:25)
var_order  <- fly_select[[2]]
var_perf   <- fly_select[[3]]

dev.new(width=5, height=4)
plot(var_perf,type="b",pch=16,lwd=2,xaxt="n",xlab="",ylab="",
     ylim=c(0,1),col="red",panel.first={grid(lwd=1.5)})
axis(1,1:length(var_order),labels=var_order)
mtext(1,text = "Added Features (from left to right)",line = 2.5,cex=1)
mtext(2,text = "Estimated Dissimilarity",line = 2.5,cex=1)

## End(Not run)

Morisita-Based Filter for Regression Problems (Parallel)

Description

Executes the MBFR algorithm on a chosen number of workers (CPU parallel computing).

Usage

MBFR_parallel(XY, scaleQ, m=2, C=NULL, ncores=4)

Arguments

Details

1. \ell is the edge length of the grid cells (or quadrats). Since the data (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.

6. The values of \ell^{-1} in scaleQ must be chosen according to the linear part of the \log-\log plot relating the \log values of the multipoint Morisita index to the \log values of \delta (or, equivalently, to the \log values of \ell^{-1}) (see logMINDEX).

Value

A list of five elements:

1. a vector containing the identifier numbers of the original features in the order they are selected through the Sequential Forward Selection (SFS) search procedure.

2. the names of the corresponding features.

3. the corresponding values of Diss.

4. the ID estimate of the output variable.

5. a C \times 3 matrix containing: (column 1) the ID estimates of the subsets retained by the SFS procedure with the target variable; (column 2) the ID estimates of the subsets retained by the SFS procedure without the output variable; (column 3) the values of Diss of the subsets retained by the SFS procedure.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay, M. Leuenberger and M. Kanevski (2017). Feature selection for regression problems based on the Morisita estimator of intrinsic dimension, Pattern Recognition 70:126–138.

J. Golay, M. Leuenberger and M. Kanevski (2015). Morisita-based feature selection for regression problems.Proceedings of the 23rd European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN), Bruges (Belgium).

Examples

## Not run: 
bf <- Butterfly(10000)

fly_select <- MBFR_parallel(bf, 5:25, ncores=2)
var_order  <- fly_select[[2]]
var_perf   <- fly_select[[3]]

dev.new(width=5, height=4)
plot(var_perf,type="b",pch=16,lwd=2,xaxt="n",xlab="",ylab="",
     ylim=c(0,1),col="red",panel.first={grid(lwd=1.5)})
axis(1,1:length(var_order),labels=var_order)
mtext(1,text = "Added Features (from left to right)",line = 2.5,cex=1)
mtext(2,text = "Estimated Dissimilarity",line = 2.5,cex=1)

bf_large <- Butterfly(10^5)
system.time(MBFR(bf_large, 5:25))
system.time(MBFR_parallel(bf_large, 5:25))

## End(Not run)

Morisita-Based Filter for Redundancy Minimization

Description

Executes the MBRM algorithm for unsupervised feature selection.

Usage

MBRM(X, scaleQ, m=2, C=NULL, ID_tot=NULL)

Arguments

Details

1. \ell is the edge length of the grid cells (or quadrats). Since the 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.

6. The values of \ell^{-1} in scaleQ must be chosen according to the linear part of the \log-\log plot relating the \log values of the multipoint Morisita index to the \log values of \delta (or, equivalently, to the \log values of \ell^{-1}) (see logMINDEX).

Value

A list of four elements:

1. a vector containing the identifier numbers of the original features in the order they are selected through the Sequential Forward Selection (SFS) search procedure.

2. the names of the corresponding features.

3. the corresponding ID estimates.

4. the ID estimate of the full data set.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay and M. Kanevski (2017). Unsupervised feature selection based on the Morisita estimator of intrinsic dimension, Knowledge-Based Systems 135:125-134.

Examples

## Not run: 
bf <- Butterfly(10000)

bf_select <- MBRM(bf[,-9], 5:25)
var_order <- bf_select[[2]]
var_perf  <- bf_select[[3]]

dev.new(width=5, height=4)
plot(var_perf,type="b",pch=16,lwd=2,xaxt="n",xlab="", ylab="",
     col="red",ylim=c(0,max(var_perf)),panel.first={grid(lwd=1.5)})
axis(1,1:length(var_order),labels=var_order)
mtext(1,text="Added Features (from left to right)",line=2.5,cex=1)
mtext(2,text="Estimated ID",line=2.5,cex=1)

## End(Not run)

Morisita-Based Filter for Redundancy Minimization (Parallel)

Description

Executes the MBRM algorithm for unsupervised feature selection (CPU parallel computing).

Usage

MBRM_parallel(X, scaleQ, m=2, C=NULL, ID_tot=NULL, ncores=4)

Arguments

Details

1. \ell is the edge length of the grid cells (or quadrats). Since the 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.

6. The values of \ell^{-1} in scaleQ must be chosen according to the linear part of the \log-\log plot relating the \log values of the multipoint Morisita index to the \log values of \delta (or, equivalently, to the \log values of \ell^{-1}) (see logMINDEX).

