Friday, January 23, 2009

Why do we need MDS?

Initially, researchers want to interpret a set of objects in terms of their relationships. However, the proximities (typically, dissimilarity) among them are in a high-dimensional space, which is beyond human's capacity of comprehension. Being troubled, the researchers think,

Heck! Why don't we try to project the objects into a 2D space and display them on a X-Y plane? As human beings, we are much more familiar with a X-Y plane and such an interpretation will be more exciting!

Thus, dimension reduction and therefore information loss is involved in MDS, and the general purpose of MDS program is to preserve the proximities between objects in the high-dimensional space as much as possible. An example of MDS in social psychology is that the 11 factors of the Aspiration Index are visually represented in an 2D plane. (And Don't you like it more when you are familiar with the way of interpreting the results?!)

Some notes:

1. MDS is a visualization tool. The goal is to reduce the observed complexity in the data matrix to lower dimensions (2 or 3) for humans to visualize.

2. MDS is a descriptive tool, rather than an inferential tool (de Leeuw, 2001). However, a representative sample should be recruited in order to generalize the description to the population.

3. MDS is more flexible than factor analysis: (a) it doesn't require that the underlying data are distributed as multivariate normal, and (b) it can be applied to any kind of distances or similarities, rather than just the computed correlation matrix.

4. MDS is different from cluster analysis. The goal of MDS is not to group/partition objects, but users can still visually cluster objects based on MDS.

5. MDS is related to self-organizing map (SOM) because they both enable visualizing low-dimensional views of high-dimensional data. However, SOM preserves data neighorhood, wheres MDS does not.

6. Besides dimensional representation (more exploratory), another goal of MDS is configural verification (more confirmatory).

7. The labeling of a dimension in MDS is arbitrary. The only requirement is that the two ends sum to zero at the center. It is similar to, but not the same as, bipolar, because it doesn't say anything about mutual exclusivity of the two ends in reality.

8. The number of dimensions is usually 2 (at best 3). On the one hand, the number should not be just 1; otherwise, all gradient-based methods in one-dimension will typically result in local optima. On the other hand, the number should not exceed 3; otherwise, visualization could be very difficult.

9. Another example of MDS would be to visualize the travel-times between cities. In the matrix, each row and each column would correspond to a city. MDS could then recreate a map containing the cities, solely from the matrix. This map would look similar to the actual map of city locations, but would differ in interesting ways. Cities connected by faster than average transportation passageways would appear closer together, while roadblocks would move cities apart.

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