PLS Regression Models

PC A to reduce the effect of noise and to optimize the predictive power of the PCR model. This is generally done by using cross-validation. Then, 

To include information about process dynamics, lagged variables can be included in X. The (auto)correlograms of all x variables should be developed to determine first how many lagged values are relevant for each variable. Then the data matrix should be augmented accordingly and used to determine the principal components that will be used in the regression step. 

Nonlinear extensions of PCA have been proposed by using autoasso-ciative neural networks discussed in Section 3.6.1 (an illustrative example is provided in Section 7.7.1) or by using principal curves and surfaces [106, 161]. 

Partial least squares (PLS) regression, develops a biased regression model between X and Y. In the context of chemical process operations, usually X denotes the process variables and Y the quality variables. PLS selects latent variables so that variation in X which is most predictive of the product quality data Y is extracted. PLS works on the sample covariance matrix (X Y)(Y X) [86, 87, 111, 172, 188, 334, 338[. Measurements of m process variables taken at n different times are arranged into a (n x m) process data matrix X. The q quality variables are given by the corresponding 

For the first latent variable, PLS decomposition is started by selecting yj, an arbitrary column of Y as the initial estimate for ui. Usually, the 

---

The most predictive PLS regression model for these data makes use of two PLS-components ... 

All algorithms described so far solve the linear PLS regression models (Equations 4.61 through 4.65). For Nonlinear PLS we assume nonlinear relations, and two major approaches are mentioned here (Rosipal and Kramer 2006). 

In reviews on the use of in situ sensors" or optical sensor systems" for bioprocesses, UV-vis does not play a major role. An example for the application of at-line UV-vis spectroscopy was presented by Noui et al. The selective flocculation processes of Saccharomyces cerevisiae homogenates were followed with a newly developed direct UV spectrophotometer. The results from a PLS regression model were in good agreement with those from off-line chemical assays. 

A PLS regression model based on X (acoustic spectra from sensor A) and Y (crystallization temperature) was established. The X matrix contains 13 objects, each with 1024 variables (frequencies 0-25kHz). An overview of the X data is shown in Figure 9.8, in which one can observe systematic changes in the acoustic signatures following the object (samples) succession. 

Performance of PLS regression models to predict benzene and total aromatics by GC methods were equally robust. Model maintenance was minimal and the spectrometer operated within validation limits for periods of over a year. 

Figure 12.13 The percentage of explained variance in both the x data (solid line) and y data (dotted line), as a function of the number of latent variables in a PLS regression model for cis-butadiene content in styrene-butadiene copolymers.

The above yields the overall conclusion that the PLS regression model con-... 

Nord and Jacobsson [97] proposed several approaches to interpret ANN models. The results were compared with those derived from a PLS regression model where the contributions of variables were studied. Notably, they employed simple architectures composed of a unique hidden layer. They discovered that the variable contribution term in ANN models is similar to that in PLS models for linear relationships, although this may not be the case for nonlinear relations. In such cases, the proposed algorithms can give additional information about the models. 

Partial least square (PLS) regression model describes the dependences between two variables blocks, e.g. sensor responses and time variables. Let the X matrix represent the sensor responses and the Y matrix represent time, the X and Y matrices could be approximated to few orthogonal score vectors, respectively. These components are then rotated in order to get as good a prediction of y variables as possible [25], Linear discriminant analysis (LDA) is among the most used classification techniques. The method maximises the variance between... 

Various transporter proteins were assessed including P-glycoprotein (P-gp). The K-PLS classification models for P-gp substrates (Table 15.2, panel A) and K-PLS regression models for P-gp inhibitors (Table 15.2, panel B) were found to represent a slight improvement over PLS. The difference in favor of K-PLS is more pronounced for the noradrenaline transporter and serotonin transporter dissociation constant models (Table 15.2, panels C, D). 

Huwyler, J. (2007) In silico prediction of brain and CSF permeation of small molecules using PLS regression models. Oral lecture at the 9fh Blood—Brain Barrier Expert Meeting, May 22, 2007, Bad Herrenalb, Germany. 

The method of partial least squares (PLS) is also a regression technique which makes use of quantities like PCs derived from the set of independent variables. The PCs in PLS regression models are called latent variables (LV), " as shown in the PLS equation, eqn (7.9). 

Plethora of simplex descriptors is usually generated in SiRMS. The PLS method proved efficient at the work with a large number of variables and was described well elsewhere [44, 45]. Briefly, a PLS regression model could be represented as Eq. 14.3 [45] ... 

The intact approach has also been applied to cereal food products (6, 23), which encompass a very wide range of particle sizes and shapes from flour to flakes, puffed grains, extruded products, crackers, and flnes. A PLS regression model was developed to predict fat in intact cereal products. The model had a root mean squared standard error of performance of 11 g kg (range 1-205 g kg ) and coefficient of determination of 0.98 (23). When contrasted with models developed with similar products in a ground state, the latter had comparable standard errors of performance of 10-11 g kg fat and coefficients of determination of 0.98-0.99 (29,40,45). 

Table 8.1 Percentages of explained variance for the predictor matrix (A, the average of the predicted matrix (7) and each of the predicted sensory variables for the first five components (comps) of the PLS regression model...

Tetteh and co-workers described the application of radial basis function (RBF) neural network models for property prediction and screening (114). They employed a network optimization strategy based on biharmonic spline interpolation for the selection of an optimum number of RBF neurons in the hidden layer and their associated spread parameter. Comparisons with the performance of a PLS regression model showed the superior predictive ability of the RBF neural model. 

LIBS combined with PLS was applied for the quantitative analysis of the ash content of coal. In order to construct a PLS model and reduce the calculation time, different spectral range data were used, and then the performances of these models were compared. The results show that good agreement was observed between the ash content provided by the thermo-gravimetric analyser and the LIBS measurements coupled to the PLS regression model for the unknown samples. The feasibility of calculating coal ash content from LIBS spectra was proved. 

Coefficients QSAR (3) coefficients derived from the non-cross-validated (or cross-validated) PLS regression models, which are used directly in the QSAR equation, together with the corresponding field values at the lattice point, to predict the target property. 

FTIR spectroscopy was used in combination with partial least square (PLS) to differentiate and quantify these two oils. The calibration plot of PLS regression model was shown a good linearity between the actual value and FTIR predicted value of percentage of palm kernel olein in virgin coconut oil. The differenees between the actual adulteration concentration and the calculated adulteration predicted from the model were very small, with a determination coefficient (R ) of 0.9973 and root mean error of calibration of 0.0838. 

It can be seen from the figure that the minimum classification error in cross-validation is obtained when 9 LVs are chosen to calculate the inner PLS regression model in Equation (32) therefore, 9 LVs are chosen as the optimal complexity to compute the ECVs. A representation of the training data onto the space spanned by the two canonical variates calculated based on the optimal complexity PLS model is reported in Eigure 11 (filled symbols). 

Application of PLS-DA differs somewhat to criterion based methods such as LDA. In this case, a PLS regression model is built to predict the categorical variable (0 and 1 in our case). The predicted values vary around 0 and 1. In order to turn this regression model into a discriminant model, it is necessary to choose a threshold, so that any predicted values above the threshold would be classified as belonging to class 1 (plastic) and all predicted values below the threshold would be classified as belonging to class 0 (cheese). It is common to apply a threshold value of 0.5 when the class values are in the [0,1] range and the number of spectra in each class is equal. However, when the number of spectra in each class is not the same, an alternate threshold may be required. This is usually selected by trial and error. 

---