Effective rank for multivariate calibration methods

Latent root regression analysis: an alternative method to PLS

Diagnostics are fundamental to multivariate calibration (MC). Two common diagnostics are leverages and spectralF‐ratios and these have been formulated for many MC methods such as partial least square (PLS), principal component regression (PCR) and classical least squares (CLS). While these are some of the most common methods of calibration in analytical chemistry, ridge regression is also common place and yet spectralF‐ratios have not been developed for it. Noting that ridge regression is a form of Tikhonov regularization (TR) and using the unifying filter factor representation for MC, this paper develops the filter factor form of leverages and spectralF‐ratios. The approach is applied to a spectral data set to demonstrate computational speed‐up advantages and ease of implementation for the filter factor representation. Copyright © 2010 John Wiley & Sons, Ltd.

The multivariate methods, principal component regression (PCR) and partial least squares (PLS) were tested as a calibration procedure for simultaneous ultraviolet spectrophotometric determination of acetaminophen (AC) and ascorbic acid (AA). Determination of these compounds is important because of their pharmacotherapeutic advantages. Due to spectral overlapping of AC and AA, PCR and PLS were used for construction of the calibration sets. The concentration linear range of AC and AA were 1.5-24.2 and 1.8-21.1 microg mL(-1) respectively. The absorption spectra were recorded from 215-310 nm. The minimum root mean square error of prediction (RMSEP) was 1.3507 and 0.4088 for AC and AA, by PLS, 0.7525 and 0.4015 by PCR in original data and 0.9454 and 0.2875, by PLS and 1.0386 and 0.4000 by PCR in derivative data. The procedure allows the simultaneous determination of AC and AA in synthetic mixtures and real sample solutions made up from pharmaceutical products, human serum and urine.

The objective of using regression is to explain the variation in one or more response variables by associating the variation with proportional variation in one or more explanatory variables. However, if the number of independent variables is multiple, they tend to be highly collinear and this contributes to multicollinearity problem. For instance, Ridge Regression (RR), Principal Component Regression (PCR), and Partial Least Squares Regression (PLSR) are some of the prediction methods used to handle dataset with multicollinearity. In addition, another problem that arises is the existence of outlying objects in a dataset. The effect of outlying data points in the presence of multicollinearity problem could be reduced with the implementation of robust regression method. A recently studied robust PLSR, which is called Partial Robust M-Regression (PRM) is found to be able in dealing with multicollinearity and outliers simultaneously. This method was employed in this study. Throughout this study, five methods of regression were chosen; OLS, RR, PCR, PLSR, and PRM, to compare which is the best method in their predictive ability. To compare these five regression methods, a simulation study had been conducted, and the mean square error of β estimate (MSE(β)) was calculated. The simulation results show that PRM outperforms other method in the presence of multicollinearity, outliers and leverage points.

The issue of multicollinearity has long been acknowledged in statistical modelling; however, it is often untreated in the most of published papers. Indeed, the use of methods for multicollinearity correction is still scarce. One important reason is that despite many proposed methods, little is known about their strength or performance. We compare the statistical properties and performance of four main techniques to correct multicollinearity, i.e., Ridge Regression (R-R), Principal Components Regression (PC-R), Partial Least Squares Regression (PLS-R), and Lasso Regression (L-R), in both a simulation study and two real data examples used for modelling volumes of heart and Thyroid as a function of clinical and anthropometric parameters. We find that when the statistical approaches were used to address different levels of collinearity, we observed that R-R, PC-R and PLS-R appeared to have a somewhat similar behavior, with a slight advantage for the PLS-R. Indeed, in all implemented cases, the PLS-R always provided the smallest value of root mean square error (RMSE). When the degree of collinearity was moderate, low or very low, the L-R method had also somewhat similar performance to other methods. Furthermore, correction methods allowed us to provide stable and trustworthy parameter estimates for predictors in the modelling of heart and Thyroid volumes. Therefore, this work will contribute to highlighting performances of methods used only for situations ranging from low to very high multicollinearity.

A comparison of two multivariate calibration methods, partial least squares (PLS) and principal component regression (PCR), applied to high-performance liquid chromatography (HPLC) data, is presented for the resolution of a pesticide mixture. The data set showed both coeluted peaks and overlapped absorption spectra. Besides, there is an additional overlapping between the signal of the mobile phase and that of some pesticide. Multivariate calibration models were evaluated using different criteria to choose the optimum number of latent variables. It is shown that PLS yields the best predictive models. Furthermore, two methods for selecting regions were applied with the goal to achieve an improved prediction ability in the present multicomponent determination by HPLC-DAD (diode array detector) with PLS. The selection of regions associated with a large correlation to the concentration and with large values in loading-weighs (from PLS) were considered. It is concluded that feature selection can also improve the multivariate calibration results using chromatographic data.

In this paper, the influence of economic variables on the annual electricity consumption in Beijing in 2000-2013 has been investigated. The purpose is to find out their relationship with single data sets and multivariate statistic methods. The multivariate methods used in this paper include ordinary least squares regression (OLSR), principal component regression (PCR), partial least squares regression (PLSR), and modified partial least squares regression (MPLSR). In comparison of these methods, the PLSR model is shown to perform better than the others. The effectiveness of the electricity consumption prediction models is illustrated and verified based on the practical data sets of electricity consumption in Beijing.

