Principal Component Regression and Partial Least Squares

2 Principal Component Regression and Partial Least Squares 

The feature common to both of these approaches is that each spectrum is reduced to a sum of pseudospectra, or loading vectors . Each spectrum is newly represented by a unique set of scores - the set of coefficients required to reconstruct the original spectrum from the set of loading vectors. Typically, each of the spectra can be reconstructed to within the noise limits by a combination of typically 5-15 loading vectors, as compared to the hundreds or thousands of intensity values in the original spectra. The scores then provide the basis for quantitation. 

The essential relationship in both the PCR and PLS models takes the form of Equation (5)  

With m spectra in the cahbration set, each having n absorbance values, A is them x n matrix of the calibration spectra. The spectra are reconstructed as a product of B (hx n), the new basis set of loading vectors, and T (mxh), the scores. To reiterate, the key to the process is that each spectrum is reduced from a vector of length n (a row in A) to a new vector of length h (the corresponding row in T), where h is typically between 5 and 15. 

The column matrix of concentrations c is also related to the loading vectors T, according to Equation (6)  

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To gain insight into chemometric methods such as correlation analysis, Multiple Linear Regression Analysis, Principal Component Analysis, Principal Component Regression, and Partial Least Squares regression/Projection to Latent Structures... 

E. Vigneau, D. Bertrand and E.M. Qannari, Application of latent root regression for calibration in near-infrared spectroscopy. Comparison with principal component regression and partial least squares. Chemometr. Intell. Lab. Syst., 35 (1996) 231-238. 

A difficulty with Hansch analysis is to decide which parameters and functions of parameters to include in the regression equation. This problem of selection of predictor variables has been discussed in Section 10.3.3. Another problem is due to the high correlations between groups of physicochemical parameters. This is the multicollinearity problem which leads to large variances in the coefficients of the regression equations and, hence, to unreliable predictions (see Section 10.5). It can be remedied by means of multivariate techniques such as principal components regression and partial least squares regression, applications of which are discussed below. 

Faber K, Kowalski BR (1997b) Propagation of measurement errors for the validation of predictions obtained by principal component regression and partial least squares. J Chemom 11 181... 

Differences between PIS and PCR Principal component regression and partial least squares use different approaches for choosing the linear combinations of variables for the columns of U. Specifically, PCR only uses the R matrix to determine the linear combinations of variables. The concentrations are used when the regression coefficients are estimated (see Equation 5.32), but not to estimate A potential disadvantage with this approach is that variation in R that is not correlated with the concentrations of interest is used to construct U. Sometiraes the variance that is related to the concentrations is a verv... 

Factor The result of a transformation of a data matrix where the goal is to reduce the dimensionality of the data set. Estimating factors is necessary to construct principal component regression and partial least-squares models, as discussed in Section 5.3.2. (See also Principal Component.)... 

Principal component regression and partial least squares are two widely used methods in the factor analysis category. PCR decomposes the matrix of calibration spectra into orthogonal principal components that best capture the variance in the data. These new variables eliminate redundant information and, by using a subset of these principal components, filter noise from the original data. With this compacted and simplified form of the data, equation (12.7) may be inverted to arrive at b. 

Although the emphasis in this chapter is on multiple hnear regression techniques, it is important to recognise that the analysis of design experiments is not restricted to such approaches, and it is legitimate to employ multivariate methods such as principal components regression and partial least squares as described in detail in Chapter 5. 

In more recent development, chemometric or multivariate calibration techniques have been applied into spectrophotometric methods. As reported by Palabiyik and Onur [24], principal component regression and partial least square were used to determine ezetimibe in combination with simvastatin. This method offers advanfages such as no chemical prefreafmenf prior to analysis as well as no need to observe graphical spectra and calculations as with the derivative method. In addition, the instrumentation used is also simpler. 

In the first case, attention is paid to excluding variables carrying low or redundant information, in the second, to excluding variables which are not functionally related to the studied response. In the latter, besides the exclusion of specific variables, one can condense the information from all the original variables into a few significant latent variables (linear combinations) by methods such as Principal Component Regression and Partial Least Squares regression. 

Regression techniques that can deal with colinear data include stepwise regression, ridge regression, principal components regression, and partial least squares (PLS) regression. The last two approaches are discussed in Sections 4.2 and 4.3. 

In the above examples there is a natural way to order the complete data set in two blocks, where both blocks have one mode in common. In Chapter 3 the methods of multiple linear regression, principal component regression, and partial least squares regression will be discussed on an introductory level. 

G. Gauglitz, G. Kraus, Principal component regression and partial least-squares, GIT Fachz. Lab. 36 (1992) 232. 

Hasegawa, T., Principal Component Regression and Partial Least Squares Modeling , in Handbook of Vibrational Spectroscopy, Vol. 3, Chalmers, J. M. and Griffiths, P. R. (Eds), Wiley, Chichester, UK, 2002, pp. 2293-2312. 

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 ... 

BP processing of stripping analysis responses allowed the quantitative determination of heavy metals in the presence of interferences, and selectivity coefficients of berberine-selective electrodes were predicted satisfactorily. o BP networks were used to analyze piezoelectric crystal data to simultaneously determine sulfur dioxide concentration and relative humidity. 2 A BP network was used to model properties of materials. 2 x vo studies described the modeling and recognition of flow injection patterns. " Compared to principal components regression and partial least squares (PLS), BP provided a better method for multicomponent kinetic determinations. BP... 

Principal Components Regression and Partial Least Squares Testing. 147... 

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