Reproducibility,complementary measure of predictability for robustness improvement of multivariate calibration models via variable selections

In multivariate calibration with the spectral dataset, variable selection is often applied to identify relevant subset of variables, leading to improved prediction accuracy and easy interpretation of the selected fingerprint regions. Until now, numerous variable selection methods have been proposed, but a proper choice among them is not trivial. Furthermore, in many cases, a set of variables found by those methods might not be robust due to the irreproducibility and uncertainty issues, posing a great challenge in improving the reliability of the variable selection. In this study, the reproducibility of the 5 variable selection methods was investigated quantitatively for evaluating their performance. The reproducibility of variable selection was quantified by using Monte-Carlo sub-sampling (MCS) techniques together with the quantitative similarity measure designed for the highly collinear spectral dataset. The investigation of reproducibility and prediction accuracy of the several variable selection algorithms with two different near-infrared (NIR) datasets illustrated that the different variable selection methods exhibited wide variability in their performance, especially in their capabilities to identify the consistent subset of variables from the spectral datasets. Thus the thorough assessment of the reproducibility together with the predictive accuracy of the identified variables improved the statistical validity and confidence of the selection outcome, which cannot be addressed by the conventional evaluation schemes.     

Bayesian linear regression and variable selection for spectroscopic calibration

This paper presents a Bayesian approach to the development of spectroscopic calibration models. By formulating the linear regression in a probabilistic framework, a Bayesian linear regression model is derived, and a specific optimization method, i.e. Bayesian evidence approximation, is utilized to estimate the model “hyper-parameters”. The relation of the proposed approach to the calibration models in the literature is discussed, including ridge regression and Gaussian process model. The Bayesian model may be modified for the calibration of multivariate response variables. Furthermore, a variable selection strategy is implemented within the Bayesian framework, the motivation being that the predictive performance may be improved by selecting a subset of the most informative spectral variables. The Bayesian calibration models are applied to two spectroscopic data sets, and they demonstrate improved prediction results in comparison with the benchmark method of partial least squares.     

Predictive-property-ranked variable reduction in partial least squares modelling with final complexity adapted models: Comparison of properties for ranking

The calibration performance of partial least squares regression for one response (PLS1) can be improved by eliminating uninformative variables. Many variable-reduction methods are based on so-called predictor-variable properties or predictive properties, which are functions of various PLS-model parameters, and which may change during the steps of the variable-reduction process. Recently, a new predictive-property-ranked variable reduction method with final complexity adapted models, denoted as PPRVR-FCAM or simply FCAM, was introduced. It is a backward variable elimination method applied on the predictive-property-ranked variables. The variable number is first reduced, with constant PLS1 model complexity A, until A variables remain, followed by a further decrease in PLS complexity, allowing the final selection of small numbers of variables.     

GA-ACE: Alternating conditional expectations regression with selection of significant predictors by genetic algorithms

The non-linear regression technique known as alternating conditional expectations (ACE) method is only applicable when the number of objects available for calibration is considerably greater than the number of considered predictors. Alternating conditional expectations regression with selection of significant predictors by genetic algorithms (GA-ACE), the non-linear regression technique presented here, is based on the ACE algorithm but introducing several modifications to resolve the applicability limitations of the original ACE method, thus facilitating the practical implementation of a very interesting calibration tool. In order to overcome the lack of reliability displayed by the original ACE algorithm when working on data sets characterized by a too large number of variables and prior to the development of the non-linear regression model, GA-ACE applies genetic algorithms as a variable selection technique to select a reduced subset of significant predictors able to accurately model and predict a considered variable response. Furthermore, GA-ACE actually provides two alternative application approaches, since it allows either the performance of prior data compression computing a number of principal components to be subsequently subjected to GA-selection, or working directly on original variables.In this study, GA-ACE was applied to two real calibration problems, with a very low observation/variable ratio (NIR data), and the results were compared with those obtained by several linear regression techniques usually employed. When using the GA-ACE non-linear method, notably improved regression models were developed for the two response variables modeled, with root mean square errors of the residuals in external prediction (RMSEP) equal to 11.51 and 6.03% for moisture and lipid contents of roasted coffee samples, respectively. The improvement achieved by applying the new non-linear method introduced is even more remarkable taking into account the results obtained with the best performance linear method (IPW-PLS) applied to predict the studied responses (14.61 and 7.74% RMSEP, respectively).     

