Population Value Decomposition (PVD)
Population Value Decomposition is a concept that evolved naturally from our studies where data at the subject level is naturally organized as a matrix. Examples of such data include: 1) fMRI studies, where data can be represented as a V by T matrix, where V is the number of voxels in the brain (typically around 100K 45mm3 three dimensional areas) and T is the time in the scanner (typically in the hundreds of 2-second intervals); 2) EEG studies, where long time series can be Fourier transformed into an F by T matrix, where F is the number of frequencies (from 100 to 1000 or more, depending on the sampling rate) and T is the number of time windows where data are assumed to be quasi-stationary (from 100 to 1500 or more.)
Thus, subject specific data can be represented as a matrix Yi, and the sample of these matrices are the data. Some important characteristics of these matrices are that: a) the entries of each matrix has the same interpretation (e.g. the same voxel at the same time point across subjects); b) the two dimensions do not necessarily have the same interpretation (space by time in the fMRI example or frequency by time in the EEG example); and c) have the same dimension. Given a population of matrices Yi we are interested in understading their structure and, possibly, their association with health outcomes. The problem is that the dimensions of Yi are very large, which makes computation and visualization very difficult. One simple solution is to look for a decomposition of the type Yi = P Vi D + Ei, where the left dimension of P and right dimension of D are very large, but the dimensions of the Vi matrix are very small. The advantage of such a decomposition is that the matrices P and D do not depend on the subject and the entire variability of the data is governed by a much smaller space spanned by the entries of Vi. Thus, both prediction and variability decomposition can be done in a much smaller space. ANother advantage is that there is a one-to-one relationship between any linear model on Vi (the intrinsic space) and the the exact same linear model on Yi (the observed space.) The decomposition is not unique and various P and D matrices can be used. In our rejoinder to our discussed JASA paper on PVD we show that one could actually start with a P and D matrix, calculate the residuals Yi - P Vi D and continue the procedure. This procedure is akin to more standard model searches and would result in an "additive" PVD.
Papers that our group wrote on this topic are: Two-stage decompositions for the analysis of functional connectivity for fMRI with application to Alzheimer's disease risk and Population Value Decomposition, a Framework for the Analysis of Image Populations. Interestingly, in the comments of Lock, Nobel and Marron rit is shown that the Candecomp/Parafac and Tucker decompositions for multi-way data can be viewed as particular cases of PVD.