Thursday, March 5, 2009

Unfolding Models

On the top of a folded handkerchief is the ideal point, representing the highest degree of preference for a particular individual, i.e., the optimal choice within a given set of items. The closer the item is to the ideal point, the higher the preference is of this individual; thus, the individual prefers choice 1 to choice 2.

While different individuals have different ideal points on the handkerchief, unfolding the handkerchief will give us a 2D diagram showing all ideal points and all the items on a common space.

Some applications of unfolding models (adapted from this):

Applicaton 1: In American Idol, a set of judges rate a set of contestants. Unfolding would display the ideal point of each judge as a point, and each contestant as a point. Three pieces of information will be revealed: (a) Judges with similar ideal points would cluster; (b) Contestants rated similarly would cluster; (c) The closeness between the ideal point of a judge and a contestant indicates how high the judge would rate the contestant.

Application 2: A set of TV brands (e.g., Panasonic, Sony, ...) were rated on a set of attributes (e.g., price, quality, style, ...). In the matrix, the rows are the brands and the columns are the attributes. Unfolding would display (the ideal point of) each brand as a point and each attribute as a point. Three pieces of information: (a) Similar brands (in terms of ideal points) would cluster; (b) Similar attributes would cluster; (c) Brands rated highly on a particular attribute would appear close to that attribute.

Application 3: Unfolding can also be used to display relationships that may not be symmetric, such as desire between people, trade-flows between nations, and journal citation frequency. Each journal would appear as both a row and a column. The matrix would contain the citation frequency of the row-journal by the column-journal. Self-citing is excluded. Unfolding would produce a diagram in which each journal would appear as two points: citing others and being cited by others. Clusters would have the obvious interpretation, and the distance between a journal’s two points would reflect the imbalances in its citation.

Other variants of unfolding models:

  1. External unfolding models. Besides the preference data, we also have a pre-existing coordinate matrix of the choice objects.
  2. Vector model of unfolding. Representing individuals by preference vectors instead of ideal points. Because it is the direction of the vector that matters, the preference vectors are usually scaled to have equal length.
  3. Weighted unfolding.

Some terms and programs:
  1. In marketing, unfolding model is known as perceptual mapping.
  2. In marketing, MDPREF ("MultiDimensional PREFerence") performs internal unfolding analysis, whereas PREFMAP ("PREFerence MAPping") performs external unfolding analysis.

Tuesday, March 3, 2009

Procrustes analysis

The purpose of Procrustes analysis is to fit one MDS solution (configuration, map), B, to another one, A, and eliminate superficial differences between B and A, by means of rotating, mirror-reflecting, dilating/magnifying, shrinking, or shifting/moving B, without changing either's shape.

Application 1. A is the physical location map, whereas B is the travel-time map produced by MDS. In Procrustes analysis, we fit B to A, which allows us to display B on the top of A and to spot differences.

Application 2. Y is easy to interpret, whereas the initial X is not. In Procrustes analysis, we fit X to Y in order to interpret X.

Application 3. F is the result from the female participants, whereas M is that from the male participants. In Procrustes analysis, we fit M to F (or F to M) so that we can compare the results from males and females on the same page (provided that the fitting is satisfactory).

Application 4. CH is is the result from Chinese participants, whereas AM is that from American participants. In Procrustes analysis, we fit CH to AM (or AM to CH) so that we can compare the cross-cultural results on the same page (provided that the fitting is satisfactory).