Multivariate calibration models

A. Bos, M. Bos and W.E. Van der Linden, Artificial neural networks as a multivariate calibration tool modeling the ion-chromium nickel system in x-ray fluorescence spectra. Anal. Chim. Acta, 277 (1993) 289-295. 

Calibration, multivariate, n - a process for creating a model that relates component concentrations or properties to the absorbances of a set of known reference samples at more than one wavelength or frequency. 

A note of caution is needed here. The figures of merit presented in this section refer to the multivariate calibration model. This multivariate model, built with standards, is then applied to future real samples. If standards and real samples match, as should be the case in most applications, the calibration model is the essential step of the overall analytical procedure. However, if real samples require additional steps (basically preprocessing steps such as extractions, preconcentrations, etc.) different from those of the standards, then the calibration model is just one more step in the whole procedure. If the previous steps are not the same, this means that the figures of merit calculated for the model do not refer to the whole analytical procedure and, therefore, other approaches should be undertaken to calculate them [56]. 

Physical and chemical effects can be combined for identification as sample matrix effects. Matrix effects alter the slope of calibration curves, while spectral interferences cause parallel shifts in the calibration curve. The water-methanol data set contains matrix effects stemming from chemical interferences. As already noted in Section 5.2, using the univariate calibration defined in Equation 5.4 requires an interference-free wavelength. Going to multivariate models can correct for spectral interferences and some matrix effects. The standard addition method described in Section 5.7 can be used in some cases to correct for matrix effects. Severe matrix effects can cause nonlinear responses requiring a nonlinear modeling method. 

Fig. 11 The calibration model relating IR spectra to solution concentration. The multivariate model relates IR absorbances of a selected frequency range and temperature or solvent-antisolvent ratio to solution concentration.

Another issue is that of transferability of the calibration model among instruments. This has been a significant obstacle to more widespread use of NIR methods. Transferability is especially important to multisite facilities, because it is needed to avoid time-consuming recalibration procedures. Calibration errors may occur among instruments because of slight differences in instrument response, especially if full-spectrum multivariate models are used. Shenk and Westerhaus addressed the problem and proposed a standardization algorithm, which was modified by others. ... 

Many scientists hesitate to rely on multivariate calibration models due to the complexity of the statistics involved. The quality of a multivariate model is highly dependent upon the factors chosen. Incorporation of too many factors leads to an overfit model which provides a low calibration error value, but predicts unknown samples poorly. Too few factors lead to a model that neither adequately fits the calibration data nor accurately predicts new samples. Typically, factors are included or excluded from the calibration model based upon their statistical significance, but this may not provide the opti-... 

Multivariate Regression Methods. The main goal of this study was to build a multivariate model for the reliable prediction of a property of interest y (cheese ripening time) from a number of predictor variables, xi, X2. .. (peak area of casein and peptide obtained by CE). This model should describe the measured x and y data of the calibration set (cheese samples at different ripening time). In particular, in this research, the PCR and PLS methods were evaluated. 

The specified number and character of performance parameters and variables, i.c.. the operational conditions, is defined as a /ev/mg domain. Simple domains for any project may require only a univariate model, while complex project measurement systems may require multivariate models. Some projects may have multiple response parameters, each of which may require a multivariate (independent) variable model The calibration and validation operations are discussed below in Section 2.3. 

To use a multivariate regression technique, you would need another way of measuring or assessing the apple s texture as a golden standard to calibrate your model against. The measured impedance values will in this case often be called X-variables and the measurements from the calibration instmment is called Y-variables. 

Einbu et al. (4) also assessed the use of a combination of FTIR spectroscopy and a multivariate model for composition predictions, but applied to a CO2 absorption process using aqueous MEA. They constructed a model based on a very extensive calibration set of 86 samples, covering MEA concentration of 10 to 80 wt% and CO2 eoneentration of 0.0 to 0.5 mol CO2 per mol amine. Based on these calibration samples, the model was calculated to have a relative predictive uncertainty of 1.4% for MEA and 3.0% for CO2. It has also successfully been use for continuous in-line monitoring of an operating pilot plant, but no quantitative results for the prediction accuracy ate given. 

Follow-up work described by Van Eckeveld et al. (2) expanded the analytical techniques used during screening to also include the sonic speed, viscosity and near-infrared (NIR) spectroscopy. Several multivariate models were constructed based on a dedicated set of 29 lab-prepared calibration samples, covering MEA concentrations of 0 to 36 wt% and CO2 concentrations of 0.0 to 0.5 mol CO2 per mol MEA at temperatures of 40 and 55 °C. The models differed in the eombination of analytioal techniques... 

Methods such as standard addition only provide good results with a relatively simple matrices. One of the main problems when a first-order multivariate model is used is the presence of unknown interferences. Mathematical models have become very important for solving this problem an example is the determination of five pollutants of the chlorophenol family in urine. The effect of the matrix is minimized by including, in the calibration step, standard samples containing the analytes in the presence of the interfering matrix. The calibration set includes 60 standard samples 50 samples of chlorophenols in water and 10 of lyophilized urine. 

