In sci.stat.edu Eric Zivot <[EMAIL PROTECTED]> wrote: : In the finance literature, it is common to do pca in a situation in which : there are more variables than observation (e.g returns on 1000 assets and : 500 observations). This says alot about the finance literature
In this case, one uses what has been called "asymptotic : principal component analysis". In stead of eigenvalue analysis on the : non-invertible N x N covariance RR' (R is N x T, N >> T), do eigen value : analysis on the smaller T x T matrix R'R. There is nothing asymptotic about it; it is simple mathematics (A'A)x = kx (x an eigen vector, k the value) (AA')(Ax) = k(Ax) the eigenvalues are the same and you solve for the original vectors. so what? it's still dumb to do anything with more variables than observations I describe this case in my recent : book Modeling Financial Time Series. The standard reference in finance is : Chamberlain and Rothschild (1983) "Arbitrage, Factor Structure and : Mean-Variance Analysis in Large Asset Markets", Econometrica, 51. : "Ronald Bloom" <[EMAIL PROTECTED]> wrote in message : ar469c$8ul$[EMAIL PROTECTED]">news:ar469c$8ul$[EMAIL PROTECTED]... :> In sci.stat.consult Paige Miller <[EMAIL PROTECTED]> wrote: :> > Ronald Bloom wrote: :> >> Suppose I have more observations than I have variables. :> >> E.g. Say 51 variables and 30 observations apiece. :> >> :> >> The data matrix is shall we say "wide" (columns correspond :> >> to variables). :> >> :> >> The resulting sample covariance matrix on the covariance among :> >> variables is necessarily rank deficient. :> >> :> >> Is there anything useful I can learn nevertheless from :> >> that matrix? :> >> :> >> E.g.: :> >> Is there anything to be garnered from a principle components :> >> decomposition of this covariance matrix in this circumstance? :> :> > Yes. Go for it. :> :> :> So let's say I have 51 variables. I collect them 30 days at a time. :> :> Every time I collect a new set of observations I want to assess :> its "surprise value", in respect of the recent past. :> :> So I have a monthly covariance matrix of rank 30. There should be :> 30 non-zero eigenvalues. If I get one more observation (of 51 items) :> can I map it into the reduced dimension subspace of size 30, :> and evaluate it for "extremity" using a multivariate prediction :> "region" using the 30x30 diagonal matrix of eigenvalues in :> the standard "hotelling form" on the reduced space? :> :> I'm just trying to translate theory into something I can :> actually practise. :> :> -Ron :> :> > -- :> > Paige Miller :> > [EMAIL PROTECTED] :> > http://www.kodak.com :> :> > "It's nothing until I call it!" -- Bill Klem, NL Umpire :> > "When you get the choice to sit it out or dance, I hope you dance" -- :> > Lee Ann Womack :> . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
