Optimal singular correlation matrices estimated when the sample size is less than or equal to the number of random variables

• Numerical examples and explicit formulas of optimal correlation matrices.
• An explicit formula for the root mean square error ρrms of the optimal matrices.
• Tight bounds for the error ρmaxρmax of the optimal matrices.
• A mechanical analogy between the correlation matrices and a model of bars.

This paper presents a number of theoretical and numerical results for two norms of optimal correlation matrices in relation to correlation control in Monte Carlo type sampling and the designs of experiments. The optimal correlation matrices are constructed for cases when the number of simulations (experiments) Nsim is less than or equal to the stochastic dimension, i.e. the number of random variables (factors) Nvar. In such cases the estimated correlation matrix can not be positive definite and must be singular. However, the correlation matrix may be required to be as close to the unit matrix as possible (optimal). The paper presents a simple mechanical analogy for such optimal singular positive semidefinite correlation matrices. Many examples of optimal correlation matrices are given, both analytically and numerically.