Latent ordering variable

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

The biplot of Fig. 31.9 shows that both the centroids of the compounds and of the methods coincide with the origin (the small cross in the middle of the plot). The first two latent variables account for 83 and 14% of the inertia, respectively. Three percent of the inertia is carried by higher order latent variables. In this biplot we can only make interpretations of the bipolar axes directly in terms of the original data in X. Three prominent poles appear on this biplot DMSO, methylene-dichloride and ethylalcohol. They are called poles because they are at a large distance from the origin and from one another. They are also representative for the three clusters that have been identified already on the column-standardized biplot in Fig. 31.7. 

A typical performance behaviour is shown in Fig. 44.16b. The increase of the NSE for the monitoring set is a phenomenon that is called overtraining. This phenomenon can be compared to fitting a curve with a polynomial of a too high order or with a PCR or PLS model with too many latent variables. It is caused by the fact that after a certain number of iterations, the noise present in the training set is modelled by the network. The network acts then as a memory, able to recall... 

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... 

In order to assess the optimal complexity of a model, the RMSEP statistics for a series of different models with different complexity can be compared. In the case of PLS models, it is most common to plot the RMSEP as a function of the number of latent variables in the PLS model. In the styrene—butadiene copolymer example, an external validation set of 7 samples was extracted from the data set, and the remaining 63 samples were used to build a series of PLS models for ris-butadicne with 1 to 10 latent variables. These models were then used to predict the ris-butadicne of the seven samples in the external validation set. Figure 8.19 shows both the calibration fit error (in RMSEE) and the validation prediction error (RMSEP) as a function of the number of... 

In order to handle multiple Y-variables, an extension of the PLS regression method discussed earlier, called PLS-2, must be used.1 The algorithm for the PLS-2 method is quite similar to the PLS algorithms discussed earlier. Just like the PLS method, this method determines each compressed variable (latent variable) based on the maximum variance explained in both X and Y. The only difference is that Y is now a matrix that contains several Y-variables. For PLS-2, the second equation in the PLS model (Equation 8.36) can be replaced with the following ... 

It should be noted that there are other multivariate variable selection methods that one could consider for their application. For example, the interactive variable selection (IVS) method71 is an actual modification of the PLS method itself, where different sets of X-variables are removed from the PLS weights (W, see Equation 8.37) of each latent variable in order to assess the usefulness at each X-variable in the final PLS model. 

When applying Eq. (11.19) or Eq. (11.21) the effects of temperature dependent liquid properties can normally be adequately accounted for by evaluating these liquid properties at the mean temperature in the film, i.e., at (Ts + Tw)/2. The latent heat, hfgt is evaluated at the saturation temperature. In some cases, in order to achieve greater accuracy, it may be necessary to account for the effect of variable liquid properties by using a more complex procedure to find the temperature at which the properties are evaluated. The appropriate temperature will then depend on the specific liquid involved. This is discussed in references [41] to [43]. 

Stimulation of the peripheral nerve trunk of intact animals leads to generation of muscle action potentials of three types. According to the duration of latent periods, they fall into the following order M-response (the result of the direct stimulation of a-motor neuron axons), Fl-response (the monosynaptic response), and polysynaptic responses with the variable latent period from 8-12 up to about 40 ms. In test animals of the III group, the changes of temporal parameters refer mainly to the latent period and duration of M-response (Table 7.4). Polysynaptic responses occur at all intensities of excitation and have a more pronounced character than in intact rats. A marked level and more distinct differentiation of the peaks of the complex action potential were noted. 

A large number of substituent descriptors have been reported in the literature. In order to use this information for substituent selection, appropriate statistical methods may be used. Pattern recognition or data reduction techniques, such as PCA or CA are good choices. As explained in Section III.B.3. in more detail, PCA consists of condensing the information in a data table into a few new descriptors made of linear combinations of the original ones. These new descriptors are called PCs or latent variables. This technique has been applied to define new descriptors for amino acids, as well as for aromatic or aliphatic substituents, which are called principal properties (PPs). These PPs can be used in FD methods or as variables in QSAR analysis. ... 

Since PLS technique is sensitive to outliers and scaling, outliers should be removed and data should be scaled prior to modeling. After data pretreatment, the number of latent variables (PLS dimensions) to be retained in the model is determined. Cumulative prediction sum of squares (CUM-PRESS) versus the number of latent variables or prediction sum of squares (PRESS) versus the number of latent variables plots are used for this purpose. It is usually enough to consider the first few PLS dimensions for monitoring activities, while more PLS dimensions are needed for prediction in order to improve the accuracy of predictions. 

Nowadays, the most favored regression technique is Partial Least Squares Regression (PLS or PLSR). As happens with PCR, PLS is based on components (or latent variables ). The PLS components are computed by taking into account both the x and the y variables, and therefore they are slightly rotated versions of the Principal Components. As a consequence, their ranking order corresponds to the importance in the modeling of the response. A further difference with OLS and PCR is that, while the former must work on each response variable separately, PLS can be applied to multiple responses at the same time. 

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