What the quality metrics mean#
Every quality_report() evaluates the design on six geometric metrics
plus the optimised criterion, and ranks each against a Monte Carlo
baseline. This page defines the metrics precisely and explains what
each one can and cannot tell you. Coordinates are normalised to
\([0,1]^d\) throughout, and \(d_{ij}\) denotes the Euclidean distance
between design points \(x_i\) and \(x_j\).
The baseline and the rank#
Raw metric values are scale-dependent and, on their own, impossible to judge. The report therefore generates 300 random designs of the same size from the same feasible space (respecting all constraints and exclusions) and computes every metric for each. The Rank column is direction-adjusted: it is the share of these random designs that your design outperforms on that metric, so higher is always better. A design at the 90th percentile on minimum distance beats 90 percent of random attempts; one at the 6th percentile beats almost none.
Because the baseline is generated at runtime from your actual space, the comparison remains honest under constraints: a heavily constrained space is harder for the random designs too.
Min distance (\(d_{\min}\))#
The maximin quantity (Johnson, Moore & Ylvisaker, 1990): the distance
between the two closest runs. It answers “am I wasting budget on
near-duplicate experiments?” and is the metric most directly targeted
by phi_p.
Minimax distance (\(d_{\mathrm{mM}}\))#
The covering radius (Johnson, Moore & Ylvisaker, 1990): the distance from the most neglected point of the space to its nearest design point, estimated over a dense reference set. It answers “how far can the truth hide from my nearest run?” Low minimax means no region is left unwatched.
Max correlation (\(|\rho|_{\max}\))#
The largest absolute pairwise Pearson correlation between design columns. Correlated columns make the effects of two inputs partially indistinguishable in any subsequent model fit; near-zero correlation keeps them separable.
2D projection discrepancy (\(\mathrm{CD}_2^{\mathrm{proj}}\))#
The centred \(L_2\)-discrepancy (Hickernell, 1998) averaged over all \(\binom{d}{2}\) two-dimensional projections of the design (lower is better). Full-dimensional uniformity does not guarantee that pairs of inputs look uniform together; this metric checks exactly that, and it is the geometric face of what the projection-based criteria optimise.
CV of distances (\(\mathrm{CV}_d\))#
The coefficient of variation of all pairwise distances. A low value means the distance spectrum is narrow, the signature of a regular, mesh-like arrangement; a high value indicates a mix of tight pairs and long gaps.
Mean distance (\(\bar{d}\))#
The average of all pairwise distances (higher is better). A coarse but useful summary of overall spread. It should be read together with the other metrics: a design can inflate mean distance by pushing points into corners while leaving the interior empty.
The criterion row#
The final row reports the optimised criterion itself against the same baseline. This is the most direct measurement of optimisation success, and its scale is meaningful only relative to the baseline, which is why both numbers are always printed. If this row ranks high while a geometric metric you care about ranks low, the message is not that the optimiser failed but that you optimised a criterion that does not serve that metric; see the page on choosing a criterion.
References#
Hickernell, F. J. (1998). A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67(221), 299-322.
Johnson, M. E., Moore, L. M., & Ylvisaker, D. (1990). Minimax and maximin distance designs. Journal of Statistical Planning and Inference, 26(2), 131-148.