Functional Linear Regression

2018 ◽  
Author(s):  
Hervé Cardot ◽  
Pascal Sarda
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Linear Models ◽  
Estimation Error ◽  
Asymptotic Properties ◽  
Principal Component Regression ◽  
Principal Component ◽  
Penalized Least Squares ◽  
Open Problems ◽  
Functional Linear Regression

This article presents a selected bibliography on functional linear regression (FLR) and highlights the key contributions from both applied and theoretical points of view. It first defines FLR in the case of a scalar response and shows how its modelization can also be extended to the case of a functional response. It then considers two kinds of estimation procedures for this slope parameter: projection-based estimators in which regularization is performed through dimension reduction, such as functional principal component regression, and penalized least squares estimators that take into account a penalized least squares minimization problem. The article proceeds by discussing the main asymptotic properties separating results on mean square prediction error and results on L2 estimation error. It also describes some related models, including generalized functional linear models and FLR on quantiles, and concludes with a complementary bibliography and some open problems.

scholarly journals A Quantitative Analysis of Glucose from Enhanced NIR Spectra through Linear Regression Model Coupled with Optimized Bandpass Filtering

Proceedings ◽  
2018 ◽  
Vol 2 (13) ◽  
pp. 1010
Author(s):  
Mahbubur Rahman Mishal ◽  
Tanvir Tazul Islam ◽  
Shahadat Hossain Antor ◽  
Tanzilur Rahman
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Near Infrared ◽  
Phosphate Buffer Solution ◽  
Principal Component Regression ◽  
Principal Component ◽  
Calibration Model ◽  
Linear Regression Models ◽  
Buffer Solution ◽  
Correction Technique

This study proposes a new preprocessing technique that combines Chebyshev filtering with baseline correction technique Asymmetric Least Squares (ALS) and Savitzky-Golay transformation (SGT) to improve the prediction of Glucose from near Infrared (NIR) spectra through linear regression models Partial Least Squares (PLS) and Principal Component Regression (PCR). To investigate the performance of the proposed technique, a calibration model was first developed and then validated through prediction of Glucose from NIR spectra of a mixture of glucose, urea, and triacetin in a phosphate buffer solution where the component concentrations are within their physiological range in blood. Results indicate that the proposed technique improves the performance of both PLS and PCR and achieves standard error of prediction (SEP) as low as 12.76 mg/dL which is in the clinically acceptable level and comparable to the existing literature.

scholarly journals Application of Derivative, Derivative Ratio, and Multivariate Spectral Analysis and Thin-Layer Chomatography-Densitometry for Determination of a Ternary Mixture Containing Drotaverine Hydrochloride, Caffeine, and Paracetamol

2007 ◽  
Vol 90 (2) ◽  
pp. 391-404 ◽  
Author(s):  
Fadia H Metwally ◽  
Yasser S El-Saharty ◽  
Mohamed Refaat ◽  
Sonia Z El-Khateeb
Keyword(s):  
Thin Layer ◽  
Least Squares ◽  
Ternary Mixture ◽  
Principal Component Regression ◽  
Pharmaceutical Preparations ◽  
Principal Component ◽  
Standard Addition ◽  
Drotaverine Hydrochloride ◽  
Assay Precision

Abstract New selective, precise, and accurate methods are described for the determination of a ternary mixture containing drotaverine hydrochloride (I), caffeine (II), and paracetamol (III). The first method uses the first (D1) and third (D3) derivative spectrophotometry at 331 and 315 nm for the determination of (I) and (III), respectively, without interference from (II). The second method depends on the simultaneous use of the first derivative of the ratio spectra (DD1) with measurement at 312.4 nm for determination of (I) using the spectrum of 40 μg/mL (III) as a divisor or measurement at 286.4 and 304 nm after using the spectrum of 4 μg/mL (I) as a divisor for the determination of (II) and (III), respectively. In the third method, the predictive abilities of the classical least-squares, principal component regression, and partial least-squares were examined for the simultaneous determination of the ternary mixture. The last method depends on thin-layer chromatography-densitometry after separation of the mixture on silica gel plates using ethyl acetatechloroformmethanol (16 + 3 + 1, v/v/v) as the mobile phase. The spots were scanned at 281, 272, and 248 nm for the determination of (I), (II), and (III), respectively. Regression analysis showed good correlation in the selected ranges with excellent percentage recoveries. The chemical variables affecting the analytical performance of the methodology were studied and optimized. The methods showed no significant interferences from excipients. Intraday and interday assay precision and accuracy values were within regulatory limits. The suggested procedures were checked using laboratory-prepared mixtures and were successfully applied for the analysis of their pharmaceutical preparations. The validity of the proposed methods was further assessed by applying a standard addition technique. The results obtained by applying the proposed methods were statistically analyzed and compared with those obtained by the manufacturer's method.

