Classical least square

From now on, we adopt a notation that reflects the chemical nature of the data, rather than the statistical nature. Let us assume one attempts to analyze a solution containing p components using UV-VIS transmission spectroscopy. There are n calibration samples ( standards ), hence n spectra. The spectra are recorded at q wavelengths ( sensors ), digitized and collected in an nx.q matrix S. The information on the known concentrations of the chemical constituents in the calibration set is stored in an nxp matrix C. Each column of C contains the concentrations of one of the p analytes, each row the concentrations of the analytes for a particular calibration standard. 

Concentrations and digitized spectra (xlOO) of four calibration samples (compare Fig. 36.1a) 

Note that eq. (36.5) is a collection of many univariate multiple regression models for each wavelength j the multiple regression of the corresponding spectral channel , i.e. Sj, on the concentration matrix C yields a vector of regression coefficients, ky (the yth column of K). For K to be estimable C C must be invertible, i.e. the number of calibration standards should at least be as large as the number of analytes. It is clearly not possible to obtain, directly or indirectly, say 3 pure spectra from recording the spectra of just 1 or 2 standards of known composition. In practice, the condition n p, or more precisely rank(C)=p, is hardly a restriction. 

As an example, we give in Table 36.1 data on calibration spectra at 10 wavelengths of 4 calibration standards for 3 analytes. The corresponding spectra are shown in Fig. 36.1a at a 10-fold higher resolution. The pure spectra calculated according to eq. (36.5), using all 100 wavelengths, are displayed in Fig. 36.1b. 

In many cases, one may measure spectra of solutions of the pure components directly, and the above estimation procedure is not needed. For the further development of the theory of multicomponent analysis we will therefore abandon the hat-notation in K. Given the pure spectra, i.e. given K (pxq), one may try and estimate the vector of concentrations (pxl) of a new sample from its measured 

This quantification method is based on the linear additive model applied to many instrumental responses. The model assumes that, for a sample i, the signal (measured response) at sensor j results from the sum of the responses of all the analytes (A) that contribute to the signal in this sensor (the term sensor will be used to refer globally to wavelength, time of the atomic peak, m/z ratio, etc.)  

Note that c, true counts how many times each column of S can be found in r. Hence, if the columns in S are the spectra at unit concentration and unit pathlength, c, true is the concentration of the analytes in the sample, expressed in the same concentration units as those used in the columns of S. 

Equation (3.17) is the fundamental equation of the CLS model that allows calibration and prediction. The calibration step consists of calculating S, which is the matrix of coefficients that will allow the quantification of future samples. S is found by entering the spectra and the known concentration of a set of calibration samples in eqn (3.17). These calibration samples, which can be either pure standards or mixtures of the analytes, must contain in total all the analytes that will be found in future samples to be predicted. Then, eqn (3.17) for I calibration samples becomes 

In addition, if each calibration sample contains only one analyte, then R contains already the spectra of the pure components and C K x K) is a diagonal matrix (i.e. non-zero values only in the main diagonal). Hence eqn (3.20) simply calculates S by dividing the spectrum of each pure calibration sample by its analyte concentration. To obtain a better estimate of S, the number of calibration samples is usually larger than the number of components, so eqn (3.19) is used. We still have two further requirements in order to use the previous equations. First, the relative amounts of constituents in at least K calibration samples must change from one sample to another. This means that, unlike the common practice in laboratories when the calibration standards are prepared, the dilutions of one concentrated calibration sample carmot be used alone. Second, in order to obtain the necessary number of linearly independent equations, the number of wavelengths must be equal to or larger than the number of constituents in the mixtures (7 A). Usually the entire spectrum is used. 

Note that eqn (3.17) has the same form as eqn (3.6). Hence, by introducing in r the vector of instrumental responses (e.g. a spectrum) of an unknown sample and in S the matrix of sensitivities that we obtained in the calibration step, the OLS solution [see the similarity with eqn (3.7)] is 

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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 ... 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

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. 

CLS Classical least-squares G (1) Free enthalpy (Gibbs free energy) ... 

Kinetic analysis usually employs concentration as the independent variable in equations that express the relationships between the parameter being measured and initial concentrations of the components. Such is the case with simultaneous determinations based on the use of the classical least-squares method but not for nonlinear multicomponent analyses. However, the problem is simplified if the measured parameter is used as the independent variable also, this method resolves for the concentration of the components of interest being measured as a function of a measurable quantity. This model, which can be used to fit data that are far from linear, has been used for the resolution of mixtures of protocatechuic... 

Clarke-Othmer process, for acetic acid-water for ethanol separation, 5 834 Claros Diagnostics, 26 976 Class 4A inert ingredients, 14 126 Classes A-C radioactive waste, 25 853 disposal of, 25 857 Classical least squares, 6 39-41 Classical thermodynamics, 24 641-642 Classification bauxite, 2 353... 

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... 

Instead of converting the step or pulse responses of a system into frequency response curves, it is fairly easy to use classical least-squares methods to solve for the best values of parameters of a model that fit the time-domain data. 

Figure 12.8 displays an organization chart of various quantitative methods, in an effort to better understand their similarities and differences. Note that the first discriminator between these methods is the direct versus inverse property. Inverse methods, such as MLR and partial least squares (PLS), have had a great deal of success in PAT over the past few decades. However, direct methods, such as classical least squares (CLS) and extensions thereof, have seen a recent resurgence [46-51]. The criterion used to distinguish between a direct and an inverse method is the general form of the model, as shown below ... 

D.M. Haaland and D.K. Melgaard, New prediction-augmented classical least squares (PACLS) methods application to unmodeled interferents, Appl. Spectrosc. 54, 1303 (2000). 

D.K. Melgaard and D.M. Haaland, Comparisons of prediction abilities of augmented classical least squares and partial least squares with realistic simulated data effects of uncorrelated and correlated errors with nonlinearities, Appl. Spectrosc., 58, 1065-73 (2004). 

Haaland and coworkers (5) discussed other problems with classical least-squares (CLS) and its performance relative to partial least-squares (PLS) and factor analysis (in the form of principal component regression). One of the disadvantages of CLS is that interferences from overlapping spectra are not handled well, and all the components in a sample must be included for a good analysis. For a material such as coal LTA, this is a significant limitation. 

The next section of this paper describes the use of classical least-squares analysis of FTIR data to determine coal mineralogy. This is followed by promising preliminary results obtained using factor analysis techniques. 

Results of classical least-squares analysis of FTIR spectra of ten coals using forty-two reference minerals were evaluated with regard to reproducibility and accuracy as described below. 

Table III. FTIR Mineralogical Results Using Classical Least-Squares, wt % of LTA...

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. 

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