Genetic algorithms applied to the selection of factors in principal component regression

Using principal component regression (PCR) as a multivariate calibration tool, always brings up the question what subset of factors, i.e. principal components (PCs) gives the best calibration model. Normally factor selection is based on deterministic methods like top–down procedures, forward–backward-stepwise variable selection or correlated principal component regression (CPCR). In contrast to this, we applied a stochastic method, i.e. a genetic algorithm (GA) for factor selection in this paper. A new kind of fitness function was applied which combined the prediction error of the calibration and an independent validation set. The performance of eigenvalue and correlation ranking was compared. A general statistical criterion for judging the significance of differences between individual calibration models is introduced. In this context it could be shown that for the uncertainties of the standard deviations representing the prediction errors a very simple approximation formula holds which only includes the number of standards. For the current applications it is shown that the GA gives a result very close to CPCR-solutions.     

Using variable combination population analysis for variable selection in multivariate calibration

Variable (wavelength or feature) selection techniques have become a critical step for the analysis of datasets with high number of variables and relatively few samples. In this study, a novel variable selection strategy, variable combination population analysis (VCPA), was proposed. This strategy consists of two crucial procedures. First, the exponentially decreasing function (EDF), which is the simple and effective principle of ‘survival of the fittest’ from Darwin’s natural evolution theory, is employed to determine the number of variables to keep and continuously shrink the variable space. Second, in each EDF run, binary matrix sampling (BMS) strategy that gives each variable the same chance to be selected and generates different variable combinations, is used to produce a population of subsets to construct a population of sub-models. Then, model population analysis (MPA) is employed to find the variable subsets with the lower root mean squares error of cross validation (RMSECV). The frequency of each variable appearing in the best 10% sub-models is computed. The higher the frequency is, the more important the variable is. The performance of the proposed procedure was investigated using three real NIR datasets. The results indicate that VCPA is a good variable selection strategy when compared with four high performing variable selection methods: genetic algorithm–partial least squares (GA–PLS), Monte Carlo uninformative variable elimination by PLS (MC-UVE-PLS), competitive adaptive reweighted sampling (CARS) and iteratively retains informative variables (IRIV). The MATLAB source code of VCPA is available for academic research on the website: http://www.mathworks.com/matlabcentral/fileexchange/authors/498750.     

A new and efficient variable selection algorithm based on ant colony optimization. Applications to near infrared spectroscopy/partial least-squares analysis

A new variable selection algorithm is described, based on ant colony optimization (ACO). The algorithm aim is to choose, from a large number of available spectral wavelengths, those relevant to the estimation of analyte concentrations or sample properties when spectroscopic analysis is combined with multivariate calibration techniques such as partial least-squares (PLS) regression. The new algorithm employs the concept of cooperative pheromone accumulation, which is typical of ACO selection methods, and optimizes PLS models using a pre-defined number of variables, employing a Monte Carlo approach to discard irrelevant sensors. The performance has been tested on a simulated system, where it shows a significant superiority over other commonly employed selection methods, such as genetic algorithms. Several near infrared spectroscopic experimental data sets have been subjected to the present ACO algorithm, with PLS leading to improved analytical figures of merit upon wavelength selection. The method could be helpful in other chemometric activities such as classification or quantitative structure-activity relationship (QSAR) problems.     

Variable importance analysis based on rank aggregation with applications in metabolomics for biomarker discovery

Biomarker discovery is one important goal in metabolomics, which is typically modeled as selecting the most discriminating metabolites for classification and often referred to as variable importance analysis or variable selection. Until now, a number of variable importance analysis methods to discover biomarkers in the metabolomics studies have been proposed. However, different methods are mostly likely to generate different variable ranking results due to their different principles. Each method generates a variable ranking list just as an expert presents an opinion. The problem of inconsistency between different variable ranking methods is often ignored. To address this problem, a simple and ideal solution is that every ranking should be taken into account. In this study, a strategy, called rank aggregation, was employed. It is an indispensable tool for merging individual ranking lists into a single “super”-list reflective of the overall preference or importance within the population. This “super”-list is regarded as the final ranking for biomarker discovery. Finally, it was used for biomarkers discovery and selecting the best variable subset with the highest predictive classification accuracy. Nine methods were used, including three univariate filtering and six multivariate methods. When applied to two metabolic datasets (Childhood overweight dataset and Tubulointerstitial lesions dataset), the results show that the performance of rank aggregation has improved greatly with higher prediction accuracy compared with using all variables. Moreover, it is also better than penalized method, least absolute shrinkage and selectionator operator (LASSO), with higher prediction accuracy or less number of selected variables which are more interpretable.     

