Between-class scattering

Prior to the actual classification, the FLDC performs a linear mapping to a lower dimensional subspace optimised for class separability, based on the between-class scatter and the within-class scatter of the training set. In classification, each sample is assigned to the class giving the highest log-likelihood using a linear classifier. 

A classification technique based on Fisher linear discrimination (FED) was applied to features extracted from the phenomenological model just described. To illustrate how this method works, Fig 3 presents a scatter plot of two features (F2 versus FI) for several fictional events, each a member from one of three distinct classes (A, B and C). While neither feature alone provides ample separation between all three classes, a combination of these features can be found which clearly separates all classes. FED finds the line in this feature space that maximizes between-class scatter and minimizes within-class scatter upon projection. The mean and standard deviation of each class projected onto this Fisher line can be used to define a probability distribution function (PDF). When projecting a new event onto the Fisher line, the PDF assigns a probability that it belongs to a particular class. Feature saliency and PDF stability are critical in assessing the confidence level associated with a prediction, and are thoroughly examined in Dills dissertation. ... 

Fig 3. Example data consisting of three classes (A, B C) with two descriptors (FI F2) illustrates the Fisher linear discrimination technique. When the data is projected onto the Fisher line, between-class scatter (dashed arrows) is maximized and within-class scatter (solid arrows) is minimized. Gaussian curves for each class, defined by the mean and standard deviation upon projection onto the Fisher line, are also provided. 

Equation 3.54 can be rewritten by adding —Xj + Xj to each term and rearranging the sums so that the total scatter is the sum of the within-class scatter and the between-class scatter as [63] ... 

Between-class scatter matrix Within-class scatter matrix Total scatter matrix Scores matrix (n x a)... 

Define the between-class scatter matrix Sb and the total scatter matrix St as [63, 118] ... 

The PC scores can be used to assess the relationships between different samples in the model. If this information is combined with known class information about the samples, an assessment of the effectiveness of the analytical method for distinguishing between classes can be made. Furthermore, the scores can be used to detect trends in the samples that might not be expected. It is common to plot PC scores using a two-dimensional scatter plot, where the axes represent different PC numbers. 

In the general case, for several classes, the sums of squares representing between-class and within-class scattering are then extended to matrices B and W ... 

The first FDA vector wi that maximizes the scatter between classes (Sb) while minimizing the scatter within classes Sw) is obtained as... 

Unlike the typical laser source, the zero-point blackbody field is spectrally white , providing all colours, CO2, that seek out all co - CO2 = coj resonances available in a given sample. Thus all possible Raman lines can be seen with a single incident source at tOp Such multiplex capability is now found in the Class II spectroscopies where broadband excitation is obtained either by using modeless lasers, or a femtosecond pulse, which on first principles must be spectrally broad [32]. Another distinction between a coherent laser source and the blackbody radiation is that the zero-point field is spatially isotropic. By perfonuing the simple wavevector algebra for SR, we find that the scattered radiation is isotropic as well. This concept of spatial incoherence will be used to explain a certain stimulated Raman scattering event in a subsequent section. 

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