Elliot is entitled to his opinion. It may be senseless to him be it is not to others who have to deal with the data that they are given. The statistics/finance literature is clear about usefullness of what I described as asymptotic pca. If the data are generated by a factor structure then eigenvalue analysis on the T x T covariance matrix (where N >> T) recovers the factor structure. Read the published papers. If you don't like it, then write a paper saying its useless and get it published. "Elliot Cramer" <[EMAIL PROTECTED]> wrote in message [EMAIL PROTECTED]">news:[EMAIL PROTECTED]... > 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 > :> > >
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