Class distance plots

Figure 6. Class Distance Plot Coomans Plot). Models are fitted separately to each class (Etu resp. Tellus). The distances for each object to the two classes are plotted. The dashed line indicates equal class distance. The soil samples (object 1-3) are located close to this line.

Class distance plot horizontal axis = class A, vertical axis = class B... 

For each of the 13 objects in the training set /, calculate the distance from the PC model of each class c by determine dic = yJ Ej=l(cxij — ci, )2, where 7=8 and corresponds to the measurements, and the superscript c indicates a model of for class c. For these objects produce a class distance plot. 

Extend the class distance plot to include the two samples in the test set using the method of steps 6 and 7 to determine the distance from the PC models. Are they predicted correctly ... 

Calculate the centroids of class A (excluding the outlier) and class B. Calculate the Euclidean distance of the 58 samples to both these centroids. Produce a class distance plot of distance to centroid of class A against class B, indicating the classes using different symbols, and comment. 

Determine the variance-covariance matrix for the 11 elements and each of the classes (so there should be two matrices of dimensions 11 x 11) remove the outlier first. Hence calculate the Mahalanobis distance to each of the class centroids. What is die reason for using Mahalanobis distance rather than Euclidean distance Produce a class distance plot for diis new measure, and comment. 

Figure 10.13 is a bar-chart representation of the Class Projection output for factor 2 from a SIMCA, in which the modulus of the class distinctions is 10 or greater for all spectra except numbers 105, 115 and 125. The values for these spectra are about 7, 4 and 1 respectively, suggesting much less distinction (particularly for spectrum number 125), as might be expected from spectra recorded along a boundary region. A class distance plot of for the two main categories is shown in Figure 10.14. The transition from stroma to tumour along a row of spectra is clearly evident in the regions of spectra numbers 105, 115 and 125 the other crossovers occur at the end of each row of spectra in the grid-map, z.e., at spectra numbers 111 and 121.

Figure 10.14 Class Distance plot output from a SIMCA analysis of the pre-processed absorbance spectra numbered 101-130, see text for details.

Various partitions, resulted from the different combinations of clustering parameters. The estimation of the number of classes and the selection of optimum clustering is based on separability criteria such as the one defined by the ratio of the minimum between clusters distance to the maximum of the average within-class distances. In that case the higher the criterion value the more separable the clustering. By plotting the criterion value vs. the number of classes and/or the algorithm parameters, the partitions which maximise the criterion value is identified and the number of classes is estimated. 

Distance plot to class centroids of the projection in Figure 4.26... 

Figure 6- The Pasquill-Gifford dispersioo coefficients versus downwind distance for various dispersion classes, (a) The lateral dispersion coefficient and b) the vertical dispersion coefficient are plotted against x.

A useful tool in the interpretation of SIMCA is the so-called Coomans plot [32]. It is applied to the discrimination of two classes (Fig. 33.18). The distance from the model for class 1 is plotted against that from model 2. On both axes, one indicates the critical distances. In this way, one defines four zones class 1, class 2, overlap of class 1 and 2 and neither class 1 nor class 2. By plotting objects in this plot, their classification is immediately clear. It is also easy to visualize how certain a classification is. In Fig. 33.18, object a is very clearly within class 1, object b is on the border of that class but is not close to class 2 and object c clearly belongs to neither class. 

The PET class has many samples that are very closely clustered (linked at a distance of 0.1). This can also be seen in the plot of the preprocessed data (Figure 4.20). 

Goodness Value Plot (Model and Sample Diagnostic) In prediction, a unanimous classification does not guarantee that an unknown is close to the samples in the predicted class even if the classes were found to be well separated (refer back to Figure 4.44). Therefore, the goodness value in Equation 4.4 is used to evaluate the quality of the classification using a relative distance measure. The approach for validating the prediction is to evaluate the distance of the unknown to the predicted class relative to an internal measure of how diffu.se the samples are in that chiss. 

PCA Residual and Distance from Boundary Table (Model and Sainple Diii nostic) The samples that were identified in the F plot as needing further investigation are the two class A. samples that are not predicted as belonging to c .iS.s A vi.e.. their values are larger titan F,. ,). Referring to rLjbk ... 

Distance from the class A model FIGURE 2.13 Example of Coomans plot. 

Figure 6.2. Relations between shear stress, deformation rate, and viscosity of several classes of fluids, (a) Distribution of velocities of a fluid between two layers of areas A which are moving relatively to each other at a distance x wider influence of a force F. In the simplest case, F/A = fi(du/dx) with ju constant, (b) Linear plot of shear stress against deformation, (c) Logarithmic plot of shear stress against deformation rate, (d) Viscosity as a function of shear stress, (e) Time-dependent viscosity behavior of a rheopectic fluid (thixotropic behavior is shown by the dashed line). (1) Hysteresis loops of time-dependent fluids (arrows show the chronology of imposed shear stress).

In the preceding description of the Mahalanobis distance, the number of coordinates in the distance metric is equal to the number of spectral frequencies. As discussed earlier in the section on principal component analysis, the intensities at many frequencies are dependent, and by using the full spectrum, we fit the noise in addition to the real information. In recent years, Mahalanobis distance has been defined with PCA or PLS scores instead of the spectral frequencies because these techniques eliminate or at least reduce most of the overfitting problem. The overall application of the Mahalanobis distance metric is the same except that the rt intensity values are replaced by the scores from PCA or PLS. An example of a Mahalanobis distance calculation on a set of Raman spectra for 25 carbohydrates is shown in Fig. 5-11. The 25 spectra were first subjected to PCA, and it was found that the first three principal components could account for most of the variance in the spectra. It was first assumed that all 25 spectra belonged to the same class because they were all carbohydrates. However, as shown in the three-dimensional plot in Fig. 5-11, the spectra can be clearly divided into three separate classes, with two of the spectra almost equal distance from each of the three classes. Most of the components in the upper left class in the two-dimensional plot were sugars however, some sugars were found in the other two classes. For unknowns, scores have to be calculated from the principal components and processed in the same way as the spectral intensities. 

A very useful method of discriminating between samples from different classes is to plot PCA or PLS scores in two or three dimensions. This is very similar to the Mahalanobis distance discussed earlier in Fig. 5-11, except that it is limited to two or three dimensions, and the Mahalanobis distance can be constructed for n dimensions. Score plots do provide a good visual understanding of the underlying differences between data from samples belonging to different classes. 

The calculation of the linear discriminant function is presented in Table 4.25 and the values are plotted in Figure 4.30. It can be seen that objects 5, 6, 12 and 18 are not easy to classify. The centroids of each class in this new dataset using the linear discriminant function can be calculated, and the distance from these values could be calculated however, this would result in a diagram comparable to Figure 4.27, missing information obtained by taking two measurements. 

---