Abstract
Metrology as the science about measurement is highly intertwined with statistical point estimation. Evaluating and controling uncertainty of measurements and analyzing them by means of exploratory data analysis (EDA) or predictive data mining requires to exploit advanced tools of statistical estimation. The main focus of the chapter is devoted to nonstandard approaches to the analysis of measurements in two fundamental models, namely, the location model and linear regression. Robust regression methods, which are resistant to the presence of outlying (anomalous) measurements, are discussed here. An illustration of their performance over a real dataset related to thyroid disease and a Monte Carlo simulation reveal here the least weighted squares estimator, which has remained quite neglected so far, outperforms much more renowned robust regression estimators in terms of the variability. Further, Bayesian estimation in the location model is revealed here to have the ability to incorporate previous measurements in a very intuitive way. Finally, the chapter gives a warning that linear regression performed on data contaminated by measurement errors yields biased estimates and requires specific estimation tools for the so-called measurement error model.
References
Abdellaoui M, Bleichrodt H, Kemel E, L’Haridon O (2021) Measuring beliefs under ambiguity. Oper Res 69:599–612
Banks D, Cron A, Raskind A (2021) Bayesian metrology in metabolomics. Chemom Intell Lab Syst 208:104208
Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton
Broniatowski M, Jurečková J, Kalina J (2018) Likelihood ratio testing under measurement errors. Entropy 20:966
Buonaccorsi JP (2010) Measurement error. Models, methods, and applications. Chapman & Hall/CRC, Boca Raton
Carroll RJ, Ruppert D, Stefanski LA, Crainiceanu CM (2006) Measurement error in nonlinear models. A modern perspective, 2nd edn. Chapman & Hall/CRC, Boca Raton
Cheng YB, Chen XH, Li HL, Cheng ZY, Jiang R, Lü J, Fu HD (2018) Analysis and comparison of Bayesian methods for measurement uncertainty evaluation. Math Probl Eng 2018:7509046
Cox MG, Forbes AB, Harris PH (2004) Bayesian estimation methods in metrology. In: Fisher R, Preuss R, Toussaint U (eds) Bayesian inference and maximum entropy methods in science and engineering. Springer, New York
Crowder S, Delker C, Forrest E, Martin N (2020) Introduction to statistics in metrology. Springer, Cham
Davies L (1990) The asymptotics of S-estimators in the linear regression model. Ann Stat 18:1651–1675
Evans IG (1965) Bayesian estimation of parameters of a multivariate normal distribution. J R Stat Soc 27:279–283
Ferrero A, Salicone S, Jetti HV (2019) Bayesian approach to uncertainty evaluation: is it always working? In: 19th international congress of metrology. EDP Sciences, Les Ulis, p 16002
Ghosh SK (2019) Bayesian statistical methods. Chapman & Hall/CRC, Boca Raton
Greene WH (2018) Econometric analysis, 8th edn. Pearson Education Limited, Harlow
Hald A (2006) A history of parametric statistical inference from Bernoulli to Fisher, 1713 to 1935. Springer, New York
Hebra A (2010) The physics of metrology. All about instruments: from tundle wheels to atomic clocks. Springer, Vienna
Hnětynková I, Plešinger M, Sima DM, Strakoš Z, Van Huffel S (2011) The total least squares problem in Ax ≈ B: a new classification with the relationship to the classical works. SIAM J Matrix Anal Appl 32:748–770
Huber PJ, Ronchetti EM (2009) Robust statistics, 2nd edn. Wiley, Hoboken
Hynek M, Zvárová J, Smetanová D, Stejskal D, Kalina J (2021) Real-time quality control of nuchal translucency measurements using the exponentially weighted moving average chart. Taiwan J Obstet Gynecol 60:84–89
Jaynes ET (2003) Probability theory. The logic of science. Cambridge University Press, Cambridge, UK
JCGM (Joint Committee for Guides in Metrology) (2012) International Vocabulary of Metrology – basic and general concepts and associated terms (VIM), 3rd edn. Available online https://www.bipm.org/en/home
Jurečková J, Koul HL, Navrátil R, Picek J (2016) Behavior of R-estimators under measurement errors. Bernoulli 22:1093–1112
Jurečková J, Picek J, Schindler M (2019) Robust statistical methods with R, 2nd edn. CRC Press, Boca Raton
Kahneman D (2011) Thinking, fast and slow. Farrar, Straus and Giroux, New York
Kalina J (2014) On robust information extraction from high-dimensional data. Serb J Manag 9:131–144
Kalina J (2015) Three contributions to robust regression diagnostics. J Appl Math Stat Inform 11(2):69–78
Kalina J (2018) A robust pre-processing of BeadChip microarray images. Biocybern Biomed Eng 38:556–563
Kalina J, Hlinka J (2017) Implicitly weighted robust classification applied to brain activity research. In: Fred A, Gamboa H (eds) Biomedical engineering systems and technologies BIOSTEC 2016. Springer, Cham, pp 87–107
Kalina J, Tichavský J (2020) On robust estimation of error variance in (highly) robust regression. Meas Sci Rev 20:6–14
