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Because variables are usually centered and standardized, the correlation between each variable and a component is simply the projection of the variable on the axis defined by the component.
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Projection of the variables on the factor plane revealed that the first and the second axes of the PCs explained 60.1 and 21.8 % of the total variance.
Projection of the variables on the plane F1/F2, which accounted for 34% of the total variance, showed a correlation between resistance to ciprofloxacin and type of ESBL produced.
After analyzed of the factors by projection of the variables on the factor-plane (see Additional file 4) and consulted the correlations of these factors (see Additional file 5), 11 independent ecological factors were chosen.
The two sets of variables can be associated with a pair of linear transforms (projectors) such that the correlation between the projections of the variables in lower dimensional space through these linear transforms are mutually maximized.
Also, we present in this paper a smoothed version of the CCA problem based on rank-1 matrix approximation where we impose some smoothness on the projections of the variables in order to avoid abrupt or sudden variations.
CCA finds the basis vectors for two sets of multidimensional variables such that the correlations between the projections of the variables onto these basis vectors are mutually maximized.
Projections of the variables on the plane F1/F2, which accounted for 34.5% of the total variance, showed a correlation between the type of ESBL produced and several phylogenetic group/subgroups of E. coli.
In CCA, and denote the two basis vectors, such that the correlation between the projections of the variables – columns in X and Y – onto these basis vectors given by and are mutually maximized:.
PCA leads to a linear combination (projection) of the original variables of high-dimensional data and thereby, is used to calculate orthogonal variables from raw data matrices.
The projection of the original variables on the plane of the two first PCs could clearly illustrate such relationship as shown in the plot of loadings (Fig. 2a).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com