Orthogonal projection

An orthogonal projection is obtained when the projection matrices V or U are orthonormal ... 

F.C. Sanchez, J. Toft, B. van den Bogaert and D.L. Massart, Orthogonal projection approach applied to peak purity assessment. Anal. Chem., 68 (1996) 79-85. 

The geometric meaning of Eq. (35.14) is that the best fit is obtained by projecting all responses orthogonally onto the space defined by the columns of X, using the orthogonal projection matrix X(X X) X (see Section 29.8). 

PI. 3.1 A MUP peptide backbone ribbon representation, shown in two orthogonal projections (left p-sheet framework right binding-cavity, highlighted) (from Luckc et at., 1999). 

The orthogonal projection of the epitaxial poly(DMDA) could not be indexed using the unit cell data for the bulk polymerized crystal (8). However, poly(DMDA) cannot usually be polymerized to completion or to high crystallinity in the bulk due to cross-linking. The use of an epitaxial substrate may have controlled the polymerization process that led to oriented single crystals. 

By applying the orthogonal projection approach, the following Qui, QU2, Rui and Ru2 matrices are obtained ... 

Matrix D is calculated and the application of the orthogonal projection approach leads to QD, QD2, RDi, RD2, Ilfu matrices. 

Figure 5.1 The least-square estimate y of the solution to equation (5.1.4) is the orthogonal projection of the observation vector y onto the column-space 1 a2, of matrix A.

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). 

The last part of Equation 3.2 expresses this orthogonal projection of the data on the latent variable. For all n objects arranged as rows in the matrix X the score vector, f1 of PCI is obtained by... 

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. 

The hermitian metric and the quaternion module structure on M descends to Mp. In particular, M " is a hyper-Kahler manifold. There is a natural action on M " of a Lie group Ur(F) = rifcU(Ffc). This action preserves the hyper-Kahler structure. The corresponding hyper-Kahler moment map is p o o where i is the inclusion M " C M, /r is the hyper-Kahler moment map for U(F)-action on M, and p is the orthogonal projection to 0 u Vk) in u(F). We denote this hyper-Kahler moment map also by p = (/ri, /T2, / s)- This increases the flexibility of the choice of parameters. Take = (Co> Cn > Cn) ( = 1) 2, 3) such that (I is a scalar matrix in u(14)- Then we can consider a hyper-Kahler quotient... 

OPA orthogonal projection approach Q number of increments per sample... 

---