Inverse least-squares regression

Table 22.7 Advantages and disadvantages of inverse least squares regression.

Thus, the name for this type of model is principal components regression it combines principal components analysis and inverse least squares regression to solve the calibration equation for the model. All that remains is to come up with a single unified equation that represents the PCR model. Therefore, rearranging the previous matrix model equation to represent the scores as a function of the spectral absorbances and the eigenvectors produces... 

Inverse least-squares (ILS), sometimes known as P-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the inverse expression of the Beer-Lam be rt Law of spectroscopy ... 

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

The model of eq. (36.3) has the considerable advantage that X, the quantity of interest, now is treated as depending on Y. Given the model, it can be estimated directly from Y, which is precisely what is required in future application. For this reason one has also employed model (36.3) to the controlled calibration situation. This case of inverse calibration via Inverse Least Squares (ILS) estimation will be treated in Section 36.2.3 and has been treated in Section 8.2.6 for the case of simple straight line regression. 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

X is the generalised inverse performed by some regression method (e.g. partial least squares regression). Inserting for X... 

The relationship between the estimated and measured y values can be described by a fundamental matrix, the hat matrix, H. As explained in Ordinary Least Squares Regression Section, the regression parameters are estimated by the general inverse as... 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

Usually, only two or three colorants are added. Due to this reason and taking into account the economic feasibility and rapidity, double division-ratio spectra derivative, inverse least-squares, and principal component regression methods are reliable for the simultaneous determination of the colorants in the drinks without a priority procedure such as separation, extraction, and preconcentration. 

Studies on binary mixture samples frequently deal with classical least-squares, inverse least-squares, principal component regression and partial least-squares methods. These methods have been used for resolving mixtures of hydrochlorothiazide and spironolactone in tablets cyproterone acetate and estradiol valerate amiloride and hydrochlorothiazide ... 

Spectrophotometric monitoring with the aid of chemometrics has also been applied to more complex mixtures. To solve the mixtures of corticosteroid de-xamethasone sodium phosphate and vitamins Bg and Bi2, the method involves multivariate calibration with the aid of partial least-squares regression. The model is evaluated by cross-validation on a number of synthetic mixtures. The compensation method and orthogonal function and difference spectrophotometry are applied to the direct determination of omeprazole, lansoprazole, and pantoprazole in grastroresistant formulations. Inverse least squares and PCA techniques are proposed for the spectrophotometric analyses of metamizol, acetaminophen, and caffeine, without prior separation. Ternary and quaternary mixtures have also been solved using iterative algorithms. 

We discuss here the use to which calibrations obtained by least-squares regression are put, i.e., their use in determining an unknown value (x ) of concentration x from a measured instrumental response Y moreover, importantly we can also obtain a value for the corresponding uncertainty (variance) in X. Such determination of x involves inversion of the calibration fitting function. Equation [8.19a] in our case inversion of this equation gives ... 

Notice that even if the concentrations of all the other constituents in the mixture are not known, the matrix of coefficients (P) can still be calculated correctly. This model, known as inverse least squares (ILS), multiple linear regression (MLR), or P matrix, seems to be the best approach for almost all quantitative analyses because no knowledge of the sample composition is needed beyond the concentrations of the constituents of interest. 

Once the PCA has been calculated from the spectral data, the concentration data can be regressed against the scores matrix using the inverse least squares method to generate the matrix of constituent calibration coefficients. A usual practice in performing PCR regression is to add an extra unit vector column to the scores matrix to allow for inclusion of an offset coefficient in the regression. 

Inverse least squares (ILS) is a least-squares method that assumes the inverse calibration model given in eqn (4.4). For this reason it is often also termed multiple linear regression (MLR). In the literature this calibration approach is... 

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