Y-block data

PLS is more complex than PCR because we are simultaneously using degrees of fieedom in both the x-block and the y-block data. In the absence of a rigourous derivation of the proper number of degrees of freedom to use for PLS a simple approximation is the number of samples, n, minus the number of factors (latent variables), f, minus 1. 

The decision whether or not to variance scale the x-block data is independent from the decision about scaling the y-block data. We can decide to scale either, both, or neither. 

Variance (cont) of prediction, 167 Variance scaling, 100, 174 Vectors basis, 94 Weighting of data, 100 Whole spectrum method, 71 x-block data, 7 x-data, 7 XE, 94 y-block data, 7 y-data, 7... 

Table IVb. Sensory variables included as y-block data in the PLS regression analysis, maximum rating of the 12 samples analyzed, and the percentage explained variance from the first two components extracted from the PLS model. ...

As we will soon see, the nature of the work makes it extremely convenient to organize our data into matrices. (If you are not familiar with data matrices, please see the explanation of matrices in Appendix A before continuing.) In particular, it is useful to organize the dependent and independent variables into separate matrices. In the case of spectroscopy, if we measure the absorbance spectra of a number of samples of known composition, we assemble all of these spectra into one matrix which we will call the absorbance matrix. We also assemble all of the concentration values for the sample s components into a separate matrix called the concentration matrix. For those who are keeping score, the absorbance matrix contains the independent variables (also known as the x-data or the x-block), and the concentration matrix contains the dependent variables (also called the y-data or the y-block). 

In addition to the set of new coordinate axes (basis space) for the spectral data (the x-block), we also find a set of new coordinate axes (basis space) for the concentration data (the y-block). 

It is often helpful to examine the regression errors for each data point in a calibration or validation set with respect to the leverage of each data point or its distance from the origin or from the centroid of the data set. In this context, errors can be considered as the difference between expected and predicted (concentration, or y-block) values for the regression, or, for PCA, PCR, or PLS, errors can instead be considered in terms of the magnitude of the spectral... 

PLS should have, in principle, rejected a portion of the non-linear variance resulting in a better, although not completely exact, fit to the data with just 1 factor. The PLS does tend to reject (exclude) those portions of the x-data which do not correlate linearly to the y-block. (Richard Kramer)... 

In principle, in the absence of noise, the PLS factor should completely reject the nonlinear data by rotating the first factor into orthogonality with the dimensions of the x-data space which are spawned by the nonlinearity. The PLS algorithm is supposed to find the (first) factor which maximizes the linear relationship between the x-block scores and the y-block scores. So clearly, in the absence of noise, a good implementation of PLS should completely reject all of the nonlinearity and return a factor which is exactly linearly related to the y-block variances. (Richard Kramer)... 

B program, PLS-2, uses the partial least squares (PLS) method. This method has been proposed by H. Wold (37) and was discussed by S. Wold (25). In such a problem there are two blocks of data, T and X. It is assumed that T is related to X by latent variables u and t is derived from the X block and u is derived from the Y block. 

To test the potential of PLS to predict odour quality, it was used in a QSAR study of volatile phenols. A group of trained sensory panelists used descriptive analysis (28) to provide odour profiles for 17 phenols. The vocabulary consisted of 44 descriptive terms, and a scale fiom 0 (absent) to S (very strong) was used. The panel average sensory scores for the term sweet were extracted and used as the Y-block of data, to be predicted from physico-chemical data. 

Step 6 In order to extract the second factor (or latent variable), the information linked to the first factor has to be subtracted from the original data and a sort of residual matrices are obtained for the X- and Y-blocks as... 

Most look at how well the concentration is predicted, or the c (or according to some authors y) block of data. 

PLS is a modelling and computational method for establishing quantitative relations between blocks of variables. Such blocks may, for instance, comprise a block of descriptor variables of a set of test systems (X block) and a block of measured responses obtained with these sytems (Y block). A quantitative relation between these blocks will make it possible to enter data, x, for a new systems and make predictions of the expected responses, y, for these systems. 

Besides the descriptive values, it is also interesting to know the correlations between the two groups of variables (rXi,Yj)- The multivariate statistical methods for this data matrix are Canonical Correlation Analysis (CCA) to investigate the relationship between both sets of variables, and Multivariate Regression with a view to predicting the values of the response variables in the Y-block in function of the variables in the X-block, using a mathematical model. 

Quantitative relations between the reaction space and the observed responses can be described as follows The data are divided into two blocks X and Y The X block contains the descriptors of the reaction system, and the Y block contains the responses observed with these systems. We are thus looking for a relation between the data structure of these blocks. 

PLS is a modelling and computational method, by which quantitative relations can be established between blocks of variables, e.g. a block of descriptor data for a series of reaction systems (X block) and a block of response data measured on these systems (Y block). By the quantitative relation between the blocks, it is possible to enter data for a new system to the X block and make predictions of the expected responses. For example, if a reaction has been run in a series of solvents, we can use a PLS model to relate the properties of the solvents to the observed optimum conditions in these solvents. By subsequently entering the property descriptors of a new solvent to the PLS model it is possible to predict the optimum conditions of the reaction in the new solvent. For this to be efficient, it is necessary that the solvents used to determine the PLS model have a sufficient spread in their properties. To ensure this, a design in the principal properties is most useful. 

The experimental design matrix and the yields of the enamine and the by-product were used to define the X block and the Y block, respectively. These data are recapitulated in Table 17.5. 

However, for many practical applications, data are at least mean centred (see Chapter 3), that is, an offset term is subtracted from both the X- and y-blocks from a regression standpoint, centring leads to model without intercept, so that Equation (17) becomes ... 

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