Orthogonalization vectors methods

Direct use of this equation is not advisable, however, since it is only slowly convergent. A more efficient method is obtained by constructing a set of orthogonal vectors x as... 

Besides the aforementioned time-domain approaches, many frequency-domain methods have also been developed and widely used. Examples are the complex curve fitting method [153], the maximum entropy method [4,263], the pole/zero assignment technique [271], the simultaneous frequency-domain approach [62], the rational fraction polynomial approach [219], the orthogonal polynomial approach [264], the polyreference frequency-domain approach [73], the multi-reference simultaneous frequency-domain approach [64] and the best-fit reciprocal vectors method [173]. 

The PCA method borrows from Equation (7.14) the idea of expanding A into the products of two vectors, and expands A as the products of two mutually orthogonal vectors, tj and pj, as... 

Two factor analysis methods that are used are the principal components regression (PCR) and the partial least squares (PLS). In the PRC method, the concentrations are expressed as functions of the principal components (PC) instead of absorbances as in ILS. The PC are orthogonal vectors that are linear combinations of the original spectral data of the standards. Here, PCI accounts for the maximum variability in the data, and PC2 accounts for the maximum variability not accounted for by PCI, etc. The other method PLS, is similar to the PCR method except that the PCs are weighted. The weighting is based on the correlation of the PCs with concentration. These are full-spectrum methods like CLS, but like ILS, one can analyze one component at a time. These methods are most often used for quantitative analysis in the near infrared region because of the broadness and overlapping nature of the bands here. 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

Because of its oscillatory component wave motion requires a related, but more complicated description than linear motion. The methods of particle mechanics use vectors to describe displacements, velocities and other quantities of motion in terms of orthogonal unit vectors, e.g. 

Therefore, when trying a line dynamics, an antisymmetric rank 2 tensor appears. As is seen, E B = 0 or, equivalently, det = 0, as the electric and the vector fields are orthogonal, which means that, with this method, the Faraday 2-form is degenerate and the field is of radiation type (also called singular, degenerate, or null). 

---