Weighted metric

A weighted Euclidean metric is defined by the weighted scalar product  

It has been shown in Chapter 29 that the set of vectors of the same dimension defines a multidimensional space S in which the vectors can be represented as points (or as directed line segments). If this space is equipped with a weighted metric defined by W, it will be denoted by the symbol S. The squared weighted distance between two points representing the vectors x and y in is defined by the weighted scalar product  

A similar effect in is observed on angular distances (or angles), which are also different from those in the unweighted space S  

Generally, two vectors that are orthogonal in S will be oblique in 5 , unless the vectors are parallel to the coordinate axes. This is illustrated in Fig. 32.4. Furthermore, if X and y are orthogonal vectors in S, then the vectors W x and W y are orthogonal in 5 ,. This follows from the definition of orthogonality in the metric W (eq. 32.12)  

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P . Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W.  

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Distances in are different from those in the usual space S. A weighted space can be represented graphically by means of stretched coordinate axes [2]. The latter result when the basis vectors of the space are scaled by means of the corresponding quantities in Vw, where the vector w contains the main diagonal elements of W. Figure 32.3 shows that a circle is deformed into an ellipse if one passes from usual coordinate axes in the usual metric I to stretched coordinate axes in the weighted metric W. In this example, the horizontal axis in 5, is stretched by a factor. l-6 = 1.26 and the vertical axis is shrunk by a factor Vo.4 = 0.63. 

Fig. 32.3. Effect of a weighted metric on distances, (a) representation of a circle in the space 5 defined by the usual Euclidean metric, (b) representation of the same circle in the space S defined by a weighted Euclidean metric. The metric is defined by the metric matrix W.

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

Here we develop the method specifically from the point of view of contingency tables and within the context of weighted metrics. We will show that LLM differs only from CFA in the type of preprocessing that is applied to the contingency table. The results of both approaches are often similar when there are no extreme contrasts in the data. 

Indirect Metricization of the Concept of Weight Metricization can also be accomplished indirectly. For example, the weight of an object can be determined... 

Weight, metric tons Amine Plant 1,605 Membrane Plant 503... 

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