An ensemble of Monte Carlo uninformative variable elimination for wavelength selection

An improved method based on an ensemble of Monte Carlo uninformative variable elimination (EMCUVE) is presented for wavelength selection in multivariate calibration of spectral data. The proposed algorithm introduces Monte Carlo (MC) strategy to uninformative variable elimination-PLS (UVE-PLS) instead of leave-one-out strategy for estimating the contributions of each wavelength variable in the PLS model. In EMCUVE wavelength variables are evaluated by different Monte Carlo uninformative variable elimination (MCUVE) models. Moreover, a fusion of MCUVE and the vote rule can obtain an improvement over the original uninformative variable elimination method. Results obtained from simulated data and real data sets demonstrate that EMCUVE can properly carry out wavelength selection in the course of data analysis and improve predictive ability for multivariate calibration model.     

A consensus successive projections algorithm – multiple linear regression method for analyzing near infrared spectra

The successive projections algorithm (SPA) is widely used to select variables for multiple linear regression (MLR) modeling. However, SPA used only once may not obtain all the useful information of the full spectra, because the number of selected variables cannot exceed the number of calibration samples in the SPA algorithm. Therefore, the SPA-MLR method risks the loss of useful information. To make a full use of the useful information in the spectra, a new method named “consensus SPA-MLR” (C-SPA-MLR) is proposed herein. This method is the combination of consensus strategy and SPA-MLR method. In the C-SPA-MLR method, SPA-MLR is used to construct member models with different subsets of variables, which are selected from the remaining variables iteratively. A consensus prediction is obtained by combining the predictions of the member models. The proposed method is evaluated by analyzing the near infrared (NIR) spectra of corn and diesel. The results of C-SPA-MLR method showed a better prediction performance compared with the SPA-MLR and full-spectra PLS methods. Moreover, these results could serve as a reference for combination the consensus strategy and other variable selection methods when analyzing NIR spectra and other spectroscopic techniques.     

Variable-weighted least-squares support vector machine for multivariate spectral analysis

Multivariate spectral analysis has been widely applied in chemistry and other fields. Spectral data consisting of measurements at hundreds and even thousands of analytical channels can now be obtained in a few seconds. It is widely accepted that before a multivariate regression model is built, a well-performed variable selection can be helpful to improve the predictive ability of the model. In this paper, the concept of traditional wavelength variable selection has been extended and the idea of variable weighting is incorporated into least-squares support vector machine (LS-SVM). A recently proposed global optimization method, particle swarm optimization (PSO) algorithm is used to search for the weights of variables and the hyper-parameters involved in LS-SVM optimizing the training of a calibration set and the prediction of an independent validation set. All the computation process of this method is automatic. Two real data sets are investigated and the results are compared those of PLS, uninformative variable elimination-PLS (UVE-PLS) and LS-SVM models to demonstrate the advantages of the proposed method.     

New cut-off criterion for uninformative variable elimination in multivariate calibration of near-infrared spectra for the determination of heroin in illicit street drugs

A new cut-off criterion has been proposed for the selection of uninformative variables prior to chemometric partial least squares (PLS) modelling. After variable elimination, PLS regressions were made and assessed comparing the results with those obtained by PLS models based on the full spectral range. To assess the prediction capabilities, uninformative variable elimination (UVE)-PLS and PLS were applied to diffuse reflectance near-infrared spectra of heroin samples. The application of the proposed new cut-off criterion, based on the t-Students distribution, provided similar predictive capabilities of the PLS models than those obtained using the original criteria based on quantile value. However, the repeatability of the number of selected variables was improved significantly.     

