Multidimensional Scaling is a technique for visualizing distances between objects where the distance is known between the pairs of objects. When the input type is variables, the probability that each point has the same class as its nearest neighbor is calculated. This article describes How to Create a Dimension Reduction Scatterplot, specifically using one of the MDS algorithms:
Requirements
- Either:
- High dimension variables in your data set (many variables)
- A distance matrix, see How to Create a Distance Matrix or you can paste one in as a table.
Method
1. From the menus, select Anything > Advanced Analysis > Dimension Reduction > Multidimensional Scaling (MDS).
2. In the object inspector under Inputs > Algorithm select MDS - Metric or MDS - Non-metric.
Note: In practical application of MDS, there tend to be contradictions in the data and it is impossible to show all the distances on the map accurately. Different MDS algorithms have been developed which make different decisions about how to reconcile these contradictions.
- Metric MDS minimizes the difference between distances in input and output spaces, which attempts to show the distances so that they are, on average, correct.
- Non-metric MDS aims to preserve the ranking or relative ordering of distances between input and output spaces.
Often we care more about relative positioning than absolute differences, in which case non-metric is preferred to metric MDS. For the example above, I used Metric.
3. Follow the instructions on How to Create a Dimension Reduction Scatterplot to further configure the output. If using variables, start on Step 3 of that section. If using a distance matrix, follow from Step 2 of that section.
Technical Details
Classic (metric) multidimensional scaling of the distance matrix (see Torgerson, W. S. (1958). Theory and Methods of Scaling. New York: Wiley).
Distances are computed between the mth and pth variables using:
\[\begin{align} d_{mp}=\sqrt{\frac{1}{n} \sum^n_{i=1} (x_{im}-x_{ip})^2 } \end{align}\]
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How to Create a Dimension Reduction Scatterplot