Classical least squares model

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

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

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. 

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... 

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 ... 

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. 

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... 

Examples are ordinary least squares (OLS) and classical least squares (CLS). Explicit methods provide transparent models with easily interpretable results. However, highly controlled experimental conditions, high-quality spectra, and accurate concentration measurements of all components in the sample matrix may be difficult to obtain, particularly in biomedical applications. 

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). 

Whatever the form of the input, the basic idea is to use a difference equation model for the process in which the current output y is related to previous values of the output yn- >yn-2> ) 3nd present and past values of the input (m , Wn-i,.. . ) hi the simple model structures, the relationship is linear, so classical least-squares can be used to solve for the best values of the unknown coefficients. These difference equation models occur naturally in sampled-data systems (see Chapter 15) and can be easily converted to Laplace-domain transfer function models. 

The fourth-derivative spectra of molybdenum complexes of tetramethyldithiocarbamate (tiram) fungicide were used for its quantification in commercial samples and in wheat grains [41], Atrazine and cyanazine were assayed in food samples by first- derivative spectrophotometry [42]. In order to improve results of assay, the first-derivative spectra of the binary mixture were subjected to chemometiic treatment (classical least squares, CLS principal component regression, PCR and p>artial least squares, PLS). A combination of first-derivative with PCR and PLS models were applied for determination of both herbicides in biological samples [42]. A first-derivative spectrophotometry was used as a reference method for simultaneous determination BriUant Blue, Simset Yellow and Tartrazine in food [43]. 

Parameter estimation was performed using a simplex-based method (Htmmelblau et al., 2002) focusing on niinirnizing the classical least square objective function based on the difference between experimental and predicted CO conversion values. The experimental industrial data set was divided into two group , the first for pjarameter estimation and the second for model validation. 

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. 

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 ... 

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. 

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

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