Principal Component Regression calibration

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

Table V. Principal Component Regression Calibrations For Selected Ash Properties...

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

The suffix in T (nxA) and P (< xA) indicates that only the first A columns of T and P are used, A being much smaller than n and q. In principal component regression we use the PC scores as regressors for the concentrations. Thus, we apply inverse calibration of the property of interest on the selected set of factor scores ... 

E. Vigneau, D. Bertrand and E.M. Qannari, Application of latent root regression for calibration in near-infrared spectroscopy. Comparison with principal component regression and partial least squares. Chemometr. Intell. Lab. Syst., 35 (1996) 231-238. 

This chapter ends with a short description of the important methods, Principal Component Regression (PCR) and Partial Least-Squares (PLS). Attention is drawn to the similarity of the two methods. Both methods aim at predicting properties of samples based on spectroscopic information. The required information is extracted from a calibration set of samples with known spectrum and property. 

P.J. Gemperhne, J.R. Long and V.G. Gregoriov, Nonlinear multivariate calibration using principal components regression and artificial neural networks. Anal Chem., 63, 2313-2323 (1991). 

Experience in this laboratory has shown that even with careful attention to detail, determination of coal mineralogy by classical least-squares analysis of FTIR data may have several limitations. Factor analysis and related techniques have the potential to remove or lessen some of these limitations. Calibration models based on partial least-squares or principal component regression may allow prediction of useful properties or empirical behavior directly from FTIR spectra of low-temperature ashes. Wider application of these techniques to coal mineralogical studies is recommended. 

Principal Component Regression (PCR) was used by Tuchbreiter and MueUiaupt to determine the composition of a number of random ethane/propene, ethane/1-hexene, and ethane/l-octene copolymers [120]. After polymerization, the polymers were characterized by both Attenuated Total Reflection Fourier Transform Infrared Spectroscopy (ATR-FT-IR) and C NMR and multivariate calibration models using PCR were subsequently developed to estimate the co-monomer content. 

Partial least squares (PLS) and principal component regression (PCR) are the most widely used multivariate calibration methods in chemometrics. Both of these methods make use of the inverse calibration approach, where it i.s... 

R. Marbach and H. M. Heise, Calibration modeling by partial least-squares and principal component regression and its optimisation using an improved leverage correction for prediction testing, Chemom. Intell. Lab. Syst., 9(1), 1990, 45-63. 

Tablet hardness is a property that, when measured, destroys the sample. The destructive nature of the test, coupled with the variability of the test itself does not contribute to an incentive to test a large number of samples. Morisseau and Rhodes99 correlated the diffuse reflectance NIR spectra of tablets pressed at different pressures and subsequently tested the tablet hardness with an Erweka Hardness Tester. The tablet hardness, as predicted by the NIR method, was at least as precise as the laboratory test method. Kirsch and Drennen100 evaluated NIR as a method to determine potency and tablet hardness of Cimetidine tablets over a range of 1-20% potency and 107-kPa compaction pressure. Hardness at different potency levels was used to build calibration models using PCA/ principal component regression and a new spectral best-fit algorithm. Both methods provided acceptable predictions of tablet hardness.

The calibration methods most frequently used to relate the property to be measured to the analytical signals acquired in NIR spectroscopy are MLR,59 60 principal component regression (PCR)61 and partial least-squares regression (PLSR).61 Most of the earliest quantitative applications of NIR spectroscopy were based on MLR because spectra were then recorded on filter instruments, which afforded measurements at a relatively small number of discrete wavelengths only. However, applications involving PCR and PLSR... 

When all of the individual component spectra are not known, implicit calibration methods are often adopted. Among these, factor analysis methods such as principal component regression (PCR)24 and partial least squares (PLS)25 are frequently used because they can function under conditions in which the number of spectra used for calibration is less than the number of wavelengths sampled. For example, a calibration set may include 30 spectra with each spectrum having 500 data points (wavelengths). 

Principal component regression and partial least squares are two widely used methods in the factor analysis category. PCR decomposes the matrix of calibration spectra into orthogonal principal components that best capture the variance in the data. These new variables eliminate redundant information and, by using a subset of these principal components, filter noise from the original data. With this compacted and simplified form of the data, equation (12.7) may be inverted to arrive at b. 

Principal component regression is accomplished in two steps, a calibration step and an unknown prediction step. In the calibration step, concentrations of the constituent(s) to be quantitated in each calibration standard sample are assembled into a matrix, y, and mean-centered. Spectra of standards are measured, assembled into a matrix X, mean-centered, and then an SVD is performed. Calibration spectra are projected onto the d principal components (basis vectors) retained and are used to determine a vector of regression coefficients that can be then used to estimate the concentration of the calibrated constituent(s). 

In more recent development, chemometric or multivariate calibration techniques have been applied into spectrophotometric methods. As reported by Palabiyik and Onur [24], principal component regression and partial least square were used to determine ezetimibe in combination with simvastatin. This method offers advanfages such as no chemical prefreafmenf prior to analysis as well as no need to observe graphical spectra and calculations as with the derivative method. In addition, the instrumentation used is also simpler. 

Fig. 9 Steps for automated determination of metastable zone using ATR-FTIR and FBRM. While automatically collecting the IR spectra for calibration, the metastable limit is determined using FBRM. Then the model for relating the IR spectra to solution concentration is constructed using multivariate analysis such as principal component regression (PCR) or partial least squares (PLS). Using this model, the solubility curve can be obtained from the IR spectra of saturated slurry.

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