Partial least squares coefficient matrix

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

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... 

Partial least squares regression (PLSR) was used again. Leave one out cross validation was carried between the new matrix XRa and concentration matrix Y. After each step of leave one out cross validation, a regression coefficient b was obtained. 

Partial least squares (PLS) analysis allows the simultaneous investigation of the relationships between a multitude of activity data (F matrix) and a set of chemical descriptors (X matrix) through latent variables (Wold et aL, 1984 Geladi and Kowalski, 1986 Hellberg, 1986 Geladi and Tosato, 1990). The latent variables correspond to the component scores in PCA and the respective coefficients to the PCA loading vectors. The PLS model can also be applied when the number of (collinear) descriptors exceeds the number of compounds in the data set. The main difference between PCA and PLS concerns the criteria for extracting the principal components and the latent variables, respectively PCA is based on the maximum variance criterion, whereas PLS uses covariance with another set of variables (X matrix). 

B being a matrix of regression coefficients, the PLS approach (see Chapter 4) can be used to calculate the model even in the cases where the matrix X is ill-conditioned (presence of collinearity or low samples/variables ratio). The corresponding classification method is then called partial least squares-discriminant analysis (PLS-DA). 

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