{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:02:52Z","timestamp":1760144572486,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T00:00:00Z","timestamp":1714608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"UK\u2019s National Measurement System programme for Data Science"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper addresses the following problem: given m potential observations to determine n parameters, m&gt;n, what is the best choice of n observations. The problem can be formulated as finding the n\u00d7n submatrix of the complete m\u00d7n observation matrix that has maximum determinant. An algorithm by Gu and Eisenstat for a determining a strongly rank-revealing QR factorisation of a matrix can be adapted to address this latter formulation. The algorithm starts with an initial selection of n rows of the observation matrix and then performs a sequence of row interchanges, with the determinant of the current submatrix strictly increasing at each step until no further improvement can be made. The algorithm implements rank-one updating strategies, which leads to a compact and efficient algorithm. The algorithm does not necessarily determine the global optimum but provides a practical approach to designing an effective measurement strategy. In this paper, we describe how the Gu\u2013Eisenstat algorithm can be adapted to address the problem of optimal experimental design and used with the QR algorithm with column pivoting to provide effective designs. We also describe implementations of sequential algorithms to add further measurements that optimise the information gain at each step. We illustrate performance on several metrology examples.<\/jats:p>","DOI":"10.3390\/a17050193","type":"journal-article","created":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T03:57:56Z","timestamp":1714622276000},"page":"193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Algorithm of Gu and Eisenstat and D-Optimal Design of Experiments"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6640-9691","authenticated-orcid":false,"given":"Alistair","family":"Forbes","sequence":"first","affiliation":[{"name":"National Physical Laboratory, Teddington TW11 0LW, UK"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Atkinson, A.C., Donev, A.N., and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press.","DOI":"10.1093\/oso\/9780199296590.001.0001"},{"key":"ref_2","unstructured":"Box, G.E.P., Hunter, W.G., and Hunter, J.S. (2005). Statistics for Experimenters: Design, Innovation and Discovery, Wiley. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1214\/aos\/1176346407","article-title":"Optimal Bayesian experimental design for linear models","volume":"12","author":"Chaloner","year":"1984","journal-title":"Ann. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3730","DOI":"10.1016\/j.measurement.2013.04.002","article-title":"Design of linear calibration experiments","volume":"46","author":"Forbes","year":"2013","journal-title":"Measurement"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Goos, P. (2002). The Optimal Design of Blocked and Split-Plot Experiments, Springer.","DOI":"10.1007\/978-1-4613-0051-9"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Goos, P., and Jones, B. (2011). Optimal Design of Experiments: A Case Study Approach, John Wiley & Sons.","DOI":"10.1002\/9781119974017"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1080\/00401706.2016.1234979","article-title":"Effective Model Selection for Definitive Screening Designs","volume":"59","author":"Jones","year":"2017","journal-title":"Technometrics"},{"key":"ref_8","unstructured":"Montgomery, D.C. (2013). Design and Analysis of Experiments, John Wiley & Sons. [8th ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"431","DOI":"10.3934\/jimo.2018161","article-title":"A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids","volume":"16","author":"Chretien","year":"2020","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kekatos, V., and Giannakis, G.B. (2011, January 13\u201316). A convex relaxation approach to optimal placement of phasor measurement units. Proceedings of the 2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), San Juan, PR, USA.","DOI":"10.1109\/CAMSAP.2011.6135909"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1080\/17415977.2018.1495720","article-title":"An efficient method to estimate sorption isotherm curve coefficients","volume":"27","author":"Berger","year":"2019","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.expthermflusci.2016.10.008","article-title":"On the optimal experiment design for heat and moisture parameter estimation","volume":"81","author":"Berger","year":"2017","journal-title":"Exp. Therm. Fluid Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chernoff, H. (1972). Sequential Analysis and Optimal Design, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611970593"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1080\/00401706.1995.10485889","article-title":"The Coordinate Exchange Algorithm for Constructing Exact Optimal Designs","volume":"37","author":"Meyer","year":"1995","journal-title":"Technometrics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1214\/aoms\/1177731118","article-title":"Sequential Tests of Statistical Hypotheses","volume":"16","author":"Wald","year":"1945","journal-title":"Ann. Math. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Boyd, S., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_17","unstructured":"Gill, P.E., Murray, W., and Wright, M.H. (1981). Practical Optimization, Academic Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1137\/1038003","article-title":"Semidefinite Programming","volume":"38","author":"Vandenberghe","year":"1996","journal-title":"SIAM Rev."