Quadratic class

The centric index C belongs to the quadratic class of indices, together with Mi and N2 indices. If the number of vertices of degree v, is denoted by Vb one obtains for graphs with Vj g 4 ... 

In LDA, it is assumed that the class covariance matrices are equal, that is, S j = S for all / = 1 to. Different class covariances are allowed in quadratic discriminant analysis (QDA). The results are quadratic class boundaries based on unbiased estimates of the covariance matrix. The most powerful method is based on regularized discriminant analysis (RDA) [7]. This method seeks biased estimates of the covariance matrices, Sy, to reduce their variances. This is done by introducing two regularization parameters A and Y according to... 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

Equation (33.10) is applied in what is called quadratic discriminant analysis (QDA). The equations can be shown to describe a quadratic boundary separating the regions where is minimal for the classes considered. 

Another class of methods of unidimensional minimization locates a point x near x, the value of the independent variable corresponding to the minimum of /(x), by extrapolation and interpolation using polynomial approximations as models of/(x). Both quadratic and cubic approximation have been proposed using function values only and using both function and derivative values. In functions where/ (x) is continuous, these methods are much more efficient than other methods and are now widely used to do line searches within multivariable optimizers. 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

Note that there are n + m equations in the n + m unknowns x and A. In Section 8.6 we describe an important class of NLP algorithms called successive quadratic programming (SQP), which solve (8.17)—(8.18) by a variant of Newton s method. 

KACSYKA can also solve quadratic, cubic and quartic equations as well as some classes of higher degree equations. However, it obviously cannot solve equations analytically in closed form when methods are not known, e.g. a general fifth degree (or higher) equation. 

The inset to Fig. 6 exhibits as depending sensitively on the polymer class, but relatively weakly on molar mass. The temperature variation of Sc T)/sl is roughly linear for small 6T near Tq, consistent with the empirical VFTH equation, as noted earlier. On the other hand, this dependence becomes roughly quadratic in 6T at higher temperatures, where Sc achieves a maximum s at 7a-Attention in this chapter is primarily restricted to the broad temperature range (To 7 7a), vdiere a decrease of Sq vith T is expected to correspond to an... 

Quadratic discriminant analysis (QDA) is a probabilistic parametric classification technique which represents an evolution of EDA for nonlinear class separations. Also QDA, like EDA, is based on the hypothesis that the probability density distributions are multivariate normal but, in this case, the dispersion is not the same for all of the categories. It follows that the categories differ for the position of their centroid and also for the variance-covariance matrix (different location and dispersion), as it is represented in Fig. 2.16A. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). By coimecting the intersection points of each couple of corresponding ellipses (at the same Mahalanobis distance from the respective centroids), a parabolic delimiter is identified (see Fig. 2.16B). The name quadratic discriminant analysis is derived from this feature. 

In the following, we describe two prominent types of spectral phase modulation, each of which plays an important role in coherent control. Both types, namely sinusoidal (Section 6.2.1) and quadratic (Section 6.2.2) spectral phase modulation, are relevant for the experiments and simulations presented in this contribution. We provide analytic expressions for the modulated laser fields in the time domain and briefly discuss the main characteristics of both classes of pulse shapes. 

Conduct both linear and quadratic regression analyses for each calibration curve. Choose the method which provides the higher/ 2 value for quantification of each lipid class in the sample. 

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). 

C.L. Chen, Ph.D. thesis, A class of successive quadratic programming methods for flowsheet optimization, University of London, 1988. 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

Chen, C.L., A Class of Successive Quadratic Programming Methods for Flowsheet Optimisation. PhD Thesis, (Imperial College, London, 1988). 

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