Quadratic minimisation problems in statistics

We consider the problem minx(x−t) A(x−t) subject to x Bx + 2b x = k where A is positive definite or positive semidefinite. Commonly occurring statistical variants of this problem are discussed within the framework of a general unifying methodology. These include non-trivial considerations that arise when (i) A and/or B are not of full rank and (ii) t takes special forms (especially t = 0 which, under further conditions, reduces to the well-known two-sided eigenvalue solution). Special emphasis is placed on insights provided by geometrical interpretations. Algorithmic considerations are discussed and examples given. Keywords. canonical analysis, constraints, geometry, Hardy-Weinberg, minimisation, optimal scaling, Procrustes analysis, quadratic forms, ratios, reduced rank, splines. 1 

• Type: Journal
• Year: 2011
• Where: MA
• Authors: Casper J. Albers, Frank Critchley, John C. Gower
• Keywords: Communications, General Unifying Methodology, MA 2011, Non-trivial Considerations, Well-known Two-sided Eigenvalue