# How to do primary component analysis on multi-mode data with non-orthogonal primary components?

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): 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: 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?

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.

• 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.
– chl
Jan 27 '11 at 10:56

Independent component analysis is suitable for separating non-orthogonal basis. Check out this paper. I guess figure 1 is what you want. Choi S. (2009) Independent Component Analysis. In: Li S.Z., Jain A. (eds) Encyclopedia of Biometrics. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-73003-5_305