Predictor variables least-squares method

Ridge regression is also used extensively to remedy multicollinearity between the X, predictor variables. It does this by modifying the least-squares method of computing the coefficients with the addition of a biasing component. 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

Eor multivariate calibration in analytical chemistry, the partial least squares (PLS) method [19], is very efficient. Here, the relations between a set of predictors and a set (not just one) of response variables are modeled. In multicomponent calibration the known concentrations of / components in n calibration samples are collected to constitute the response matrix Y (n rows, / columns). Digitization of the spectra of calibration samples using p wavelengths yields the predictor matrix X (n rows, p columns). The relations between X and Y are modeled by latent variables for both data sets. These latent variables (PLS components) are constructed to exhaust maximal variance (information) within both data sets on the one hand and to be maximally correlated for the purpose of good prediction on the other hand. From the computational viewpoint, solutions are obtained by a simple iterative procedure. Having established the model for calibration samples. comp>o-nent concentrations for future mixtures can be predicted from their spectra. A survey of multi-component regression is contained in [20],... 

The method of multiple linear regression analysis derives a least-squares fit of the predictor (independent) variables, molecular properties in this case, to biological activity. Usually the investigator examines the effect of including or not including particular variables. Although this approach is often used in 3D-QSAR, we will not discuss it further because it is already well documented. ... 

Partial least squares projection of latent structures (PLS) is a method for relating the variahons in one or several response variables (Y variables or dependent variables) to the variations of several predictors (X variables), with explanatory or predictive purposes [12-14]. PLS performs particularly well when the various X variables express common information, i.e., when there is a large amount of correlation or even collinearity among them. PLS is a bilinear method where information in the original X data is projected onto a small number of underlying ( latent ) variables to ensure that the first components are those that are most relevant for predicting the Y variables. Interpretahon of the relationship between X data and Y data is then simplified, as this relahonship is concentrated on the smallest possible number of components [15]. 

Partial least squares projection of latent structures (PLS) is a method for relating the variations in one or several response variables (Y variables or dependent variables) to the variations of several predictors (X variables), with explanatory or predictive purposes. 

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