Dynamic PCA

[Nonius]

@afekz...yes, bingo. it's to look at the complement as a residual in a regression against the higher, more stable stuff; of necessity, that residual is a weighted sum of all that lower crap, but you don't necessarily explicitly compute those weights.

[ronin]

Oh I agree with that - compute what you need, cut off the rest - no argument.



The point I was making was about snapshots of non-stationary systems. Especially intraday vols are seasonal and random. In my mind the biggest problem is not taking that into account, much more than what ever time you can save by not calculating residual eigenvalues.

[tbrown122387]

saw this: https://jwindle.github.io/doc/JSM-presentation.pdf seems relevant. closed form, which is nice. never implemented it though

[Nonius]

thanks. I just thumbed through it but it looks kind of interesting.

[akimon]

I found the concept of using Variational Bayes and Auto Encoders quite fascinating, and I view it as an extension to PCA.



Basically, rather than assuming the data is representable as a linear combination of eigenvectors, we get a neural network to represent the data as a highly non-linear function of simple gaussian "Latent Vectors" (which work like PCA components).



Links to the original paper:

Auto-Encoding Variational Bayes



Link to something I did:

Generating Large Images from Latent Vectors

[doomanx]

Reviving an old thread, but I attended a talk by Bouchard yesterday where he discussed something quite similar to the issues raised here re: diffusion of eigenvectors. Link to the talk is here (slides aren't great without his soothing voice talking over them) https://krzakala.github.io/cargese.io/Bouchaud.pdf and the paper here https://arxiv.org/pdf/1603.04364.pdf. It's fairly interesting as they're taking the RMT approach but working directly with eigenvectors. My summary:



Suppose we have a random matrix model for our covariance matrix (think true * noise, where the noise is rotationally invariant). The first topic is about cleaning covariance matrices.



The object they are focusing on is overlap between the eigenvectors of the true matrix and the perturbed matrix, that is, a quantity proportional to the inner product [img]/User%20Files/12517/Latex-Equation-14555.gif[/img] where u_i is an eigenvector of the noisy cov matrix and v_j is an eigenvector of the true covariance matrix.



They first proved some analytic results that give a shrinkage function for the eigenvalues, which shows good performance on the cross validation experiments (in the case of empirical covariance matrices it's Ledoit and Peche non-linear shrinkage, but they extend this to more general RMT).



The part I though was really interesting was the second application. Suppose we now have two independent realisations from the random model for our covariance matrix. What are the overlaps between these eigenvectors? Think two sample covariance matrices computed over different intervals. Using their formula one can perform a kind of 'eyeballed' hypothesis test for whether the eigenvectors are statistically significantly different or not, and he said in the talk they're working on a formal statistical test for this problem. I think it's quite an interesting line of work.

[Maggette]

Thanks doomax, for both. The interesting Bouchaud paper and reviving the old thread. I somehow missed akimons post too. ALso quite interesting to me.

[phopstar]

The fair N day forward value should simply be the current value scaled by the risk free rate.



F = Se^{rT}