},{"key":"ref_19","unstructured":"Barker, R.M., Cox, M.G., Forbes, A.B., and Harris, P.M. (2007). Software Support for Metrology Best Practice Guide No. 4: Modelling Discrete Data and Experimental Data Analysis. Technical Report DEM-ES 018."},{"key":"ref_20","unstructured":"BIPM (2019). The International System of Units (SI Brochure (EN)), BIPM. [9th ed.]."},{"key":"ref_21","unstructured":"BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML (2008). Evaluation of Measurement Data\u2014Guide to the Expression of Uncertainty in Measurement, JCGM 100:2008."},{"key":"ref_22","unstructured":"BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML (2011). Evaluation of Measurement Data\u2014Supplement 2 to the \u201cGuide to the Expression of Uncertainty in Measurement\u201d\u2014Extension to Any Number of Output Quantities, JCGM 102:2011."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Golub, G., and Van Loan, C. (2013). Matrix Computations, Johns Hopkins University Press. [4th ed.].","DOI":"10.56021\/9781421407944"},{"key":"ref_24","unstructured":"Hart, G.W. (1995). Multidimensional Analysis: Algebras and Systems for Science and Engineering, Springer."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1137\/0917055","article-title":"Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization","volume":"17","author":"Gu","year":"1996","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1214\/aoms\/1177729893","article-title":"Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix","volume":"21","author":"Sherman","year":"1950","journal-title":"Ann. Math. Stat."},{"key":"ref_27","unstructured":"Wilkinson, J.H. (1988). The Algebraic Eigenvalue Problem, Oxford University Press, Inc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1088\/0026-1394\/27\/1\/002","article-title":"The International Temperature Scale of 1990 (ITS-90)","volume":"27","year":"1990","journal-title":"Metrologia"},{"key":"ref_29","unstructured":"Bartlett, G., Forbes, A., Heaps, E., Raby, A.C., and Yacoot, A. (2022, January 16\u201321). Spatial positioning correction for multi-axis nanopositioning stages. Proceedings of the ASPE Convention and Expo, Indianapolis, IN, USA."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Pukelsheim, F. (2006). Optimal Design of Experiments, SIAM. Reproduction of 1993 book published by John Wiley and Sons, New York.","DOI":"10.1137\/1.9780898719109"},{"key":"ref_31","unstructured":"Handscombe, D.C., and Mason, J.C. (2003). Chebyshev Polynomials, Chapman & Hall\/CRC Press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"100128","DOI":"10.1016\/j.measen.2021.100128","article-title":"Optimization of sensor distribution using Gaussian processes","volume":"18","author":"Forbes","year":"2021","journal-title":"Meas. Sens."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E., and Williams, C.K.I. (2006). Gaussian Processes for Machine Learning, MIT Press.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"99","DOI":"10.14311\/AP.2021.61.0099","article-title":"Application of ruled surfaces in freeform and gear metrology","volume":"61","year":"2021","journal-title":"Acta Polytech."},{"key":"ref_35","unstructured":"Zelen\u00fd, V., Linkeov\u00e1, I., and Skalnik, P. (2015, January 1\u20135). Calibration of freeform standard. Proceedings of the 15th International Conference of the European Society for Precision Engineering and Nanotechnology, EUSPEN 2015, Leuven, Belgium."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Pavese, F., and Forbes, A.B. (2009). Data Modeling for Metrology and Testing in Measurement Science, Birkh\u00e4user-Boston.","DOI":"10.1007\/978-0-8176-4804-6"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Forbes, A.B. (2022). Sensitivity analysis for Gaussian associated features. Appl. Sci., 12.","DOI":"10.3390\/app12062808"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1088\/0026-1394\/14\/4\/002","article-title":"Note on the Application of the Method of Least Squares","volume":"14","author":"Grabe","year":"1978","journal-title":"Metrologia"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1214\/aoms\/1177731236","article-title":"Some Improvements in Weighing and Other Experimental Techniques","volume":"15","author":"Hotelling","year":"1944","journal-title":"Ann. Math. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"S183","DOI":"10.1088\/0026-1394\/51\/4\/S183","article-title":"Evaluation of mass measurements in accordance with the GUM","volume":"51","author":"Nielsen","year":"2014","journal-title":"Metrologia"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1088\/0026-1394\/47\/4\/014","article-title":"Long term study of gauge block interferometer performance and gauge blocks stability","volume":"47","author":"Lewis","year":"2010","journal-title":"Metrologia"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1016\/j.aml.2006.11.016","article-title":"Eigenvalues of rank-one updated matrices with some applications","volume":"20","author":"Ding","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Hogben, L. (2007). Handbook of Linear Algebra, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420010572"},{"key":"ref_44","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions, Dover."},{"key":"ref_45","unstructured":"Fletcher, R. (1987). Practical Methods of Optimization, John Wiley and Sons. [2nd ed.]."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/5\/193\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:38:30Z","timestamp":1760107110000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/5\/193"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,2]]},"references-count":45,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["a17050193"],"URL":"https:\/\/doi.org\/10.3390\/a17050193","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,5,2]]}}}