Canonical Measure of Correlation (CMC) and Canonical Measure of Distance (CMD) between sets of data. Part 3. Variable selection in classification

In multivariate regression and classification issues variable selection is an important procedure used to select an optimal subset of variables with the aim of producing more parsimonious and eventually more predictive models. Variable selection is often necessary when dealing with methodologies that produce thousands of variables, such as Quantitative Structure-Activity Relationships (QSARs) and highly dimensional analytical procedures.In this paper a novel method for variable selection for classification purposes is introduced. This method exploits the recently proposed Canonical Measure of Correlation between two sets of variables (CMC index). The CMC index is in this case calculated for two specific sets of variables, the former being comprised of the independent variables and the latter of the unfolded class matrix. The CMC values, calculated by considering one variable at a time, can be sorted and a ranking of the variables on the basis of their class discrimination capabilities results. Alternatively, CMC index can be calculated for all the possible combinations of variables and the variable subset with the maximal CMC can be selected, but this procedure is computationally more demanding and classification performance of the selected subset is not always the best one.The effectiveness of the CMC index in selecting variables with discriminative ability was compared with that of other well-known strategies for variable selection, such as the Wilks’ Lambda, the VIP index based on the Partial Least Squares-Discriminant Analysis, and the selection provided by classification trees.A variable Forward Selection based on the CMC index was finally used in conjunction of Linear Discriminant Analysis. This approach was tested on several chemical data sets. Obtained results were encouraging.     

Multivariate calibration on NIR data: development of a model for the rapid evaluation of ethanol content in bakery products

A new NIR method based on multivariate calibration for determination of ethanol in industrially packed wholemeal bread was developed and validated. GC-FID was used as reference method for the determination of actual ethanol concentration of different samples of wholemeal bread with proper content of added ethanol, ranging from 0 to 3.5% (w/w). Stepwise discriminant analysis was carried out on the NIR dataset, in order to reduce the number of original variables by selecting those that were able to discriminate between the samples of different ethanol concentrations. With the so selected variables a multivariate calibration model was then obtained by multiple linear regression. The prediction power of the linear model was optimized by a new “leave one out” method, so that the number of original variables resulted further reduced.     

Variable selection in multivariate calibration based on clustering of variable concept

Recently we have proposed a new variable selection algorithm, based on clustering of variable concept (CLoVA) in classification problem. With the same idea, this new concept has been applied to a regression problem and then the obtained results have been compared with conventional variable selection strategies for PLS. The basic idea behind the clustering of variable is that, the instrument channels are clustered into different clusters via clustering algorithms. Then, the spectral data of each cluster are subjected to PLS regression. Different real data sets (Cargill corn, Biscuit dough, ACE QSAR, Soy, and Tablet) have been used to evaluate the influence of the clustering of variables on the prediction performances of PLS. Almost in the all cases, the statistical parameter especially in prediction error shows the superiority of CLoVA-PLS respect to other variable selection strategies. Finally the synergy clustering of variable (sCLoVA-PLS), which is used the combination of cluster, has been proposed as an efficient and modification of CLoVA algorithm. The obtained statistical parameter indicates that variable clustering can split useful part from redundant ones, and then based on informative cluster; stable model can be reached.     

A bootstrapping soft shrinkage approach for variable selection in chemical modeling

In this study, a new variable selection method called bootstrapping soft shrinkage (BOSS) method is developed. It is derived from the idea of weighted bootstrap sampling (WBS) and model population analysis (MPA). The weights of variables are determined based on the absolute values of regression coefficients. WBS is applied according to the weights to generate sub-models and MPA is used to analyze the sub-models to update weights for variables. The optimization procedure follows the rule of soft shrinkage, in which less important variables are not eliminated directly but are assigned smaller weights. The algorithm runs iteratively and terminates until the number of variables reaches one. The optimal variable set with the lowest root mean squared error of cross-validation (RMSECV) is selected. The method was tested on three groups of near infrared (NIR) spectroscopic datasets, i.e. corn datasets, diesel fuels datasets and soy datasets. Three high performing variable selection methods, i.e. Monte Carlo uninformative variable elimination (MCUVE), competitive adaptive reweighted sampling (CARS) and genetic algorithm partial least squares (GA-PLS) are used for comparison. The results show that BOSS is promising with improved prediction performance. The Matlab codes for implementing BOSS are freely available on the website: http://www.mathworks.com/matlabcentral/fileexchange/52770-boss.     

