Inverse least-squares method

Brown, C.W., "Classical and Inverse Least-Squares Methods in Quantitative Spectral Analysis", Spectrosc. 1986 (1) 23-37. 

Haaland, D.M. "Classical versus Inverse Least-Squares Methods in Quantitative Spectral Analyses", Spectrosc. 1987 (2) 56-57. 

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. 

PLS is a powerful technique that shares the advantages of both the CLS and ILS methods hut does not suffer from the limitations of either these methods. A PLS calibration can, in principle, he based on the whole spectrum, although in practice the analysis is restricted to regions of the spectrum that exhibit variations with changes in the concentrations of the components of interest. As such, the use of PLS can provide significant improvements in precision relative to methods that use only a limited number of frequencies [9]. In addition, like the inverse least squares method, PLS treats concentration rather than spectral intensity as the independent variable. Thus, PLS is able to compensate for unidentified sources of spectral interference, although all such interferences that may be present in the samples to be analysed must also be present in the calibration standards. The utility of PLS will be demonstrated by several examples of food analysis applications presented in Section 4.7. 

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. 

The inverse least-squares method (ILS) assumes that concentration is a function of absorbance. For m calibration standards and n digitized absorbances,... 

This means a quantitative analysis can be performed even if the concentration of only one component in the calibration mixtures is known. The disadvantage of the inverse least-squares method is that the analysis is restricted to a small number of frequencies because the matrix that must be inverted has dimensions equal to the number of frequencies, and this number cannot exceed the number of calibration mixtures used in the analysis. 

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. 

ILS is a least-squares method that assumes the inverse calibration model given in eqn (3.4). For this reason it is often also termed multiple linear regression (MLR). In this model, the concentration of the analyte of interest, k, in sample i is regressed as a linear combination of the instrumental measurements at J selected sensors [5,16-19] ... 

Knowing Ax, (r) and mx, (r), we can find Act (r) from equation (10.92), as in the QL inversion scheme. Note that equation (10.92) should hold for any frequency, because the electrical reflectivity and the material property parameters are functions of frequency as well Ax, = Ax, (r,uj), mx, = mx, (r,cj). In reality, of course, it holds only approximately. Therefore the conductivity Act (r) can be found by using the least squares method to solve equation (10.92) with respect to the logarithm of the total conductivity a (r), similar to equation (10.90) ... 

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). 

Least-squares methods have been used to determine molecular structures successfiilly first by Nosberger et al. [31], Schwendeman [28], and Typke [32]. Nosberger et al. fitted structural parameters (internal coordinates) to isotopic differences of moments of inertia. To solve the normal equations, they used the singular value decomposition of real matrices to calculate the pseudo-inverse of such matrices with the option to omit nearzero singular values in illdetermined systems. Schwendeman [28] fitted internal coordinates to moments of inertia or isotopic differences of these... 

X-ray diffraction (XRD) and magnetic susceptibility measurements were carried out on polycrystalline materials that had been calcined in oxygen. XRD powder patterns were recorded with Cu Ka radiation using a Ni filter on a Rigaku diffractometer. For reference all patterns were recorded with an internal NBS Si standard. After correcting the peak position based on the observed silicon line positions, the lattice constants were then refined by a least squares method weighted proportional to the square root of the height and inversely with the square of the width of the peak. 

The inverse least squares (ILS) method is sometimes referred to as the P-matrix method. The calibration model is transformed so that component concentrations are defined as a function of the recorded response values,... 

The amplitudes of the histogram of the distribution function are calculated by a non-negative least square method. This procedure is known as the exponential sampling method and is applicable to both monomodal and bimodal distributions. However, in view of the limitations of the Laplace inversion, it is difficult to resolve bimodal distributions with a ratio between the two particle species below 2. 

To reduce the requirements of long experimental data records and improve the kernel estimation accuracy, least-squares methods also can be used to solve the classical linear inverse problem described earlier in Equation 13.6, where the parameter vector 9 includes all discrete kernel values of the finite Volterra model of Equation 13.17, which is Knear in these unknown parameters (i.e., kernel values). Least-squares methods also can be used in connection with orthogonal expansions of the kernels to reduce the number of unknown parameters, as outlined below. Note that solution of this inverse problem via OLS requires inversion of a large square matrix with dimensions [(M -I- f -I- 1) /((M -F 1) / )], where M is... 

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. 

Franke, J. E., Inverse Least Squares and Oassical Least Squares Methods for (Juantitative fibrational Spectroscopy , in Handbook of Vibrational Spectroscopy, Vol. 3, Chalmers, J. M. and Griffiths, P. R. (Eds), Wiley, Chichester, UK, 2002, pp. 2276-2292. 

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. 

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