Recycled polypropylene (rPP) often contains a small amount of polyethylene (PE). Since polypropylene (PP) and PE are incompatible, the presence of PE compromises the performance of rPP materials and needs to be closely monitored. In our previous work, Raman and near-infrared (NIR) spectrometries were evaluated to monitor PE content in rPP with partial least square regression (PLSR) modeling. The NIR spectrometry exhibited a wider application range, but the accuracy of the prediction models might be further improved. In the current work, a different modeling method, principal component regression (PCR) was employed to analyze PE content in rPP with NIR spectrometry. Spectrum pretreatment methods, including multivariate scatter correction (MSC), standard normal variate transformation (SNV), smoothing, and first derivative, were investigated to improve the NIR spectrum quality. Forward and backward interval methods were used to optimize spectral range selection. The outcomes were compared with our previous PLSR modeling results. The highest accuracy in independent validation was achieved by a PCR model with an R 2 of 0.9991 and a root-mean-square error of prediction (RMSEP) of 0.1596 PE%. On the other hand, a PLSR model achieved the lowest RMSEP of 0.9712 PE% for a non-colored post-consumer rPP sample. The PCR models might be sensitive to interference and more suitable for post-industrial materials, which have a simpler chemical composition. The PLSR models might have better stability and be more suitable for complicated post-consumer samples. Both the PCR and PLSR models were successfully applied to a gray commercial rPP sample.

In principal component regression (PCR) and partial least‐squares regression (PLSR), the use of unlabeled data, in addition to labeled data, helps stabilize the latent subspaces in the calibration step, typically leading to a lower prediction error. For using unlabeled data in PLSR, a non‐sequential approach based on optimal filtering (OF) has been proposed in the literature. In this work, a sequential version of the OF‐based PLSR and a PCA‐based PLSR (PLSR applied to PCA‐preprocessed data) are proposed. It is shown analytically that the sequential version of the OF‐based PLSR is equivalent to that of PCA‐based PLSR, which leads to a new interpretation of OF. Simulated and experimental data sets are used to point out the usefulness and pitfalls of using unlabeled data. Unlabeled data can replace labeled data to some extent, thereby leading to an economic benefit. However, in the presence of drift, the use of unlabeled data can result in an increase in prediction error compared to that obtained with a model based on labeled data alone. Copyright © 2011 John Wiley & Sons, Ltd.

In this study, a model for prediction of lignocellulose components of agricultural residues has been developed with Fourier Transformed Near Infrared (FT-NIR) spectroscopy data. Two calibration techniques (Principal Component Regression (PCR) and Partial Least Square Regression (PLSR)) were assessed for prediction of lignin, holocellulose, α-cellulose, pentosan and ash, and found the PLSR better for lignin, holocellulose and α-cellulose. The PCR also produced better results for quantification of pentosan and ash. Spectral range (7000–5000 cm–1) showed more informative than other parts of the spectral data. The PLSR showed maximum value of R2 (R2 = 0.91%) for prediction of holocellulose. For the prediction of pentosan, the PCR was better (R2 = 0.68%). The PCR also showed better results (R2 = 86%) for quantification of ash. To determine amount of lignin, the PLSR was the best (R2 = 0.83%) when the spectral data were de-trained and smoothed with Savitzky-Golay (S-G) filtering simultaneously. For prediction of α-cellulose, the PLSR was the best model (R2 = 0.94%) when the data were pretreated with mean normalization. Considering the best alternatives inNear Infrared (NIR) data preprocessing and calibration techniques, methods for quantification of lignocellulose components of agricultural residues have been developed which is rapid, cost effective, and less chemical intensive and easily usable in pulp and paper industries and pulp testing laboratories.

Weather index insurance is a relatively new alternative to traditional agricultural insurance such as individual yield-based crop insurance. It is still mostly at the experimental stage, rather than in widespread use like traditional crop insurance. A major challenge for weather insurance is basis risk, where the loss estimated by the index differs from the actual loss, and this is generally believed to be the main limitation in the use of weather index insurance for crops. Variable basis risk is an important type of basis risk that occurs when there are incorrect variables or missing variables for the design of the weather index. In agriculture, there is a relatively small sample size of yields, and therefore, as the number of considered weather variables increases, the problems of limited degrees of freedom for predictive models must be overcome. The objective of this article is to demonstrate two possible approaches that could be used to construct a multivariable weather index to reduce variable basis risk. Forage insurance is used as an example, and a main focus of the research is on reducing the dimensionality of the predictive model and resolving the problem of multicollinearity among weather variables. The research uses daily weather information and county-level forage yield data from Ontario, Canada. Two multivariable indices are developed based on principal component regression (PCR) and partial least squares regression (PLSR) methods, and they are evaluated against a single-variable benchmark index based on cumulative precipitation (SVCP) using several basis risk metrics. The results show that both the PCR and PLSR models are superior compared to the single-variable weather index based on cumulative rainfall (SVCR) index and can be used to achieve the objective of reducing the dimensionality of the weather variable matrix and addressing the issue of multicollinearity. Though the PLSR indices perform better than the PCR and SVCR indices in terms of average value of basis risk (),(rom) the PCR method produces a smaller percentage of mismatch, suggesting that the PCR method may be superior in correctly detecting when the insurance payment should be triggered. The methods demonstrated in this article will assist in the development of weather index–based crop insurance.

