Toward more numerically stable computations of optimal experimental designs

Abstract

The optimal experimental design theory is based on working with the so-called information matrix $\mathbf{M},$ which measures the amount of information provided by a design. As a consequence, the crucial expressions that are evaluated in algorithms for finding optimal designs generally involve the information matrix. However, we show that these expressions can be based on a matrix $\mathbf{H}$ of a simpler form, which we denote as half-information matrix. In particular, we show that by using suitable matrix decompositions, many algorithms can be formulated just in the terms of $\mathbf{H},$ with no need to compute the information matrix. Crucially, working with the half-information matrix is more stable: its condition number is the square root of that of the information matrix. As a consequence, stabilized versions of algorithms (i.e. those based on working with $\mathbf{H}$) can be applied to much more ill-conditioned models. We show on examples for both exact and approximate designs that the stabilized algorithms work on models that are unfeasible for the standard versions. We also show that the use of suitable rank-one updates of the utilized matrix decompositions can lead to a significant speed-up of the exchange-type algorithms.

Keywords:

• Algorithm
• A-optimality
• D-optimality
• Numerical stability
• Optimal design

Acknowledgments

The author would like to thank two anonymous reviewers who helped to improve the paper.

Disclosure statement

The author reports there are no competing interests to declare.

Additional information

Funding

Slovak Scientific Grant Agency (grant VEGA 1/0362/22).