Objective vector

Convergence of the iteration requires the norm of the objective vector 1g to be less than the convergence criterion, e. The initial estimates used, if not provided externally, are, in addition to Equation (7-28)... 

Another approach to eq. (44.2) is to add an extra dimension to the object vector X, on which all objects have the same value. Usually 1 is taken for this extra term. The 0 term can then be included in the weight vector, w (w,W2,-0). This is the same procedure as for MLR where an extra column of ones is added to the X-matrix to accommodate for the intercept (Chapter 10). The objects are then characterized by the vector xj (X, X2 1). Equation 44.2 can then be written ... 

FIGURE 2.10 Euclidean distance and city block distance (Manhattan distance) between objects represented by vectors or points xA and xB. The cosine of the angle between the object vectors is a similarity measure and corresponds to the correlation coefficient of the vector... 

Here, x, is an object vector, and the center is estimated by the arithmetic mean vector x, alternatively robust central values can be used. In R a vector d Mahalanobis of length n containing the Mahalanobis distances from n objects in X to the center... 

FIGURE 5.4 Linear discriminant scores dj for group j by the Bayesian classification rule based on (Equation 5.2). mj, mean vector of all objects in group j Sp1, inverse of the pooled covariance matrix (Equation 5.3) x, object vector (to be classified) defined by m variables Pj, prior probability of group j. 

The concepts described so far are usually not applied in the original data space but in a transformed and enlarged space using basis expansions (see Section 4.8.3 about nonlinear regression). Thus each observation jc, is expressed by a set of basis functions (object vectors jc, with m dimensions are replaced by vectors h(x,) with r dimensions)... 

The cosine of the angle between object vectors and the Mahalanobis distance are independent from the scaling of the variables. The latter accounts for the covariance structure of the data, but considering the overall covariance matrix of all objects might result in poor clustering. Thus usually the covariance for the objects in each cluster is taken into account. This concept is used in model-based clustering as mentioned above, and will be discussed in more detail in Section 6.6. 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

The argument / varies in the interval -n to +n, the parameters xb x2,..., x5 are the values of the different features of one object, so that one object vector is represented by /(/). Curves close to each other represent similar objects. 

In MOO, ideal and nadir objective vectors are occasionally used. The ideal objective vector contains the optimum values of the objectives, when each of them is optimized individually disregarding the other objectives. The ideal objective vector denoted by superscript (i.e., [/i /2 J) is shown in Figure 1.2a along with the nadir objective vector denoted by superscript N (i.e., [/i /2 ]). Here,/i is the value of /i(x) when /2(x) is optimized individually, and is the value of /2(x) when /i(x) is optimized individually. Components of the nadir objective vector are the upper bounds (i.e., most pessimistic values) of objectives in the Pareto-optimal set. In case of two objectives, as shown in Figure 1.2a, they correspond to the value of one objective when the other is optimized individually. This may not be the case if there are more than two objectives (Weistroffer, 1985). The ideal objective vector is not realizable unless the objectives are non-conflicting in which case the MOO problem has only a unique solution, namely, ideal objective vector. However, it tells us the best possible value for each of the... 

No Preference Methods (e.g., global criterion and neutral compromise solution) These methods, as the name indicates, do not require any inputs from the decision maker either before, during or after solving the problem. Global criterion method can find a Pareto-optimal solution, close to the ideal objective vector. 

For multi-objective optimization, theoretical background has been laid, e.g., in Edgeworth (1881) Koopmans (1951) Kuhn and Tucker (1951) Pareto (1896, 1906). Typically, there is no unique optimal solution but a set of mathematically incomparable solutions can be identified. An objective vector can be regarded as optimal if none of its components (i.e., objective values) can be improved without deterioration to at least one of the other objectives. To be more specific, a decision vector x S and the corresponding objective vector f(x ) are called Pareto optimal if there does not exist another x G S such that / (x) < /j(x ) for alH = 1,..., A and /j(x) < /j(x ) for at least one index j. In the MCDM literature, widely used synonyms of Pareto optimal solutions are nondominated, efficient, noninferior or Edgeworth-Pareto optimal solutions. 

