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). 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. 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/ . =================================================================
