A-optimality • TSDFGS

A-optimality minimizes the average variance of the ridge coefficients. It uses the training set alone.

Type	Optimal design
Direction	Smaller is better
Uses a test set	no (a test set given to `optTrain()` is ignored)
Score a given set	`aopt_score()`
Search with it	`optTrain(method = "AOpt")`

Definition

With $\lambda = 1/p$ : $\mathrm{A} = \mathrm{tr}[(X'X + \lambda I)^{-1}]$ . Akdemir and Isidro-Sánchez (2019) allow a contrast matrix; TSDFGS uses the identity. For $p > n$ the $p - n$ zero eigenvalues each add $1/\lambda$ .

Use in TSDFGS

library(TSDFGS)
data(geno)
aopt_score(geno[1:50, ])

## [1] 143016

fit <- optTrain(geno, cand = 1:404, n.train = 50, method = "AOpt",
                 min.iter = 300, console = FALSE, seed = 1)
fit$OPTtrain

##  [1] 324 316 310 299 402 314 379  85 277 332 101  46 196 311 158 132  56 122 268
## [20]  73 391 369 382 246 256 110 152 399  65 386 227 136 260  70  10 328 162  44
## [39] 253 262 135 309 149  62 370  52 350 245  33 138

The search is a random-start exchange (hill-climbing) search; see How the scores are computed. Use min.iter larger than 300 on real data and check fit$converged.

Original work

Akdemir, D., & Isidro-Sánchez, J. (2019). Design of training populations for selective phenotyping in genomic prediction. Scientific Reports, 9, 1446. https://doi.org/10.1038/s41598-018-38081-6

Akdemir, D., Sanchez, J. I., & Jannink, J.-L. (2015). Optimization of genomic selection training populations with a genetic algorithm. Genetics Selection Evolution, 47, 38. https://doi.org/10.1186/s12711-015-0116-6

Repository or website of the original authors: R package STPGA by Akdemir and colleagues: CRAN.R-project.org/package=STPGA (GPL-3).

License review

Formulas and algorithms are ideas and methods, which copyright does not protect. Where an original package is under a GPL license, we read it to check our results and credit it; TSDFGS is GPL-3, which is compatible with GPL-3 and GPL (>= 3) code. This is our reading of the public record, not legal advice; see the license review.

Original software: STPGA, GPL-3 (AOPT); compared, not copied. STPGA reports the trace divided by $p$ , which does not change the optimum.
Article license: Akdemir and Isidro-Sánchez (2019) open access (Scientific Reports).
Verdict: included.

Reviewed 2026-11-04. All criteria and the ones left out.