MDS Example 

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Mammals Milk MDS Case Study
ClustanMDS is illustrated using the Mammals Milk case study distributed with ClustanGraphics and previewed here .  A proximity matrix of squared Euclidean distances was computed and an optimum MDS configuration in two dimensions was found that preserves the rank order of the distance matrix to the input proximity matrix with less than 1% distortion.  A hierarchical cluster analysis by Ward's Method was then mapped on to the final 2-dimensional MDS configuration.

ClustanGraphics completed a series of 100 trials using different random starting configurations in less than half a second. The best configuration was obtained in trial 64 with stress reduced to 0.68% and is reproduced below.  This is an ordination of 25 mammals according to the composition of their milk, colour coded by the 7-cluster model, with rabbit and elephant as singleton clusters.  Another example of the summary 5-cluster model is shown here.

When the maximum number of iterations is increased to 5000, 9 trials fail to converge and 18 trials find a local minimum.  The remaining 82 trials (four-fifths) find configurations with less than 1.5% stress that are all close to the presumed global minimum.

Opening dialogue for ClustanMDS - showing defaults for all the parameters.  In this example the user has clicked the The ClustanMDS dialogue requires only 3 parameters, each of which has a default set by the procedure.  The parameters are defined here.

In this example a proximity matrix of 300 squared Euclidean distance coefficients has been computed for 25 cases (mammals).

ClustanMDS is run 100 times, each with a different random starting configuration.  The lowest stress value for all of the trials is recorded, and the corresponding configuration is saved as scatterplot variables in 2 dimensions.

A stress value of 0.68% was recorded for the final configuration, which is rated as an excellent fit.

Note that ClustanMDS took less than half a second to run on an Intel Pentium 4 computer with 384MB of RAM, under Windows/XP.

Details dialogue in ClustanMDS, obtained by clicking the Details button in the ClustanMDS dialogue The detailed results for the 100 trials are displayed in the MDS Details dialogue, shown right.

This lists the number of iterations completed in each trial, the resulting stress value and whether it was satisfactory, minimum or failed to converge.

Of the 100 trials, 43 resulted in satisfactory stress according to Kruskal's ratings.  Of these, the minimum stress of 0.68% was obtained at trial 64.  However, it should be noted that most of the trials resulting in satisfactory stress produced similar plots of the new configuration.

On clicking finish in the main ClustanMDS dialogue, the saved configuration is displayed (as below).

Scatterplot of new MDS variables, obtained in Cluster/Scatterplots.  The displays the 7-cluster model for Mammals Milk by Ward's Method 

This scatterplot shows the 7-cluster model for Mammals Milk, with the 5 clusters and 2 singletons (Rabbit and Elephant) colour coded.  The horizontal axis is the main principal component, saved as Scatter1, and the vertical axis is the second principal component, Scatter2.

The View Data dialogue can be used to display the scatterplot variables generated and saved by ClustanMDS, as well as the input or transformed data. 

The values for Scatter1 and Scatter2 can be displayed using View/Data.

They can also be saved to a text file, or copied to the clipboard.

The MDS results will be retained in the ClustanGraphics file, and can be used to plot any subsequent cluster analysis on the same data.

 

 

 

Excel spreadsheet containing the scatterplot variables obtained from ClustanMDS 

When the scatterplot variables are copied to the clipboard they are easily read into another Windows application such as a spreadsheet or word processor.

In this example (right) the results for ClustanMDS on the Mammals Milk case study were read into Excel with a single click - simply highlight cell A1 and click copy.

It is now possible to carry out further statistical evaluation of the MDS results in Excel.