A new accelerated algorithm for ill-conditioned ridge regression problems

Silva, Tatiane; Ribeiro, Ademir A.; Periçaro, Gislaine Aparecida

doi:10.1007/s40314-017-0430-4

Cited by 7 publications

(14 citation statements)

References 9 publications

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“…Values higher than 1 suggest a problem in the model. , In our case, the predicted value of lignin content will be exaggerated by a minimal alteration in the Raman spectra. However, research has suggested the ability of the RR and LR algorithms to depress the negative effect caused by an ill-posed or ill-conditioned problem. − This explains the lower maximum values of RMSE and RMSEP in RR and LR compared to PCR and PLSR.…”

Section: Results and Discussionmentioning

confidence: 99%

Predictive Modeling of Lignin Content for the Screening of Suitable Poplar Genotypes Based on Fourier Transform–Raman Spectrometry

et al. 2021

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The quick and non-invasive evaluation of lignin from biomass has been the focus of much attention. Several types of spectroscopies, for example, near-infrared (NIR) and Fourier transform–Raman (FT–Raman), have been successfully applied to build quantitative predictive lignin models based on chemometrics. However, due to the effect of sample moisture content and ambient humidity on its signals, NIR spectroscopy requires sophisticated pre-testing preparation. In addition, the current FT–Raman predictive models require large variations in the independent value inputs as restrictions in the corresponding mathematical algorithms prevent the effective biomass screening of suitable genotypes for lignin contents within a narrow range. In order to overcome the limitations associated with the current methods, in this paper, we employed Raman spectra excited using a 1064 nm laser, thus avoiding the impact of water and auto-fluorescence on NIR signals. The optimal baseline correction method, data type, mathematical algorithm, and internal reference were selected in order to build quantitative lignin models based on the data with limited variation. The resulting two predictive models, constructed through lasso and ridge regressions, respectively, proved to be effective in assessing the lignin content of poplar in large-scale breeding and genetic engineering programs.

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Section: Results and Discussionmentioning

confidence: 99%

Predictive Modeling of Lignin Content for the Screening of Suitable Poplar Genotypes Based on Fourier Transform–Raman Spectrometry

et al. 2021

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“…where jjx i BE jj and jjAx i BE À yjj denote the 2-norms of the solution vector x i BE and the residual vector Ax i BE À y, maxðjjx i BE jjÞ denotes the maximum value in jjx i BE jj when i is taken from 1 to g, and maxðjjAx i BE À yjjÞ denotes the maximum value in jjAx i BE À yjj when i is taken from 1 to g. Note that many optimization problems can be transformed into convex optimization models. Thus, equation (16) is one of the reasonable models to find the optimal biased estimate since it is also a convex model. The following numerical examples also show that this formula is reasonable to some extent.…”

Section: Discussion On the Special Issuesmentioning

confidence: 99%

“…The following numerical examples also show that this formula is reasonable to some extent. It must be pointed out that there may be a better optimization model than equation (16), which will be further studied in the future work. (4) Find the minimum value in all the p i and the corresponding x i BE is the desired optimal biased estimate of x.…”

Section: Discussion On the Special Issuesmentioning

confidence: 99%

“…Many methods, such as L-curve method and ridge trace method, have been proposed for choosing the ridge parameters in the last decades. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] But so far, these processes for choosing the ridge parameters all require very complex calculations, and the ridge parameters obtained are often not the optimal solutions. It is very necessary to develop a new BE method for solving the ill-posed problems more effectively.…”

Section: Introductionmentioning

confidence: 99%

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A new biased estimation method based on Neumann series for solving ill-posed problems

Yang

2019

International Journal of Advanced Robotic Systems

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The ill-posed least squares problems often arise in many engineering applications such as machine learning, intelligent navigation algorithms, surveying and mapping adjustment model, and linear regression model. A new biased estimation (BE) method based on Neumann series is proposed in this article to solve the ill-posed problems more effectively. Using Neumann series expansion, the unbiased estimate can be expressed as the sum of infinite items. When all the high-order items are omitted, the proposed method degenerates into the ridge estimation or generalized ridge estimation method, whereas a series of new biased estimates can be acquired by including some high-order items. Using the comparative analysis, the optimal biased estimate can be found out with less computation. The developed theory establishes the essential relationship between BE and unbiased estimation and can unify the existing unbiased and biased estimate formulas. Moreover, the proposed algorithm suits for not only ill-conditioned equations but also rank-defect equations. Numerical results show that the proposed BE method has improved accuracy over the existing robust estimation methods to a certain extent.

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“…To the best of our knowledge, the only work that considers a primal-dual fixed point approach to deal with ridge regression is [16], where the authors deal with ill-conditioned problems. They present an algorithm based on the gradient method and an accelerated version of this algorithm.…”

Section: Contributions and Literature Reviewmentioning

confidence: 99%