Calibration classical least-squares

There are two paradigms of multivariate calibration. Classical least squares (CLS) models the instrumental response as a function of analyte concentration. 

Different calibration models, such as classical least squares and multivariate calibration approaches have been considered. 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

To produce a calibration using classical least-squares, we start with a training set consisting of a concentration matrix, C, and an absorbance matrix, A, for known calibration samples. We then solve for the matrix, K. Each column of K will each hold the spectrum of one of the pure components. Since the data in C and A contain noise, there will, in general, be no exact solution for equation [29]. So, we must find the best least-squares solution for equation [29]. In other words, we want to find K such that the sum of the squares of the errors is minimized. The errors are the difference between the measured spectra, A, and the spectra calculated by multiplying K and C ... 

However, our preoccupation is with the opposite application given a newly measured spectrum y , what is the most likely mixture composition and, how precise is the estimate Thus, eq. (36.2) is necessary for a proper estimation of the parameters B, but we have to invert the relation y =fix) = xB into, say, x = g y) for the purpose of making future predictions about x (concentration) given y (spectrum). We will treat this case of controlled calibration using classical least squares (CLS) estimation in Section 36.2.1. 

The CLS method hinges on accurately modelling the calibration spectra as a weighted sum of the spectral contributions of the individual analytes. For this to work the concentrations of all the constituents in the calibration set have to be known. The implication is that constituents not of direct interest should be modelled as well and their concentrations should be under control in the calibration experiment. Unexpected constituents, physical interferents, non-linearities of the spectral responses or interaction between the various components all invalidate the simple additive, linear model underlying controlled calibration and classical least squares estimation. 

MLR is based on classical least squares regression. Since known samples of things like wheat cannot be prepared, some changes, demanded by statistics, must be made. In a Beer s law plot, common in calibration of UV and other solution-based tests, the equation for a straight line... 

Infrared data in the 1575-400 cm region (1218 points/spec-trum) from LTAs from 50 coals (large data set) were used as input data to both PLS and PCR routines. This is the same spe- tral region used in the classical least-squares analysis of the small data set. Calibrations were developed for the eight ASTM ash fusion temperatures and the four major ash elements as oxides (determined by ICP-AES). The program uses PLSl models, in which only one variable at a time is modeled. Cross-validation was used to select the optimum number of factors in the model. In this technique, a subset of the data (in this case five spectra) is omitted from the calibration, but predictions are made for it. The sum-of-squares residuals are computed from those samples left out. A new subset is then omitted, the first set is included in the new calibration, and additional residual errors are tallied. This process is repeated until predictions have been made and the errors summed for all 50 samples (in this case, 10 calibrations are made). This entire set of... 

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. 

Calibration of Caustic/Salt systems using NIR Spearoscopy description of 227-228 inverse classical least squares (ICLS). 227-243... 

If a calibration function is used with coefficients obtained by fitting the response of an instrument to the model in known concentrations of calibration standards, then the uncertainty of this procedure must be taken into account. A classical least squares linear regression, the default regression... 

D. M. Haaland, W. B. Chambers, M. R. Keenan and D. K. Melgaard, Multi-window classical least-squares multivariate calibration methods for quantitative ICP-AES analyses, Appl. Spectrosc., 54(9), 2000, 1291— 1302. 

Calibration and mixture analysis addresses the methods for performing standard experiments with known samples and then using that information optimally to measure unknowns later. Classical least squares, iterative least squares, principal components analysis, and partial least squares have been compared for these tasks, and the trade-offs have been discussed (Haaland,... 

When one is provided with quantitative information for the target analyte, e.g., concentration, in a series of calibration samples, and when the respective instrumental responses have been measured, there are two central approaches to stating the calibration model. These methods are often referred to as classical least squares... 

Multivariate techniques are inverse calibration methods. In normal least-squares methods, often called classical least-squares methods, the system response is modeled as a function of analyte concentration. In inverse methods, the concentrations are treated as functions of the responses. The latter has some advantages in that concentrations can be accurately predicted even in the presence of chemical and physical sources of interference. In classical methods, all components in the system need to be considered in the mathematical model produced (regression equation). 

That is, concentration is expressed as a function of absorbances at a set of frequencies. Thus, in the cahbration step, only the concentrations of the component(s) of interest need be known. However, all components that may be present in the samples to be analysed must be included in the cahbration standards. In other words, as in classical least squares, it is essential that the set of calibration standards be representative of the samples to be analysed however, unlike the classical least squares case, it is not necessary to know the concentrations of interfering component(s) in the cahbration standards. 

There are several mathematical limitations inherent in the inverse least squares method. The number of frequencies employed cannot exceed the number of calibration standards in the training set. The selection of frequencies is further limited by the problem of collinearity that is, the solution of the matrix equation tends to become unstable as more frequencies that correspond to absorptions of a particular component x are included because the absorbances measured at these frequencies will change in a collinear manner with changes in the concentration of x. Thus, the possibilities for averaging out errors through the use of over-determination are greatly reduced by comparison with the classical least squares method, in which there are no limitations on the number of frequencies employed. 

To obtain adequate statistics, the calibration range should be established by at least 10 independently prepared solutions. If a linear calibration model is assumed and the parameters of the model (slope and intercept) are determined by classical least-squares regression, the assumptions of the regression must hold. Namely ... 

H. Mark, R. Rubinovitz, D. Heaps, P. Gemperline, D. J. Dahm, and K. D. Dahm, Gomparison of the Use of Volume Fractions with Other Measures of Goncentration for Quantitative Spectroscopic Calibration Using the Classical Least-Squares Method, Appl. Spectrosc., 64,1006 (2010). 

The p and k matrix methods are two classical least squares approaches to multicomponent calibration. There are techniques based on factor analysis, however, that are increasingly popular these include the... 

However, multicomponent quantitative analysis is the area we are concerned with here. Regression on principle components, by PCR or PLS, normally gives better results than the classical least squares method in equation (10.8), where collinearity in the data can cause problems in the matrix arithmetic. Furthermore, PLS or PCR enable a significant part of the noise to be filtered out of the data, by relegating it to minor components which play no further role in the analysis. Additionally, interactions between components can be modelled if the composition of the calibration samples has been well thought out these interactions will be included in the significant components. 

This method of quantitative analysis is known as K matrix, or classic least squares (CLS). It has the advantage of being able to use large regions of the spectrum, or even the entire spectrum, for calibration to gain an averaging effect for the predictive accuracy of the final model. One interesting side effect is that if the entire spectrum is used for calibration, the rows of the K matrix are actually spectra of the absorptivities for each of the constituents. These will actually look very similar to the pure constituent spectra. 

The classical least-squares (CLS) approach for relating absorption spectra to concentrations of components of the mixture is as follows. This CLS approach assumes that Beer s law holds so that the absorption at each frequency is directly proportional to the concentrations of the components. For a mixture of I chemical components, with n digitized absorbances, and m calibration standards, the following relationship holds ... 

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