Plane of closest fit

One of the earliest interpretations of latent vectors is that of lines of closest fit [9]. Indeed, if the inertia along v, is maximal, then the inertia from all other directions perpendicular to v, must be minimal. This is similar to the regression criterion in orthogonal least squares regression which minimizes the sum of squared deviations which are perpendicular to the regression line (Section 8.2.11). In ordinary least squares regression one minimizes the sum of squared deviations from the regression line in the direction of the dependent measurement, which assumes that the independent measurement is without error. Similarly, the plane formed by v, and Vj is a plane of closest fit, in the sense that the sum of squared deviations perpendicularly to the plane is minimal. Since latent vectors v, contribute... 

Pearson, K. 1901. On lines and planes of closest fit to systems of points in space. Philosophical Magazine Journal of Science 2(6) 559-572. 

It is important to discuss the concept of uniqueness at this point. The principal components are unique but are not unique in providing a basis for the plane of closest fit. This plane can also be described using another basis, e.g., P can be rotated by Z (where Z is an R x R nonsingular matrix). Then upon using (7J) 1 to counterrotate T, the solution TP does not change TP = T(Z ) z P = T(PZ) = TP, where P is the new basis and T are the scores with respect to the new basis. This property is known as rotational freedom [Harman 1967], Summarizing, the plane found is unique, but not its basis vectors.1... 

Pearson K.. On Lines and Planes of Closest Fit to Systems of Points in Space Philosophical Magazine. 1901 2 559-572. 

In order to calculate C,- as a function of potential we need information on and 00 (which now is the potential at the plane of closest approach and not at the electrode surface). is evaluated by integrating the C vs. potential curve and 0o is found from equation 4.19. Equation 4.20 is used next to calculate Q. (a) C,- is now calculated from equation 4.30a. The resulting curve is given in Fig. 28. Once this curve is known, we can reverse this process and calculate C for a different concentration of the same electrolyte, (b) If the experimental curve fits the calculated one, it is deduced that the curve C vs. potential does not change as the concentration of the... 

The atomic structure of the nuclei of metal deposits, which have the simplest form since they involve only one atomic species, appear to be quite different from those of the bulk metals. The structures of metals fall mainly into three classes. In the face-centred cubic and the hexagonal structures each atom has 12 co-ordination with six neighbours in the plane. The repeat patterns obtained by laying one plane over another in the closest fit have two alternative arrangements. In the hexagonal structure the repeat pattern is A-B-A-B etc., whereas in the face-centred cubic structure the repeat pattern is A-B-C-A-B-C. In the body-centred cubic structure in which each atom is eight co-ordinated, the repeat pattern is A-B-A-B. (See Figure 1.4.)... 

---