Computational details

As has already been explained, the elements of statistical_properties are quantities which can be calculated starting from the information matrix $C_d$. There are three fundamental calculations: the canonical variances, the pairwise variances, and the canonical efficiency factors.

The canonical variances are the inverses of the eigenvalues of $C_d$, eigenvalues of zero corresponding to canonical variances of $\infty$. Thus we need the roots of the polynomial $\vert C_d-xI\vert=0$. As $C_d$ is a rational matrix, this polynomial admits a factorization into irreducible factors over the rational field. Thus, in theory, the multiplicities of the canonical variances can be determined exactly, even if some of the values themselves are irrational. If the eigenvalues of $C_d$ are numerically extracted directly without factoring the characteristic polynomial, then the problem of inexact counts of those eigenvalues can arise.

Pairwise variances are defined above in terms of the Moore-Penrose inverse $C_d^+$ of $C_d$: $v_{dii'} = c^+_{dii} + c^+_{di'i'} - 2 c^+_{dii'}$. In fact, any generalized inverse $C_d^-$ of $C_d$ can be used, from which $v_{dii'} = c^-_{dii} + c^-_{di'i'} - 2 c^-_{dii'}.$ Let $J$ be an all-ones matrix. If $d$ is connected, then $C_d+aJ$ is invertible for any $a\neq 0$ and $C_d^-=(C_d+aJ)^{-1}$ is a generalized inverse of $C_d$ (the same operation can be carried out for the connected components of $C_d$ if $d$ is disconnected). Thus pairwise variances can be calculated by inversion of a rational, nonsingular matrix.

Efficiency factors are defined as eigenvalues of the matrix $F_d=R^{-1} C_d R^{-1}$, which can certainly be irrational. Extracting the roots of $N_dK^{-1}N_d'$ with respect to $R^2$, that is, solving the equation $\vert N_dK^{-1}N_d' -\mu R^2\vert=0$, produces values $\mu_{di}$ for $i=1,\ldots,v$ satisfying $\mu_{dv}=0$ and otherwise $\mu_{di}=1-e_{di}$. Thus efficiency factors can be found by extracting roots of a symmetric, rational matrix, involving the same computational issues as for the canonical variances.

The number of infinite canonical variances equals the number of connected components of $d$ less 1 (this being zero for any connected designs). Numerical extraction of eigenvalues of $C_d$ can potentially produce, at a given level of precision, values indistinguishable from zero that are in actuality positive, consequently producing an erroneous number of infinite canonical variances. This approximation error is prohibited by cross-checking against the connected indicator.