Value

A list of four elements:

1. a vector containing the identifier numbers of the original features in the order they are selected through the Sequential Forward Selection (SFS) search procedure.

2. the names of the corresponding features.

3. the corresponding ID estimates.

4. the ID estimate of the full data set.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay and M. Kanevski (2017). Unsupervised feature selection based on the Morisita estimator of intrinsic dimension, Knowledge-Based Systems 135:125-134.

Examples

bf <- Butterfly(10000)

bf_select <- MBRM_parallel(bf[,-9], 5:25, ncores=2)
var_order <- bf_select[[2]]
var_perf  <- bf_select[[3]]

## Not run: 
dev.new(width=5, height=4)
plot(var_perf,type="b",pch=16,lwd=2,xaxt="n",xlab="", ylab="",
     col="red",ylim=c(0,max(var_perf)),panel.first={grid(lwd=1.5)})
axis(1,1:length(var_order),labels=var_order)
mtext(1,text="Added Features (from left to right)",line=2.5,cex=1)
mtext(2,text="Estimated ID",line=2.5,cex=1)

bf_large <- Butterfly(10^5)
system.time(MBRM(bf_large[,-9], 5:25))
system.time(MBRM_parallel(bf_large[,-9], 5:25))

## End(Not run)

The Multipoint Morisita Index for Spatial Patterns

Description

Computes the multipoint Morisita index for spatial patterns (i.e. 2-dimensional patterns).

Usage

MINDEX_SP(X, scaleQ=1:5, mMin=2, mMax=5, Wlim_x=NULL, Wlim_y=NULL)

Arguments

Details

1. Q^{(1/2)} is the number of grid cells (or quadrats) along each of the two axes.

2. Q^{(1/2)} is directly related to \delta (see References).

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

Value

A data.frame containing the value of the m-Morisita index for each value of \delta and m.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay, M. Kanevski, C. D. Vega Orozco and M. Leuenberger (2014). The multipoint Morisita index for the analysis of spatial patterns, Physica A 406:191–202.

L. Telesca, J. Golay and M. Kanevski (2015). Morisita-based space-clustering analysis of Swiss seismicity, Physica A 419:40–47.

L. Telesca, M. Lovallo, J. Golay and M. Kanevski (2016). Comparing seismicity declustering techniques by means of the joint use of Allan Factor and Morisita index, Stochastic Environmental Research and Risk Assessment 30(1):77-90.

Examples

sim_dat <- SwissRoll(1000)

m <- 2
scaleQ <- 1:15 # It starts with a grid of 1^2 cell (or quadrat).
               # It ends with a grid of 15^2 cells (or quadrats).
mMI <- MINDEX_SP(sim_dat[,c(1,2)], scaleQ, m, 5)

plot(mMI[,1],mMI[,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)

## Not run: 
require(colorRamps)
colfunc <- colorRampPalette(c("blue","red"))
color <- colfunc(4)
dev.new(width=5,height=4)
plot(mMI[5:15,1],mMI[5:15,2],pch=19,col=color[1],xlab="",ylab="",
     ylim=c(1,max(mMI[,5])))
title(xlab=expression(delta),cex.lab=1.5,line=2.5)
title(ylab=expression(I['2,'*delta]),cex.lab=1.5,line=2.5)
for(i in 3:5){
  points(mMI[5:15,1],mMI[5:15,i],pch=19,col=color[i-1])
}
legend.text<-c("m=2","m=3","m=4","m=5")
legend.pch=c(19,19,19,19)
legend.lwd=c(NA,NA,NA,NA)
legend.col=c(color[1],color[2],color[3],color[4])
legend("topright",legend=legend.text,pch=legend.pch,lwd=legend.lwd,
       col=legend.col,ncol=1,text.col="black",cex=0.9,box.lwd=1,bg="white")

xlim_l <- c(-5,5)     # By default, the spatial extent of the grid is set so
ylim_l <- c(-6,6)     # that it is the same as the spatial extent of the data.
xlim_s <- c(-0.6,0.2) # But it can be modified to cover either a larger (l)
ylim_s <- c(-1,0.5)   # or a smaller (s) study area (or validity domain).

mMI_l <- MINDEX_SP(sim_dat[,c(1,2)], scaleQ, m, 5, xlim_l, ylim_l)
mMI_s <- MINDEX_SP(sim_dat[,c(1,2)], scaleQ, m, 5, xlim_s, ylim_s)

## End(Not run)

The (Multipoint) Morisita Index for Intrinsic Dimension Estimation

Description

Estimates the intrinsic dimension of data using the Morisita estimator of intrinsic dimension.

Usage

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

Arguments

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 list of two elements:

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

2. a data.frame containing the values of S_m and M_m for each value of m.

Author(s)

Jean Golay jeangolay@gmail.com

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.

J. Golay, M. Leuenberger and M. Kanevski (2017). Feature selection for regression problems based on the Morisita estimator of intrinsic dimension, Pattern Recognition 70:126–138.

J. Golay and M. Kanevski (2017). Unsupervised feature selection based on the Morisita estimator of intrinsic dimension, Knowledge-Based Systems 135:125-134.