Fuzzy C-Regression Models

2013 ◽  
Vol 278-280 ◽  
pp. 1323-1326
Author(s):  
Yan Hua Yu ◽  
Li Xia Song ◽  
Kun Lun Zhang
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Linear Models ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Estimation Methods ◽  
Statistical Regression ◽  
Fuzzy Linear Regression ◽  
Fuzzy Degree

Fuzzy linear regression has been extensively studied since its inception symbolized by the work of Tanaka et al. in 1982. As one of the main estimation methods, fuzzy least squares approach is appealing because it corresponds, to some extent, to the well known statistical regression analysis. In this article, a restricted least squares method is proposed to fit fuzzy linear models with crisp inputs and symmetric fuzzy output. The paper puts forward a kind of fuzzy linear regression model based on structured element, This model has precise input data and fuzzy output data, Gives the regression coefficient and the fuzzy degree function determination method by using the least square method, studies the imitation degree question between the observed value and the forecast value.

Principal component analysis for functional data

2018 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Principal Component Analysis ◽  
Linear Regression ◽  
Functional Data ◽  
Weighted Least Squares ◽  
Principal Component ◽  
Adaptive Methods ◽  
Component Analysis ◽  
Infinite Dimensional ◽  
Role Of Principal ◽  
Functional Linear Regression

This article discusses the methodology and theory of principal component analysis (PCA) for functional data. It first provides an overview of PCA in the context of finite-dimensional data and infinite-dimensional data, focusing on functional linear regression, before considering the applications of PCA for functional data analysis, principally in cases of dimension reduction. It then describes adaptive methods for prediction and weighted least squares in functional linear regression. It also examines the role of principal components in the assessment of density for functional data, showing how principal component functions are linked to the amount of probability mass contained in a small ball around a given, fixed function, and how this property can be used to define a simple, easily estimable density surrogate. The article concludes by explaining the use of PCA for estimating log-density.

Development and Validation of Chemometric-Assisted Spectrophotometric Methods for Simultaneous Determination of Phenylephrine Hydrochloride and Ketorolac Tromethamine in Binary Combinations

2016 ◽  
Vol 99 (5) ◽  
pp. 1247-1251 ◽  
Author(s):  
Hamed M Elfatatry ◽  
Mokhtar M Mabrouk ◽  
Sherin F Hammad ◽  
Fotouh R Mansour ◽  
Amira H Kamal ◽  
...  
Keyword(s):  
Least Squares ◽  
Simultaneous Determination ◽  
Principal Component Regression ◽  
Principal Component ◽  
Data Matrix ◽  
Ketorolac Tromethamine ◽  
Concentration Data ◽  
Phenylephrine Hydrochloride ◽  
Spectrophotometric Methods

Abstract The present work describes new spectrophotometric methods for the simultaneous determination of phenylephrine hydrochloride and ketorolac tromethamine in their synthetic mixtures. The applied chemometric techniques are multivariate methods including classical least squares, principal component regression, and partial least squares. In these techniques, the concentration data matrix was prepared by using the synthetic mixtures containing these drugs dissolved in distilled water. The absorbance data matrix corresponding to the concentration data was obtained by measuring the absorbances at 16 wavelengths in the range 244–274 nm at 2 nm intervals in the zero-order spectra. The spectrophotometric procedures do not require any separation steps. The accuracy, precision, and linearity ranges of the methods have been determined, and analyzing synthetic mixtures containing the studied drugs has validated them. The developed methods were successfully applied to the synthetic mixtures and the results were compared to those obtained by a reported HPLC method.

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