58
Community Analysis Package
11
Non Metric Multidimensional scaling
Multidimensional scaling (MDS) is a technique for expressing the similarities
between different objects in a small number of dimensions. Hopefully, this allows
a complex set of inter relationships to be summarised in a simple figure. The
method attempts to place the most similar objects (samples) closest together.
The starting point for the calculations is a similarity or dissimilarity matrix between
all the sites or quadrates. These can be non metric distance measures for which
the relationships between the sites (columns) cannot be plotted in a Euclidean
space. The aim of non metric MDS is to find a set of metric coordinates for the
sites which most closely approximates their non metric distances.  The basic
MDS algorithm is as follows:
1.
Calculate the similarity or dissimilarity between sites.
2.
Assign to each site a set of coordinates in p dimensional space. These
coordinates can be either chosen at random or chosen using 
Principal
Coordinates Analysis. The value of p is chosen by the user.
3.
Compute the Euclidean distance between these sites using the starting
coordinates.
4.
Compare the original dissimilarity between the sites with these Euclidean
distances by calculating a stress function. The smaller the 
stress
function the closer the correspondence.
5.
Adjust the positions such as to reduce the stress.
6.
Repeat 2 to 4 until the stress is minimised or the maximum number of
iterations is reached.
CAP uses Kruskals s least squares monotonic transformation to minimise the
stress (see 
Kruskal, 1964
111
; 
Kruskal & Wish
111
, 1977). The program is designed
to find an optimal two dimensional representation of the data. It can happen that
no useful two dimensional representation can be produced. While it is possible to
produce anything up to a six dimensional solution, in practice this is of little use,
as it cannot be displayed. When requested, CAP lists solutions for 3 or more
dimensions but does not plot them.
From the ordination drop down menu select Non metric MDS. The setup for Non 
metric MDS is displayed. This offers a series of options that are described below.
The program can usually be run with the default values.
Output from Non metric MDS is presented under a number of tabbed components
that can each be viewed by clicking on the tab. These are described in turn
below.
MDS setup
59
.
MDS plots
61
Sample coordinates
60
Stress
60
Copyright 2004, PISCES CONSERVATION LTD





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