Principal component analysis maximum variance directions

An often-overlooked issue is the inherent non-orthogonality of coordinate systems used to portray data points. Almost universally a Euclidean coordinate system is used. This assumes that the original variables are orthogonal, that is, are uncorrelated, when it is well known that this is generally not the case. Typically, principal component analysis (PCA) is performed to generate a putative orthogonal coordinate system each of whose axes correspond to directions of maximum variance in the transformed space. This, however, is not quite cor-... 

The method of PLS bears some relation to principal component analysis instead of Lnding the hyperplanes of maximum variance, it Lnds a linear model describing some predicted variables in terms of other observable variables. It is used to Lnd the fundamental relations between two matrices (X andY), that is, a latent variable approach to modeling the covariance structures in these two spaces. A PLS model will try to Lnd the multidimensional direction irMIspace that explains the maximum multidimensional variance direction in flrfspace. 

A principal component analysis is reasonable only when the intrinsic dimensionality is much smaller than the dimensionality of the original data. This is the case for features related by high absolute values of the correlation coefficients. Whenever correlation between features is small, a significant direction of maximum variance cannot be found (Fig. 3.7) all principal components participate in the description of the data structure hence a reduction of data by principal component analysis is not possible. 

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