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Consider the following picture representing the experimental data sequence obtained by two 1D-sensors (each point of the sequence is plotted on XY plane according to the respective sensor reading):
experimental raw data

It's visually obvious that two modes have been registered. Let's assume that generally those two modes interfere, so there's no easy possibility to separate them by isolating certain sequence segments. I try the classic principal component analysis by finding the covariance matrix, then finding the set of eigenvalues and corresponding eigenvectors:
experimental data pca

White box dimensions represents the magnitude if the eigenvalues, box orientation represents the direction of eigenvectors.

It's clear that PCA first component deviates slightly from the high-magnitude mode direction, while the second component deviates greatly due to skewness of the lower-magnitude mode original direction.

It is known that PCA, being based on eigenvectors, results in orthogonal basis of primary components.

Is there other elegant methods (or PCA-derived methods) to obtain the non-orthogonal basis of primary skewed components?

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There are factor analysis techniques that allow oblique rotation, not just the orthogonal rotation that PCA uses. Take a look at direct oblimin rotation or promax rotation.

Not sure what statistical application you are using. In R, the psych and HDMD packages have commands that allow oblique rotations.

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  • $\begingroup$ Rotation is not limited to FA models: You can use it on any loadings or pattern matrix, provided it makes sense. In fact, there is a promax() function in base R which takes as input the loadings matrix, and other oblique rotations methods can be found in the GPArotation package. This is basically what principal() from the psych package uses. $\endgroup$ – chl Jan 27 '11 at 10:56

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