mmds Metric (classical) Multidimensional Scaling (a.k.a Principal Coordinate Analysis) of a (Euclidean) Distance Matric

Description

Perform an MMDS of a (Euclidean) distance matrix measured between a set of weighted objects.

MMDS Give factor scores that make it possible to draw a map of the objects such that the distances between objects on the map best approximate the original distances between objects.

Method: Transform the distance matrix into a (double centered) covariance matrix which is then analyze via its eigen-decomposition. The factor score of each dimension are scaled such that their variance (i.e., the sum ot their weighted squared factor scores) is equal to the eigen-value of the corresponding dimension. Note that if the masses vector is absent, equal masses (i.e. 1 divided by number of objects) are used.

Usage

mmds(DistanceMatrix,masses=NULL)
	

Arguments

Value

Sends back a list

Author(s)

Herve Abdi

References

The procedure and references are detailled in: Abdi, H. (2007). Metric multidimensional scaling. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 598–605.

(Paper available from www.utdallas.edu/~herve).

See Also

GraphDistatisCompromise distatis

Examples

# An example of MDS from Abdi (2007)
# Discriminability of Brain States
# Table 1.
# 1. Get the distance matrix
D <- matrix(c(
0.00, 3.47, 1.79, 3.00, 2.67, 2.58, 2.22, 3.08,
3.47, 0.00, 3.39, 2.18, 2.86, 2.69, 2.89, 2.62,
1.79, 3.39, 0.00, 2.18, 2.34, 2.09, 2.31, 2.88,
3.00, 2.18, 2.18, 0.00, 1.73, 1.55, 1.23, 2.07,
2.67, 2.86, 2.34, 1.73, 0.00, 1.44, 1.29, 2.38,
2.58, 2.69, 2.09, 1.55, 1.44, 0.00, 1.19, 2.15,
2.22, 2.89, 2.31, 1.23, 1.29, 1.19, 0.00, 2.07,
3.08, 2.62, 2.88, 2.07, 2.38, 2.15, 2.07, 0.00),
ncol = 8, byrow=TRUE)
rownames(D) <- c('Face','House','Cat','Chair','Shoe','Scissors','Bottle','Scramble')
colnames(D) <- rownames(D)
# 2. Call mmds
BrainRes <- mmds(D)
# Note that compared to Abdi (2007)
# the factor scores of mmds are equal to F / sqrt(nrow(D))
# the eigenvalues of mmds are equal to Lambda *{1/nrow(D)}
# (ie., the normalization differs but the results are proportional)
# 3. Now a pretty plot with the prettyPlot function from prettyGraphs
prettyPlot(BrainRes$FactorScore, 
           display_names = TRUE,
           display_points = TRUE,
           contributionCircles = TRUE,
          contributions = BrainRes$Contributions)
# 4. et Voila!

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(DistatisR)
Loading required package: prettyGraphs
Loading required package: car
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/DistatisR/mmds.Rd_%03d_medium.png", width=480, height=480)
> ### Name: mmds
> ### Title: mmds Metric (classical) Multidimensional Scaling (a.k.a
> ###   Principal Coordinate Analysis) of a (Euclidean) Distance Matric
> ### Aliases: mmds
> ### Keywords: DistatisR
> 
> ### ** Examples
> 
> # An example of MDS from Abdi (2007)
> # Discriminability of Brain States
> # Table 1.
> # 1. Get the distance matrix
> D <- matrix(c(
+ 0.00, 3.47, 1.79, 3.00, 2.67, 2.58, 2.22, 3.08,
+ 3.47, 0.00, 3.39, 2.18, 2.86, 2.69, 2.89, 2.62,
+ 1.79, 3.39, 0.00, 2.18, 2.34, 2.09, 2.31, 2.88,
+ 3.00, 2.18, 2.18, 0.00, 1.73, 1.55, 1.23, 2.07,
+ 2.67, 2.86, 2.34, 1.73, 0.00, 1.44, 1.29, 2.38,
+ 2.58, 2.69, 2.09, 1.55, 1.44, 0.00, 1.19, 2.15,
+ 2.22, 2.89, 2.31, 1.23, 1.29, 1.19, 0.00, 2.07,
+ 3.08, 2.62, 2.88, 2.07, 2.38, 2.15, 2.07, 0.00),
+ ncol = 8, byrow=TRUE)
> rownames(D) <- c('Face','House','Cat','Chair','Shoe','Scissors','Bottle','Scramble')
> colnames(D) <- rownames(D)
> # 2. Call mmds
> BrainRes <- mmds(D)
> # Note that compared to Abdi (2007)
> # the factor scores of mmds are equal to F / sqrt(nrow(D))
> # the eigenvalues of mmds are equal to Lambda *{1/nrow(D)}
> # (ie., the normalization differs but the results are proportional)
> # 3. Now a pretty plot with the prettyPlot function from prettyGraphs
> prettyPlot(BrainRes$FactorScore, 
+            display_names = TRUE,
+            display_points = TRUE,
+            contributionCircles = TRUE,
+           contributions = BrainRes$Contributions)
dev.new(): using pdf(file="Rplots135.pdf")
$col
         [,1]     
Face     "#305ABF"
House    "#305ABF"
Cat      "#305ABF"
Chair    "#305ABF"
Shoe     "#305ABF"
Scissors "#305ABF"
Bottle   "#305ABF"
Scramble "#305ABF"

$pch
     [,1]
[1,]   21
[2,]   21
[3,]   21
[4,]   21
[5,]   21
[6,]   21
[7,]   21
[8,]   21

$constraints
$constraints$minx
[1] -0.8928154

$constraints$miny
[1] -0.5828126

$constraints$maxx
[1] 1.028313

$constraints$maxy
[1] 0.9642077


> # 4. et Voila!
> 
> 
> 
> 
> 
> dev.off()
png 
  2 
>