Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics

Bauer, Frank; Gutting, Martin; Lukas, Mark A.

doi:10.1007/978-3-642-54551-1_99

Cited by 29 publications

(42 citation statements)

References 136 publications

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“…Many sophisticated approaches for selecting an appropriate value of α could be employed [15], but we restrict ourselves to a heuristic approach due to successful past experience.…”

Section: A) Regularized Linear Regressionmentioning

confidence: 99%

Reducing false positives for residual-based on-line fault detection by means of filters

Serdio

Lughofer

Pichler

et al. 2014

2014 IEEE International Conference on Systems, Man, and Cybernetics (SMC)

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Departing from a previously demonstrated Residual-Based Fault-Detection system, where the residual signals are statistically tracked by means of tolerance bands, we introduce the use of filters as an intermediate step in the online operation of the system. This step will provide smoother residual signals where the underlying trend of the signal is kept and the significant anomalies pointing to potential fault candidates are still not filtered out. The research was performed with well-known existing filters, used in other domains such as engineering, image processing or econometrics. We demonstrate how the better stability of the residual signals significantly boosts the performance of a residual-based fault detection framework, decreasing the false positives rates whereas also increasing the true positives rates.

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“…Many sophisticated approaches for selecting an appropriate value of α could be employed [15], but we restrict ourselves to a heuristic approach due to successful past experience.…”

Section: A) Regularized Linear Regressionmentioning

confidence: 99%

Reducing false positives for residual-based on-line fault detection by means of filters

Serdio

Lughofer

Pichler

et al. 2014

2014 IEEE International Conference on Systems, Man, and Cybernetics (SMC)

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“…Numerous approaches exist to determine tuning parameter values, e.g. the number of respective basis vectors for PCR and PLS, or the ridge value for RR [37][38][39][40][41][42][43][44][45][46]. Typically, a single criterion is used based on some sort of cross-validation (CV) procedure such as leave-one-out CV (LOOCV) or leave multiple out CV (LMOCV) to compute the root mean square error of CV (RMSECV) values.…”

Section: Selecting Final Tuning Parameter Values and Assessing Effectmentioning

confidence: 99%

Using the L₁ norm to select basis set vectors for multivariate calibration and calibration updating

Shahbazikhah

Kalivas

Andries

et al. 2016

Journal of Chemometrics

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dWith projection based calibration approaches, such as partial least squares (PLS) and principal component regression (PCR), the calibration space is spanned by respective basis vectors (latent vectors). Up to rank k basis vectors are formed where k ≤ min(m,n) with m and n denoting the number of calibration samples and measured variables. The user needs to decide how many and which respective basis vectors (tuning parameters). To avoid the second issue, basis vectors are selected top-down starting with the first and sequentially adding until model criteria are satisfied. Ridge regression (RR) avoids the issues by using the full set of basis vectors. Another approach is to select a subset from the total available. The presented work develops a process based on the L 1 vector norm to select basis vectors. Specifically, the L 1 norm is used to select singular value decomposition (SVD) basis set vectors for PCR (LPCR). Because PCR, PLS, RR, and others can be expressed as linear combination of the SVD basis vectors, the focus is on selection and comparison using the SVD basis set. Results based on respective tuning parameter selections and weights applied to the SVD basis vectors for LPCR, top-down PCR, correlation PCR (CPCR), PLS, and RR are compared for calibration and calibration updating using spectroscopic data sets. The methods are found to predict equivalently. In particular, the L 1 norm produces similar results to those obtained by the well-studied CPCR process. Thus, the new method provides a different theoretical framework than CPCR for selecting basis vectors.

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“…The recent literature shows an increasing interest in heuristic (or noise-level-free) parameter choice strategies, even if the well-known Bakushinsky veto [1] states that, in Hilbert spaces, all heuristic parameter choice rules, which do not make use of the knowledge about the exact noise level, will never converge in the worst-case scenario analysis. Nevertheless, heuristic parameter selection techniques are used quite frequently in practical applications, often giving good results [46,3]. The L-curve method [40,26] and the Generalized Cross Validation method [14] are likely the most popular heuristic parameter selection strategies; they have been deeply investigated by several authors [9,34,17,53].…”

Section: Introductionmentioning

confidence: 99%

“…Several minimization rules for the selection of the regularization parameter for many iterative regularization methods as the Landweber method and the CGLS method are given in [15], both in case of known and unknown noise norm. A detailed and careful comparison of many parameter choice rules is performed in [3,46].…”

Section: Introductionmentioning

confidence: 99%

A stopping criterion for iterative regularization methods

Landi

Piccolomini

Tomba

2016

Applied Numerical Mathematics

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Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics

Cited by 29 publications

References 136 publications

Reducing false positives for residual-based on-line fault detection by means of filters

Reducing false positives for residual-based on-line fault detection by means of filters

Using the L₁ norm to select basis set vectors for multivariate calibration and calibration updating

A stopping criterion for iterative regularization methods

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Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics

Cited by 29 publications

References 136 publications

Reducing false positives for residual-based on-line fault detection by means of filters

Reducing false positives for residual-based on-line fault detection by means of filters

Using the L1 norm to select basis set vectors for multivariate calibration and calibration updating

A stopping criterion for iterative regularization methods

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Using the L₁ norm to select basis set vectors for multivariate calibration and calibration updating