Load vector

For generalized Newtonian fluids the load vector (i.c. the right-hand side in Equation (5.31) is expressed as... 

The main subroutine for evaluation of the elemental stiffness equations and load vectors. 

Step 1 - the n x n coefficient matrix is augmented with the load vector on the right-hand side to form an n x n + ) matrix. 

MAXEL, EIAXST ELEMENT COI mECTIVITY (MAXDF) NODAL VELOCITIES (MAXDF) GLOBAL LOAD VECTOR... 

This implies that the scalar product of a score vector s, with a loading vector allows us to reconstruct the value in the table X. Written in full, this is equivalent... 

Here, the loading vector p, contains the coefficients of the separate univariate regressions of the individual X-variables on t,. The / element of p, p j, represents the regression coefficient of it regressed on t, py = tj j The full vector of... 

The data array Xmyriyqy which may be preprocessed by standardisation and transformation, is decomposed in a set of three-way products of score images, T, loading vectors, p, and the residual, Emytlyq ... 

FIGURE 2.14 General concept of a latent variable (left), and calculation of the score u of a linear latent variable from the transposed variable vector xT and the loading vector b as a scalar product (right). 

The loading vector is usually scaled to a length of 1, that means bT b = 1 it defines a direction in the variable space. 

Calculation of scores as described by Equations 2.20 and 2.21 can be geometrically considered as an orthogonal projection (a linear mapping) of a vector x on to a straight line defined by the loading vector b (Figure 2.15). For n objects, a score vector u is obtained containing the scores for the objects (the values of the linear latent variable for all objects). 

FIGURE 2.15 Rectangular projection of a variable vector x, on to an axis defined by loading vector b resulting in the score ut. 

If several linear latent variables are calculated, the corresponding loading vectors are collected in a loading matrix B, and the scores form a score matrix V (Figure 2.16). 

In PCA, for instance, each pair j, k of loading vectors is orthogonal (all scalar products bj h/, are zero) in this case, matrix B is called to be orthonormal and the projection corresponds to a rotation of the original coordinate system. 

Projection of the variable space on to a plane (defined by two loading vectors) is a powerful approach to visualize the distribution of the objects in the variable space, which means detection of clusters and eventually outliers. Another aim of projection can be an optimal separation of given classes of objects. The score plot shows the result of a projection to a plane it is a scatter plot with a point for each object. The corresponding loading plot (with a point for each variable) indicates the relevance of the variables for certain object clusters. 

The direction in a variable space that best preserves the relative distances between the objects is a latent variable which has maximum variance of the scores (these are the projected data values on the latent variable). This direction is called by definition the first principal component (PCI). It is defined by a loading vector... 

In chemometrics, the letter p is widely used for loadings in PCA (and partial least-squares [PLS]). It is common in chemometrics to normalize the lengths of loading vectors to 1 that means p p = 1 m is the number of variables. The corresponding... 

Data for a demo example with 10 objects and two mean-centered variables x and x2 are given in Table 3.1 the feature scatter plot in Figure 3.1. The loading vector for PCl,/>i, has the components 0.839 and 0.544 (in Section 3.6 we describe methods to calculate such values). Note that a vector in the opposite direction (—0.839, —0.544) would be equivalent. The scores C of PCI cover more than 85% of the total variance. 

FIGURE 3.1 Scatter plot of demo data from Table 3.1. The first principal component (PCI) is defined by a loading vectorp — [0.839, 0.544], The scores are the orthogonal projections of the data on the loading vector. 

All loading vectors are collected as columns in the loading matrix, P, and all score vectors in the score matrix, T (Figure 3.2). 

Resulting directions (loading vectors) are orthogonal as in classical PCA. 

Similarly, the fcth PC (3 < k < m) is defined as above by maximizing the variance under the constraints that the new loading vector has length one and is orthogonal to all previous directions. All vectors bj can be collected as columns in the matrix B. 

In general the variance of scores f, corresponding to a loading vector bj can be written as... 

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