The importance of validating the multivariate model cannot be over-emphasised. In particular, the data should be checked for outliers , that is, samples whose properties are different from the rest of the calibration set. If outliers are not detected and either removed or corrected, serious errors may be built into the model. Check for outliers by plotting actual compositions (Y data) against predicted compositions. In a good model, all the samples will lie close to the line of best fit. Outliers will be isolated and associated with poor predictive accuracy. 

Multivariate model allows calibration of very complex mixtures because only knowledge of constituents of interest... 

What exactly is collinearity and why is it a problem in multivariate models Collinearity is the effect observed when the relative amounts of two or more constituents are a constant throughout all the training samples. This causes so much trouble for multivariate models because of the way they correlate information. Remember that these models do not calibrate by creating a direct relationship between the constituent data and spectral response. Instead, they try to correlate the change in concentration to some corresponding changes in the spectra. When constituents are collinear, multivariate models cannot not differentiate them, and the calibrations for the constituents will be unstable. 

The conceptual idea is to merge the data from both standardizations and include a new independent dummy variable (let us call it z ). This takes the value of 0 for the data points associated with a regression e.g. the SAM regression) and the value of 1 for the other data points e.g. aqueous calibration). Then, a multivariate model of the type y = Po + X + 2 + 3 ... 

Wang et al proposed a multivariate dominant factor based non-linearized PLS model for LIBS measurements. In constructing such a multivariate model, non-linear transformation of multi-characteristic line intensities according to the physical mechanisms of a laser-induced plasma spectrum were made, combined with a linear-correlation-based PLS method, to model the non-linear self-absorption and inter-element interference effects. Moreover, a secondary PLS was applied, utilizing information from the whole spectrum to correct the model results further. The proposed method showed a significant improvement when compared with a conventional PLS model. Even compared with the already improved baseline dominant-factor-based PLS model, the PLS model based on the multivariate dominant factor yielded the same calibration quality while decreasing the RMSEP. 

Often, the use of a fixed reference sample is also used to help improve the precision and accuracy of multivariate models in which many spectra are used together to create a calibration or training set. In these cases, the reference spectrum is the mean of the entire set of Raman spectra used for calibration and it is subtracted from each of the individual calibration spectra as well as any subsequent spectra from which a prediction is to be made. This approach is called mean centering. ... 

The most stringent need for wavenumber axis calibration is in determinations based on band position. For this reason, qualitative analyses are likely to be affected by drifts or inaccuracy in the wavenumber axis [14]. Likewise, quantitative determinations based on band position, such as strain in diamond films [6], will be affected similarly. Other quantitative analyses may also be affected by band-position error. It is common to use the raw spectral intensities (intensity at every wavenumber) in a multivariate analysis. Although this approach can be very powerful, any unexpected shift in wavenumber calibration can cause severe error in the model. In essence, the spectral pattern to which the model has been trained has been shifted. The mathematics of the model are expecting a particular relationship of intensity between adjacent variables (wavenumbers) and cannot usually account for shifts [31], To some extent, multivariate models can be desensitized to inaccuracy and imprecision by assuring that the calibration samples also exhibit some of the same shifting features, but model sensitivity may suffer as a result. Although not in common use, other deconvolution methods have been introduced which may be applicable to removing shift effects of inaccurate wavenumber calibrations [37]. 

Multivariable calibration permits the simultaneous determination of multicomponent mixtures and it is mainly based on spectroscopy data. Full-spectrum multivariate calibration methods offer the advantage of speed in the determination of the analytes, avoiding separation steps in the analytical procedures. Partial least squares (PLS) has become the usual first-order multivariate tool because of the quality of the calibration models obtained, the ease of its implementation, and the availability of software [27]. However, all first-order methods, of which PLS is no exception, are sensitive to the presence of unmodeled interferents, that is, compounds occurring in new samples that have not been included during the training step of the multivariate model. This situation is encountered... 

In the more general case, when the preprocessed signals are not to be used for explorative purposes only, but for modelling tasks (classification, multivariate calibration) as well, the Pchemjrr can also be estimated by multivariate modelling. Eor instance, a PCA model can be built with samples measured in those different conditions that we know a priori which may introduce unwanted variability (batches, seasonality, acidity of the media, humidity content, etc.) then the loadings of the few PCs where this variability is modelled can be used as Pchemjrr-... 

In the majority of quantitative problems, as most in food chemistry are, calibration models are constructed for a large number of parameters, for example near-infrared (NIR), fluorescence spectra or chromatographic profiles. Such parameters are highly correlated and their effective modelling requires the use of latent variables. Partial least squares (PLS) regression is one of the most popular multivariate modelling techniques that can deal with the multicoUinearity problem. The classic PLS model can be presented mathematically as ... 

Calibration—Each instrument must be calibrated by the manufacturer or user in accordance with Practice E 1655. This practice serves as a guide for the multivariate calibration of infrared spectrometers used in determining the physical characteristics of petroleum and petrochemical pr ucts. The procedures describe treatment of the data, development of the calibration, and validation of the calibration. Note that bias and slope adjustments are specifically not recommended to improve calibration or prediction statistics for IR multivariate models. 

If the task is multivariate calibration, for example, the proper choice of a pre-processing method will essentially aflFect the quality of the resultant model. For more details about the use of these techniques together with PCA and PLS, readers are advised to consider the fundamental monograph by Erikson et al [8]. 

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