Some clarifications on eigenvectors and eigenvalues from PCA

Question

Could somebody tell me whether suggestions in bold true or not?

Q # 1: http://www.math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb071108.pdf

Chapter 2.2 Interpretation of the eigenvectors/eigenportfolios

This paper says that loadings in the maximal eigenvector need to be all positive and should not change sign, what if i have negative ones, can i force them to be positive always by simply taking them by module, e.g. MathAbs(Vector) ?

Q # 2 : The same paper also defines weights for eigenportfolio in this way :

Q[i] = EigVecCoef[i] / StdDev[i] // Page # 10 paragraph 2.2 in the doc above

There is also another paper that says that eigenvector is an angle (or direction) of the portfolio's spread which allows to map current portfolio's spread to initial axes (dimensions) :

http://georgemdallas.wordpress.com/2013/10/30/principal-component-analysis-4-dummies-eigenvectors-eigenvalues-and-dimension-reduction/

So i do not understand - why do i need to divide each value in eigenvector by standard deviation to calculate weights if this portfolio is already mapped to initial axes?

Answer 1

Q #1: I'm not sure if you have the answer quite right. The signs for the loadings are arbitrary, but you cannot take the absolute value. You can multiply by -1.

Q #2: It might be helpful to think about what PCA is actually doing. This paper might be helpful: http://arxiv.org/pdf/1404.1100v1.pdf (A Tutorial on Principal Component Analysis by Jonathon Shlens). The key point is towards the end, where the author explains the relationship of SVD with PCA. "We can conclude that finding the principal components amounts to finding an orthonormal basis that spans the column space..." of the data matrix. But this quote is referring to finding the principal components using the covariance matrix. The authors of the paper you attached use the correlation matrix. This is essentially finding a set of variables that spans the space of scaled returns. Thus you need to scale the weights of your eigenvectors.

You may also find this piece helpful - it seems to discuss a similar application with a little more derivation: Some clarifications on eigenvectors and eigenvalues from PCA

Answer 2

A # 1 : Several replies from the following topic answer my Q # 1 - yes, if I take only one dimension after PCA then I can simply make all vectors positive, hence take only absolute values by module.

If I take several dimensions then entire vector needs to be reverted and each value inside particular vector can be multiplied by -1 because reversion of orthogonal vectors does not change their meaning, the only requirement in this case is if one value in vector was changed then all other need to be changed too.

Time series of PCA - Sign change in factor loadings