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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

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