Mathematics for Archetypal Analysis

This is a side-bar to a longer paper on Archetypal Analysis in Marketing Research within the web site of Action Marketing Research.

Archetypal Analysis uses a form of iterative regression. This summary is based on the formulation by Professor Adele Cutler.

Assume:

• n consumers (survey respondents)
• m variables
• p archetypes to be solved for
• X is an observed data matrix of dimensions n*m
• Z is a matrix of archetypes to be solved for of dimensions p*m
• a is a matrix of mixture weights of dimensions n*p
• ß is a matrix relating consumers to archetypes of dimensions p*n

The object of the procedure is to maximally explain the data in X or more precisely, to minimize the residual sum of squares (RSS) in X given these two alternating least squares problems:

• X = aZ
• Z = ßX

Each row of a represents one respondent.

• The sum of values in that row = 1.00
• Each value >= 0.00 and <= 1.00

Each row of ß represents one archetype.

• The sum of values in that row = 1.00
• Each value >= 0.00 and <= 1.00

Algorithm:

• 1. Initialize ß matrix
• 2. Solve for a using constrained least squares
• 3. Solve for ß using constrained least squares
• 4. Check for improvement in RSS; return to 2) if needed