Quadratic discriminant function

One parametric routine implements a quadratic discriminant function using the Bayesian theorem. The equation for the discriminants are as follows ... 

The second term in the right-hand side of Equation (12) defining the discriminant function is the quadratic form of a matrix expansion. Its relevance to our discussions here can be seen with reference to Figure 3 which illustrates the division of the sample space for two groups using a simple quadratic function. This Bayes classifier is able to separate groups with very differently shaped distributions, i.e. with differing covariance matrices, and it is commonly referred to as the quadratic discriminant function. 

Tables Discriminant scores using the quadratic discriminant function as classifier (a), and the resulting confusion matrix (b)...

Figure 4 Scatter plot of the data from Table I and the calculated quadratic discriminant function...

Table 5.3 Discriminant scores using the quadratic discriminant function as...

The final classification involved a hierarchy of quadratic discriminant functions (QDFs). The first hierarchical level separated the six fuels into three fuel-groups (oil coal series oil shale/Czech black coal), whilst the second hierarchical level used separate QDFs to separate each fuel group into its component fuel-types. Some fuel separations proved to be very good with oil, oil shale, Czech black coal and brown coal being 94.1%, 89.3%, 86.0% and 84.1% correctly allocated to their fuel-types, respectively. The coal-series and, in particular peat and coal, showed considerable overlap and were less effectively separated. In total, of those particles allocated to a fuel-type, over 80% were allocated correctly. 

Discriminant analysis (DA) performs samples classification with an a priori hypothesis. This hypothesis is based on a previously determined TCA or other CA protocols. DA is also called "discriminant function analysis" and its natural extension is called MDA (multiple discriminant analysis), which sometimes is named "discriminant factor analysis" or CD A (canonical discriminant analysis). Among these type of analyses, linear discriminant analysis (LDA) has been largely used to enforce differences among samples classes. Another classification method is known as QDA (quadratic discriminant analysis) (Frank and Friedman, 1989) an extension of LDA and RDA (regularized discriminant analysis), which works better with various class distribution and in the case of high-dimensional data, being a compromise between LDA and QDA (Friedman, 1989). 

The raw and the standardized coordinates are calculated both manually and using software [STATISTICA, 1995], As in most cases where the reader of publications wants to reproduce the results, surprisingly we get a different result. In our case this is because most software (SPSS, STATISTICA,. ..) calculates a constant along with the raw coefficients. At the same time this demonstrates that there are several ways of finding discriminant functions. So, in some instances it may be convenient to use so-called elementary discriminant functions [AHRENS and LAUTER, 1981] or to try quadratic discrimination (see [FAHRMEIR and HAMERLE, 1984]). 

Several other functions such as the quadratic and regularised discriminant functions have been proposed and are suitable under certain circumstances. 

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

In the high-dimensional space, the optimized super-planes were computed by substituting dot product operation for linear kernel function and by solving the dual problem using quadratic programming. The discrimination functions of quartz, feldspar and biotite in the high-dimensional space were obtained and represented as... 

The value of the discriminant for the equation x2 - Ax + 3 = 0 is positive, and we see that there are clearly two different roots, as indicated in plot (a) of Figure 2.23, which shows the curve cutting the x-axis at x = 1 and x = 3. The curve of the function y = x2 - 4x + 4, shown in plot (b), touches the x-axis at x = 2. In this case the discriminant is zero, and we have two equal roots, given byx = + /5 = 2 0. Note that although the curve only touches the x-axis in one place, the equation x2 - 4x+4 = 0 still has two roots they just happen to be identical. Finally, in the case of curve (c), there are no values of x corresponding to y = 0, indicating that there are no real roots of the quadratic equation y -Ax + 6 = 0, as the discriminant is equal to -8. 

If the filter coefficients are to be used for discriminatory purposes, then the criterion function should strive to reflect differences among classes. In this section three suitable discriminant criterion functions are described. These discriminant criterion functions are Wilk s lambda (3a), entropy (3e), and the cross-validated quadratic probability measure (3cvqpm)-... 

The cross-validated quadratic probability measure (CVQPM) (see Chapter 12 for more details) assesses the trustworthiness of the class predictions made by the discriminant model. The CVQPM ranges from 0 to 1. Ideally, larger values of the QPM are preferred, since this implies the classes can be differentiated with a higher degree of certainty. The CVQPM criterion function based on a band of coefficients x[ ) (t) would be defined as follows... 

The function has an extremum at x = 0, y = 0. We show a plot of the function with fl = 1, c = 1, b = 3 in Fig. 2.8. If the discriminant D = 4ac — b >0, the contour plot will show ellipses, with a definite maximum, otherwise a saddle point emerges. In fact, we are dealing with a binary quadratic form in two real variables. The binary quadratic form is positive definite if its discriminant is positive. ... 

This mechanistic question is one of the examples of the success of density functional theory (DFT) methods for metal-organic chemistry. Earher work on the reaction mechanism could not discriminate between the two alternatives. Analysis of the different orbitals based on Extended Hueckel theory (EHT) calculations led to the conclusion that the (3+2) pathway is more likely, but the authors could not exclude the possibility of a (2+2) pathway [4] similar to the results of HE calculations in combination with Quadratic configuration interaction (QCISD(T)) single points [32]. 

Many functions have interesting, highly exploitable features (e.g., continuity and derivability). Specifically, many can be well approximated by means of quadratic functions as their minimum is approached. Conversely, the Fibonacci method does not discriminate between functions and takes all of them in the same way the worst one. 

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