# Discussion on random matrix theory and impact on PCA

I've written a paper for university on Random Matrices and during my research I've had an interesting idea, let me explain: Wigner's Semicircle Law has seen much advancement since its original proof in 1955, most recently I believe being Tao's proof of the Wigner-Gaudin-Mehta-Dyson conjecture showing universality. Now here's the leap, much of big data is reliant on Principle Component Analysis, or decomposing data into their respective eigenvalues and eigenvectors. Then we compare the results to similar datasets to see if there are correlations. However if we treat stock prices as Brownian motions, i.e. iterated random processes with eigenvalues and eigenvectors tending to the circular law, then doesn't that inherently create a bias in our comparison of the eigenvectors w.r.t other iterated random processes.

For example a group of commodities stocks in agriculture and another in mining we assume aren't correlated, but after batching and PCA they share similar normed eigenvalues. Is this not in part due to the fact they share the same distributive law at least for large enough batches and repetitive samplings? If so are there already methods or hypothesis tests that filter through this?

It was just a thought and I don't really have many people to discuss this idea with seeing as I'm stuck at home. I may be wrong on how PCA works or how financial products are correlated as I'm not in the field.

EDIT: I feel like some further context is needed since this isn't a result familiar to most.

From RMT, eigenvalues have a semicircle distribution for symmetric matrices each with i.i.d normally distributed entries. The restrictions on i.i.d have recently been shown to not matter so we can proceed nevertheless. If we take a covariance matrix of all stock tickets beginning in A comparing the average daily return over a period of time, each we can assume lognormal distribution forming a, lets say 10000 by 10000 symmetric matrix. Thus we get a sequence of random covariance matrices $$\Gamma_1, \Gamma_2, ..., \Gamma_n$$. Each of the entries we assume to be i.i.d since the stocks have 'nothing' to do with each other (although a weaker results holds for non-i.i.d entries). Now this series of matrices forms a chain of covariance matrices tending towards the underlying covariance matrix of the entire history of the stocks (if we sampled correctly). We known from RMT that once we decompose these matrices into their eigenvalues, the eigenvalues tend to the semicircle distribution. Since this distribution is continuous, there's a spread in results i.e. there is some underlying variance to the eigenvalue decomposition of covariance matrices. Thus when we use covariance matrices shouldn't there be some sort of hypothesis test that's able to filter out this underlying distribution, similar to comparing normal distributions where we need to account for variance when comparing two mean values. This would be dependent on how i.i.d the random variables are, the size of the matrix, the number of samples taken, and the mean/variance of the random variables themselves. What would be weird about this hypothesis test is that we'd expects as $$n$$ gets larger so does the error bound, capture by the asymptotic relationship between the size and convergence to the semicircle distribution.

TLDR: Is there some sort of hypothesis testing for PCA, or any eigenvalue method, that filters out the underlying tendency of random covariance matrices so as to account for the variance? Similar to how when you compare the mean of two normal distributions you need to perform a hypothesis test to account for the variance.

Also the more I write about this, the more I feel as though this is more related to data science as opposed to quantative finance as I realise my examples don't seem to fit very well.

• Hi, this is a bit of a follow up on your question and @mark leeds' answer: If I remember correctly, couldn't you also do some rotations in factor space after the PCA - which would add to the identification problem (in search of a better word)? Commented Feb 11, 2021 at 18:23
• Well such a translation would be isomorphic so the distribution would stay the same, i.e. not eliminating the trends of the eigenvalue towards their distribution. Commented Feb 11, 2021 at 18:37

Hi: I don't follow your question totally but I can comment on one aspect of it. ( So this is not an answer ). The ideas that A) stock returns are geometric brownian motion processes and B) that PCA captures some kind of similarity in stocks from two different sectors are pretty much two different things.

A) comes from efficient markets theory where it is posited that $$ln(P_t) = ln(P_{t-1} + \epsilon_t$$. ( random walk which, in continuous time is a brownian motion ).

