Principal coordinates analysis

In general, two related techniques may be used principal component analysis (PCA) and principal coordinate analysis (PCoorA). Both methods start from the n X m data matrix M, which holds the m coordinates defining n conformations in an m-dimensional space. That is, each matrix element Mg is equal to q, the jth coordinate of the /th conformation. From this starting point PCA and PCoorA follow different routes. 

Principal coordinate analysis (PCoorA) [37], on the other hand, operates on the square n X n matrix, reflecting the relationships between the conformations. The... 

Figure 4 The energy landscape of the pnon protein (Pi P) (residues 124-226) in vacuum, obtained by principal coordinate analysis followed by the minimal energy envelope procedure. Two large basins are seen. One basin is associated with the native Pi P conformation the other is associated with partially unfolded conformations.

Principal coordinates analysis 31.6.1 Distances defined from data... 

Principal coordinates analysis (PCoA) is applied to distance tables rather than to original data tables, as is the case with principal components analysis (PCA). 

Using D as input we apply principal coordinates analysis (PCoA) which we discussed in the previous section. This produces the nxn factor score matrix S. The next step is to define a variable point along they th coordinate axis, by means of the coefficient kj and to compute its distance d kj) from all n row-points ... 

The lower path is somewhat more complicated. The first step in the path involves either PCA (83) or principal-coordinate analysis (PCO) (83). This step can be followed by optimization of a function that minimizes the error between the proximity measure computed in the reduced-dimension and full coordinate systems if desired. Xie et al. (84) recently published an interesting paper along these lines. Kruscal stress (79) is a widely used function in this regard... 

The global minimum of this BLN model is the four-stranded P-barrel [332] shown in Figure 1.38. The hydrophobic beads prefer the core environment, while the neutral beads support the turns. Berry and co-workers have previously investigated the self-assembly of this system from separated strands and have performed a principal coordinate analysis [334,335]. 

Figure 36.3 Three-dimensional Sakmarian and Roadian-Wordian ordination plots of the selected OGUs on the three first axes of principal coordinates analysis, which respectively account for 94% and 89% of total variation.

FIGURE 16.4 Results of a similarity-based compaiison producing a distance square matrix (lower window) on which two different clustering methods were applied. The top-left window represents a UPGMA tree where one additional criterion was printed (bands of one-dimensional electrophoresis profiles). The top-right window is a three-dimensional representation of the OTUs presented on the distance matrix below that have been reordered using the principal coordinate analysis ordination method. 

Figure 19. Principal Coordinate Analysis (PCoA) of the Type III antifreeze protein in an explicit solvent simulation (right-hand side) and in an implicit solvent simulation (left-hand side). The NMR structure was used as the reference.

Over 80% of the 21 860 trees recorded across all 3 sites were assigned to a functional group. All three forests showed the tjqDically negative exponential distribution of stem diameter class distributions associated with mature forests (Uuttera et al., 2000) and apparently had similar climate, soil and topography. It was therefore hypothesised that the forests would be functionally similar. A principal coordinates analysis, however, indicated that there was significant difference in (p-diversity between the three forests despite the apparent functional similarity of all three forests. 

Metrical MDS operates on an input matrix of dissimilarities, or distances, between pairs of samples, giving as a result a matrix of coordinates whose configuration minimizes a loss function. This method presents an optimization phase, which can be performed with a variety of loss functions to be considered, and other possible variations of the methods concern the input distance matrix, which can be calculated according to different weights and criteria. When the Euclidean distance is considered, the classical MDS, also known as the principal coordinates analysis, consists in performing a PC A on the double-centred distance matrix and then rotating the solution so that the stress criterion S is minimized ... 

FIGURE 28 Comparison of (A) principal component analysis and (B) classical multidimensional scaling-principal coordinate analysis for the FlourRheo data set. See Figure 23 for the legend. 

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