J. Golay, M. Leuenberger and M. Kanevski (2015). Morisita-based feature selection for regression problems. Proceedings of the 23rd European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning (ESANN), Bruges (Belgium).

Examples

sim_dat <- SwissRoll(1000)

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).
mMI_ID <- MINDID(sim_dat, scaleQ[5:15])

print(paste("The ID estimate is equal to",round(mMI_ID[[1]][1,3],2)))

Functional Measure of Clustering Using the Morisita Estimator of ID

Description

Computes the functional m-Morisita index for a given set of threshold values.

Usage

MINDID_FMC(XY, scaleQ, m=2, thd)

Arguments

Details

1. \ell is the edge length of the grid cells (or quadrats). Since the input 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 vector containing the value(s) of the m-Morisita slope, S_m, for each threshold value.

Author(s)

Jean Golay jeangolay@gmail.com

References

J. Golay, M. Kanevski, C. D. Vega Orozco and M. Leuenberger (2014). The multipoint Morisita index for the analysis of spatial patterns, Physica A 406:191–202.

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

L. Telesca, J. Golay and M. Kanevski (2015). Morisita-based space-clustering analysis of Swiss seismicity, Physica A 419:40–47.

Examples

## Not run: 
bf    <- Butterfly(10000)
bf_SP <- bf[,c(1,2,9)]

m      <- 2
scaleQ <- 5:25
thd    <- quantile(bf_SP$Y,probs=c(0,0.1,0.2,0.3,
                                   0.4,0.5,0.6,
                                   0.7,0.8,0.9))

nbr_shuf    <- 100
Sm_thd_shuf <- matrix(0,length(thd),nbr_shuf)
for (i in 1:nbr_shuf){
  bf_SP_shuf      <- cbind(bf_SP[,1:2],sample(bf_SP$Y,length(bf_SP$Y)))
  Sm_thd_shuf[,i] <- MINDID_FMC(bf_SP_shuf, scaleQ, m, thd)
}
mean_shuf <- apply(Sm_thd_shuf,1,mean)

dev.new(width=6, height=4)
matplot(1:10,Sm_thd_shuf,type="l",lty=1,col=rgb(1,0,0,0.25),
        ylim=c(-0.05,0.05),ylab=bquote(S[.(m)]),xaxt="n",
        xlab="",cex.lab=1.2)
axis(1,1:10,labels = FALSE)
text(1:10,par("usr")[3]-0.01,srt=45,ad=1,
     labels=c("0_100", "10_100","20_100","30_100",
              "40_100","50_100","60_100",
              "70_100","80_100","90_100"),xpd=T,font=2,cex=1)
mtext("Thresholds",side=1,line=3.5,cex=1.2)
lines(1:10,mean_shuf,type="b",col="blue",pch=19)

legend.text<-c("Shuffled","mean")
legend.pch=c(NA,19)
legend.lwd=c(2,2)
legend.col=c("red","blue")
legend("topleft",legend=legend.text,pch=legend.pch,lwd=legend.lwd,
       col=legend.col,ncol=1,text.col="black",cex=1,box.lwd=1,bg="white")

## End(Not run)

Renyi's Generalized Dimensions

Description

Estimates Rényi's generalized dimensions (or Rényi's dimensions of qth order). It is mainly for q=2 that the result is used as an estimate of the intrinsic dimension of data.

Usage

RenDim(X, scaleQ=1:5, qMin=2, qMax=2)

Arguments

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 list of two elements:

1. a data.frame containing the value of Rényi's information of qth order (computed using the natural logarithm) for each value of \ln (\delta) and q. The values of \ln (\delta) are provided with regard to the [0,1] interval.

2. a data.frame containing the value of D_q for each value of q.

Author(s)

Jean Golay jeangolay@gmail.com

References

C. Traina Jr., A. J. M. Traina, L. Wu and C. Faloutsos (2000). Fast feature selection using fractal dimension. Proceedings of the 15th Brazilian Symposium on Databases (SBBD 2000), João Pessoa (Brazil).

E. P. M. De Sousa, C. Traina Jr., A. J. M. Traina, L. Wu and C. Faloutsos (2007). A fast and effective method to find correlations among attributes in databases, Data Mining and Knowledge Discovery 14(3):367-407.

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

H. Hentschel and I. Procaccia (1983). The infinite number of generalized dimensions of fractals and strange attractors, Physica D 8(3):435-444.

Examples

sim_dat <- SwissRoll(1000)

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).
qRI_ID <- RenDim(sim_dat[,c(1,2)], scaleQ[5:15])

print(paste("The ID estimate is equal to",round(qRI_ID[[1]][1,2],2)))

Swiss Roll Data Set Generator

Description

Generates random points on the Swiss Roll manifold.

Usage

SwissRoll(N=10000)

Arguments

Value

A N \times 3 data.frame containing the coordinates of the Swiss roll data points embedded in \rm I\!R^3.

References

J. A. Lee and M. Verleysen (2007). Nonlinear Dimensionality Reduction, Springer, New York.

Examples

sim_dat <- SwissRoll(1000)

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

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 jeangolay@gmail.com

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)