Klauenberg K, Martens S, Bošnjaković A, Cox MG, van der Veen AMH, Elster C (2022) The GUM perspective on straight-line errors-in-variables regression. Measurement 187:110340
Kong L, Pan H, Li X, Ma S, Xu Q, Zhou K (2019) An information entropy-based modeling methods for the measurement system. Entropy 21:691
Krystek M, Anton M (2007) A weighted total least-squares algorithm for fitting a straight line. Meas Sci Technol 18:3438–3442
Krystek M, Anton M (2011) A least-squares algorithm for fitting data points with mutually correlated coordinates to a straight line. Meas Sci Technol 22:035101
Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New York
Lesurf JCG (2002) Information and measurement, 2nd edn. Institute of Physics Publishing, Bristol
Lira I (2016) Beyond the GUM: variance-based sensitivity analysis in metrology. Meas Sci Technol 27:075006
Liu Y, Deng Z, Hu E (2021) Multi-sensor fusion positioning method based on batch inverse covariance intersection and IMM. Appl Sci 11:4908
Maronna R, Martin D, Yohai V (2006) Robust statistics: theory and methods. Wiley, New York
Marschall M, Wübbeler G, Elster C (2022) Rejection sampling for Bayesian uncertainty evaluation using the Monte Carlo techniques of GUM-S1. Metrologia 59:015004
Medina D, Li H, Vilà-Valls J, Closas P (2019) Robust statistics for GNSS positioning under harsh conditions: a useful tool? Sensors 19:5402
Nielsen L (2002) Evaluation of measurements by the method of least squares. In: Algorithms for approximation IV. University of Huddersfield, Huddersfield, pp 170–186
Ostojski MS, Gębala J, Orlińska-Woźniak P, Wilk P (2016) Implementation of robust statistics in the calibration, verification and validation step of model evaluation to better reflect processes concerning total phosphorus load occurring in the catchment. Ecol Model 332:83–93
Papageorgiou G, Bouboulis P, Theodoridis S (2017) Robust nonlinear regression: a greedy approach employing kernels with application to image denoising. IEEE Trans Signal Process 65:4309–4323
Peremans K, Van Aelst S (2018) Robust inference for seemingly unrelated regression models. J Multivar Anal 167:212–224
Pešta M (2013) Total least squares and bootstrapping with applications in calibration. Statistics 47:966–991
R Core Team (2021) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org
Raghavendra NV, Krishnamurthy L (2013) Engineering metrology and instruments. Oxford University Press, New Delhi
Ramnath V (2020) Comparison of straight line curve fit approaches for determining parameter variances and covariances. Int J Metrol Qual Eng 11:14
Robinson A (2007) The story of measurement. Thomas & Hudson, London
Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York
Rousseeuw PJ, Van Driessen K (2006) Computing LTS regression for large data sets. Data Min Knowl Disc 12:29–45
Saleh A, Picek J, Kalina J (2012) R-estimation of the parameters of a multiple regression model with measurement errors. Metrika 75:311–328
Saleh A, Arashi M, Saleh RA, Norouzirad M (2022) Rank-based methods for shrinkage and selections with application to machine learning. Wiley, Hoboken
Senin N, Catalucci S, Moretti M, Leach RK (2021) Statistical point cloud model to investigate measurement uncertainty in coordinate metrology. Precis Eng 70:44–62
Sheinin OB (1996) The history of the theory of errors. Hänsel-Hohenhausen, Egelsbach
Stigler SM (1973) Simon Newcomb, Percy Daniell, and the history of robust estimation 1885–1920. J Am Stat Assoc 68:872–879
Stigler SM (1986) The history of statistics: the measurement of uncertainty before 1900. Harvard University Press, Cambridge, MA
Tellinghuisen J (2001) Statistical error propagation. J Phys Chem A 105:3917–3921
Tin TC, Tan ST, Yong H, Kim JOH, Teo EKY, Lee CK, Than P, Tan APS, Phang SC (2021) A realizable overlay virtual metrology system in semiconductor manufacturing: proposal, challenges and future perspectives. IEEE Access 9:65418–65439
Víšek JÁ (2011) Consistency of the least weighted squares under heteroscedasticity. Kybernetika 47:179–206
Wheeler DJ (2020) Some outlier tests: part one. Comparisons and recommendations. Available at https://www.spcpress.com/pdf/DJW378.pdf
Wübbeler G, Marschall M, Elster C (2020) A simple method for Bayesian uncertainty evaluation in linear models. Metrologia 57:065010
Yohai VJ (1987) High breakdown-point and high efficiency robust estimates for regression. Ann Stat 15:642–656
Acknowledgments
The work of J. Kalina was supported by the project GA22-02067S (Approximate Neurocomputing) of the Czech Science Foundation. The authors would like to thank an anomymous reviewer for valuable suggestions.
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Kalina, J., Vidnerová, P., Soukup, L. (2022). Modern Approaches to Statistical Estimation of Measurements in the Location Model and Regression. In: Aswal, D.K., Yadav, S., Takatsuji, T., Rachakonda, P., Kumar, H. (eds) Handbook of Metrology and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-19-1550-5_125-1
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