As mentioned in the introduction, we here assume that a DM is able to participate in the solution process. (S)he is expected to know the problem domain and be able to specify preference information related to the objectives and/or different solutions. We assume that less is preferred to more in each objective for him/her. (In other words, all the objective functions are to be minimized.) If the problem is correctly formulated, the final solution of a rational DM is always Pareto optimal. Thus, we can restrict our consideration to Pareto optimal solutions. For this reason, it is important that the multi-objective optimization method used is able to find any Pareto op>-timal solution and produce only Pareto optimal solutions. However, weakly Pareto optimal solutions are sometimes used because they may be easier to generate than Pareto optimal ones. A decision vector x G S (and the corresponding objective vector) is weakly Pareto optimal if there does not exist another x G S such that /i(x) < /i(x ) for alH = 1,..., A . Note that Pareto optimality implies weak Pareto optimality but not vice versa. 

The DM may find information about the ranges of feasible Pareto optimal objective vectors useful. Lower bounds form a so-called ideal objective vector z G R. Its components zf are obtained by minimizing each ob-... 

The upper bounds of the Pareto optimal set, that is, the components of a nadir objective vector are in practice difficult to obtain. It can... 

Since most of this book is devoted to evolutionary methods for multiobjective optimization, we here only wish to discuss some differences between EMO approaches and scalarization based approaches. As mentioned before, EMO approaches are a posteriori type of methods and they try to generate an approximation of the Pareto optimal set. In bi-objective optimization problems, it is easy to plot the objective vectors produced on a plane and ask the DM to select the most preferred one. While looking at the... 

In the initialization phase of the NIMBUS method, the ranges in the Pareto optimal set, that is, the ideal and the nadir objective vectors are computed to give the DM some information about the possibilities of the problem. The starting point of the solution process can be specified by the DM or it can be a neutral compromise solution located approximately in the middle of the Pareto optimal set. To get it, we set + z )/2 as a reference point and solve (6.4). 

In what follows, we describe the interactive solution process for the standard SMB process using the IND-NIMBUS process design tool. For further details, see Hakanen et al. (2007). The aim here is to give an understanding of the nature of an interactive solution process. The DM involved was an expert in SMB processes. First, in the initialization phase, the ranges in the Pareto optimal set were computed as z = (0.891, 0.369, 97.2, 90.0) and = (0.400,2.21, 90.0, 70.0). A neutral compromise solution f(x ) = (0.569,1.58, 92.5,76.9) was the starting point for the interactive solution process. Remember that the objective functions represented throughput (T), consumption of desorbent (D), purity (P) and recovery (R) and their values are here presented in objective vectors in this order (T, D, P, R). Note that the second objective function was minimized while the others were maximized. 

Our applications involve performing the m-band DWT m > 2 for each object vector in a spectral data set containing n spectra each of dimension p. The wavelet (or scaling) coefficients produced from the DWT are used as features for some multivariate method. The m-band DWT has previously been described for a single data vector, but it is more convenient to redefine the transform using a slight change of notation. 

This is because an objective scalar a does not change its value, objective vector a is the same arrow looked at from different frames and, ultimately, an objective tensor A transforms an objective vector (say aO to an objective vector (say a2) in all frames indeed if in the original frame a2 = Aai and in the starred frame a = A aj then by (3.31) we obtain (3.32). 

As a result, we obtain If scalar a is objective (3.55) then its material derivative a and space gradient grada are objective while Gradfl and da/dt are not. If a is an objective vector, diva, a.a = a are objective, while material derivative a is not. Ultimately, with objective (second order) tensor A, the vector divA and the scalars trA, detA are objective. 

Transformation (frame change) of objective vector a (3.56) gives the objectivity of the scalar product... 

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