Latent variables definition

Depending on the aim of data analysis different mathematical criteria are applied for the definition of latent variables ... 

The direction in a variable space that best preserves the relative distances between the objects is a latent variable which has maximum variance of the scores (these are the projected data values on the latent variable). This direction is called by definition the first principal component (PCI). It is defined by a loading vector... 

Another well-established method for building predictive models is partial least squares (PLS) regression. PLS is a modem relative of MLR, having been established in the 1960 s by Wold. The method reaches beyond linear regression by replacing the descriptors with a matrix of latent variables distilled from both the structural features of the training compounds and their experimental results. In PLS, the use of the term latent variables differs from its formal definition in other regression methods. ... 

The simplest definition of model complexity is based on the number of terms in the model or, in other words, the model complexity is made up by the number of model variables from Ordinary Least Squares regression cpx = p), the number M of significant principal components from Principal Component Regression (cpx = M), and the number of significant latent variables from Partial Least Squares regression (cpx = M)... 

PCR method is a two-step process, in which the projection stage is separated and independent from the regression one. As discussed in Section 3.3, this can lead to the drawback that the components that are extracted in the decomposition step, based only on the information about the X-matrix, can be poorly predictive for the Y-block. Starting from these considerations, another method was proposed, PLS regression [2,18,19], in which information in Y is actively used also for the definition of the latent variable space. Indeed, PLS looks for components which compromise between explaining the variation in the X-block and predicting the responses in Y. This corresponds to a bilinear model, which can be summarized mathematically as ... 

Model parameters can also be used to compute indices, which reflect the relative importance of the predictors in the definition of the model itself. In particular, two indices, the variable importance in projection (VIP) [43] and the selectivity ratio (SR) [44] are often used in the ambit of latent variable-based calibration. VIP is a measure of how much the individual variables contribute to the definition of both the X- and the Y-spaces in PLS-modelling. 

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