B) comes more from economics-investment theory where it is assumed that a stocks return has various components due to its fundamental characteristics and one of these components is the "market" factor. Factor models are used to break down a stock return into factors and factor loadings. The fact that the "market" drives part of a stock's return is usually termed the "market" factor in say a PCA.

So, my point is that A) and B) are two pretty diffetrent concepts so I wouldn't lump them together. A) would be discussed in any decent derivatives text like say Hull's. ( other books are available also that get more into the math of difficusion processes etc ). B) would be discussed in a financial-econometrics text like say Zivot's or (Rudd and Clasing's). Also, an investments book like that of William Sharpe.

That's all I can say but hopefully it helps a little because, based on your question, it sounded like you were combining the two concepts and this could lead to some confusion.

• Thank you for segmenting the two ideas however the devil's advocate in me would say that the two are linked, i.e. the randomness of A emerges as a consequence of the variance of analysis for B attributing different weights to different factors, whether it be institutional growth investing or a hedge fund applying technical investing to the market. These differences in view in turn cause variance which is where the randomness comes from. Commented Feb 11, 2021 at 18:35
• Hi John: It's an interesting concept you bring up but, usually or atleast AFAIK, when people talk about stock returns being "random" , they're referring to the raw returns before any modelling is done. With the respect to factor models, the variance that you are referring to is called the "residual risk" or specific variance of the stock. Check out rudd and clasing if you can get your hands on it. I remember that book being pretty good back in the day. It's quite old now but these concepts don't change that much. Commented Feb 12, 2021 at 15:46
• As a definition that captures what randomness, at least in mathematics, is. However why we choose to add randomness is in part due to the number of actors in the market and the difference in those strategies. If everyone followed the same strategy we wouldn't model the return of a product randomly. Thanks for the recommendation, I'll take a look at the book; I need to read up on the fundamentals before I seriously consider these ideas. Commented Feb 12, 2021 at 16:33
• Hi John: as far as fundamentals, barra used everything in the kitchen sink: size, price to earningfs, capitalization, indistry, liquidity, price to book, volatility, and everything else you can think of that could possibly qualify as a factor. All told ( this is back in mid 90's, don't know if it changed since ), they use 13 factors and 5 industry weightings. The weightings were for when a company was considered to be in two or more industries. I think Rudd and Clasing discuss this some. Factor models are also discussed in Eric Zivot's Splus-Finmetrics book and he gives some examples. Commented Feb 12, 2021 at 22:45
• Also, John: if you have an edition of "Multivariate Analysis" by Johnson and Wichern, there's a PCA example in there that arrives at "the market" as a the first significant factor in a PCA of stock returns. That's all I remember ( all my books are in storage ) but that might be worth looking at. Commented Feb 13, 2021 at 1:08

I must too confess ignorance about Wigner-Kermit-Ringo processes :-) But I do know about PCA, and iterative reductive market processes,,,

I suspect (but cannot hope to prove) that you are posing a false opposition here? Yes, grains and metals are correlated. So associated stocks (eg Deere and Rio Tinto) will indeed appear linked under PCA analysis. As indeed they probably are, looking at these two and oilcos against say FANG, Microsoft and Tesla!

If you accept a statistically significant difference between these groups, getting cute about the difference between grains and industrial metals is indeed cute. your PCA might just be suggesting a difference between “old economy” (inc ALL commodities) and “new economy” tech.

So the nature of the problem isn’t clear to me... the beta to Ags might indeed be very different to the beta to Copper and Iron Ore. but that’s a distinction that maybe PC3, 4 or 5 draws out, once it has separated the resources from the Tech (and the Financials, the Consumer etc.).

Yes, both are eigen-plays. As such, they should come to identical solutions, maybe via different paths. But the underlying decomposition of returns is the same process. The key “difference” I can see is that PCA has to separate out non-Comms before it starts to worry about the difference between different kinds of Comms.

I might well have missed the point here. Sorry if so!