The use of FT-NIR for API content assay in organic solvent media: a single calibration for multiple downstream processing streams

The use of near infrared spectroscopy (NIRS) in downstream solvent based processing steps of an active pharmaceutical ingredient (API) is reported. A single quantitative method was developed for API content assessment in the organic phase of a liquid–liquid extraction process and in multiple process streams of subsequent concentration and depuration steps. A new methodology based in spectra combinations and variable selection by genetic algorithm was used with an effective improvement in calibration model prediction ability. Root mean standard error of prediction (RMSEP) of 0.05 in the range of 0.20–3.00% (w/w) was achieved. With this method, it is possible to balance the calibration data set with spectra of desired concentrations, whenever acquisition of new spectra is no longer possible or improvements in model's accuracy for a specific selected range are necessary. The inclusion of artificial spectra prior to genetic algorithms use improved RMSEP by 10%. This method gave a relative RMSEP improvement of 46% compared with a standard PLS of full spectral length.     

Genetic algorithm optimisation combined with partial least squares regression and mutual information variable selection procedures in near-infrared quantitative analysis of cotton-viscose textiles

In this work, different approaches for variable selection are studied in the context of near-infrared (NIR) multivariate calibration of textile. First, a model-based regression method is proposed. It consists in genetic algorithm optimisation combined with partial least squares regression (GA-PLS). The second approach is a relevance measure of spectral variables based on mutual information (MI), which can be performed independently of any given regression model. As MI makes no assumption on the relationship between X and Y, non-linear methods such as feed-forward artificial neural network (ANN) are thus encouraged for modelling in a prediction context (MI-ANN). GA-PLS and MI-ANN models are developed for NIR quantitative prediction of cotton content in cotton-viscose textile samples. The results are compared to full-spectrum (480 variables) PLS model (FS-PLS). The model requires 11 latent variables and yielded a 3.74% RMS prediction error in the range 0-100%. GA-PLS provides more robust model based on 120 variables and slightly enhanced prediction performance (3.44% RMS error). Considering MI variable selection procedure, great improvement can be obtained as 12 variables only are retained. On the basis of these variables, a 12 inputs ANN model is trained and the corresponding prediction error is 3.43% RMS error.     

A strategy that iteratively retains informative variables for selecting optimal variable subset in multivariate calibration

Nowadays, with a high dimensionality of dataset, it faces a great challenge in the creation of effective methods which can select an optimal variables subset. In this study, a strategy that considers the possible interaction effect among variables through random combinations was proposed, called iteratively retaining informative variables (IRIV). Moreover, the variables are classified into four categories as strongly informative, weakly informative, uninformative and interfering variables. On this basis, IRIV retains both the strongly and weakly informative variables in every iterative round until no uninformative and interfering variables exist. Three datasets were employed to investigate the performance of IRIV coupled with partial least squares (PLS). The results show that IRIV is a good alternative for variable selection strategy when compared with three outstanding and frequently used variable selection methods such as genetic algorithm-PLS, Monte Carlo uninformative variable elimination by PLS (MC-UVE-PLS) and competitive adaptive reweighted sampling (CARS). The MATLAB source code of IRIV can be freely downloaded for academy research at the website: http://code.google.com/p/multivariate-calibration/downloads/list.     

Infrared spectroscopy and multivariate calibration to monitor stability quality parameters of biodiesel

This paper evaluates analytical methods based on near infrared (NIR) and middle infrared (MIR) spectroscopy and multivariate calibration to monitor the stability of biodiesel. There was a focus on three parameters: oxidative stability index, acid number and water content. Ethylic and methylic biodiesel from different feedstocks were used in experiments of accelerated aging, in order to take into account the wide variety of oilseeds and feedstocks available in Brazil. Partial least squares (PLS) and multiple linear regression (MLR) models were developed. Different pre-processing techniques and spectral variable/regions selection algorithms were evaluated. For MLR models, the successive projection algorithm (SPA) was employed. Interval PLS (iPLS) and selection of variables taking into account the significant regression coefficients were used for PLS models. Results showed that both near and middle infrared regions, and all variable selection methods tested were efficient for predicting these three important quality parameters of B100, the root mean squares error of prediction (RMSEP) values being comparable to the reproducibility of the corresponding standard method for each property investigated.