Development of novel univariate and multivariate validated chemometric methods for the analysis of dasatinib, sorafenib, and vandetanib in pure form, dosage forms and biological fluids

This study aims to estimate simultaneously metformin hydrochloride (MET) and glyburide (GLY) in a multicomponent tablets dosage form by spectrophotometric method using chemometric approaches such as principal component regression (PCR) and partial least-squares regression (PLS). Because of highly overlapped in UV spectra and difference proportions of two active ingredients, the conventional univariate calibration methods was not allowed without previous separation. The linearity ranges used to construct the calibration matrix were selected in the ranges from 40.00 to 200.00 mg L-1for MET and from 1.00 to 10.00 mg L-1for GLY. The absorbances were measured in the wavelength range of 200-400 nm, using ethanol as solvent. The resulting UV spectra were subjected to PCR and PLS algorithms and the optimum numbers of principal components (PCs) were selected according to prediction residual error sum of squares (PRESS) values of leave-one out cross-validation. The number of PCs for MET and GLY were found to be 5, 3 by PCR and 5, 3 by PLS, respectively. A set of synthetic mixtures was employed to verify the models and the performance of the models were shown in the values of the root mean square error in prediction (RMSEP). RMSEP values of MET and GLY were 1.806, 0.256 for PCR and 1.802, 0.185 for PLS, respectively. The suitable calibration models were applied to the analysis of these compounds in pharmaceutical formulation.

The objective of this study was to compare the predictions of the protein contents and hardness values by partial least squares regression (PLSR) and principal components regression (PCR) models for bulk samples of Canadian wheat, which were obtained from different locations and crop years. Wheat samples of Canada Western Red Spring (CWRS), Canada Western Hard White Spring (CWHWS), Canada Western Soft White Spring (CWSWS), and Canada Prairie Spring Red (CPSR) classes were obtained from nearby agricultural farms in the main wheat growing locations in the Provinces of Alberta, Saskatchewan, and Manitoba from 2007, 2008, and 2009 crop years. Wheat samples were conditioned to moisture levels of 13, 16, and 19 % (wet basis) and pooled together for developing the regression models. A database of the near-infrared (NIR) hyperspectral image cubes of bulk samples of wheat classes was created in the wavelength region of 960–1,700 nm with 10 nm intervals. Reference protein contents and hardness values were determined using the Dumatherm method and single kernel characterization system (SKCS), respectively. A tenfold cross-validation was used for the ten-factor partial least squares regression (PLSR) and principal components regression (PCR) models for prediction purposes. Prediction performances of regression models were assessed by calculating the estimated mean square errors of prediction (MSEP), standard error of cross-validation (SECV), and correlation coefficient (r). Using the full data set in the protein prediction study, the ten-component PLSR model gave 1.76, 1.33, and 0.68 for the estimated MSEP, SECV, and r, respectively, which were better than the results for the ten-component PCR model (2.02, 1.42, and 0.62, respectively). For the hardness prediction, the estimated MSEP, SECV, and r values were 147.7, 12.15, and 0.82, respectively, for the ten-component PLSR model using the full data set. The PLSR models prediction performances outperformed the PCR models for predicting protein contents and hardness of wheat.

The utilization of selection methods such as genetic algorithm (GA) aims to construct better partial least squares (PLS) and principal component regression (PCR) models than those established from the full-spectrum range. Determination of paracetamol (PAR), orphenadrine citrate (Or.cit), and caffeine (CAF) in the presence of PAR nephrotoxic impurity [p-aminophenol (PAP)]. GA was applied to select the optimum wavelengths used. A calibration set was prepared in which the three drugs, together with PAP, were modeled by multilevel multifactor design. This calibration set was used to build the PLS and PCR models, either with or without preprocessing the data using GA. Results were compared with and without preprocessing, and this revealed that GA can find an optimized combination of spectral wavelengths, yielding a lower root mean square error of prediction as well as a lower number of latent variables used. The results of the two models show that simultaneous determination of the aforementioned drugs can be performed in the concentration ranges of 20-60, 3-11, and 1-9 μg/mL for PAR, Or.cit, and CAF, respectively. The proposed models were applied for the determination of the three drugs in their pharmaceutical formulations, and the results were verified by the standard addition technique. GA can be useful as a wavelength selection tool before applying multivariate PLS and PCR methods. GA gives an improvement in the predictive ability of the models with lower RMSEP and less number of latent variables (LVs). The proposed PLS, PCR, GA-PLS, and GA-PCR spectrophotometric methods were able to determine paracetamol, orphenadrine citrate, and caffeine in the presence of p-aminophenol and severe spectral overlapping.