Perform Pearson Aitken selection

Description

These functions implement the Pearson Aitken selection formulae.

Usage

mxPearsonSelCov(origCov, newCov)
mxPearsonSelMean(origCov, newCov, origMean)

Arguments

Details

Both mxPearsonSelCov and mxPearsonSelMean match the dimnames of newCov and origCov to determine which partition of origCov to replace with newCov.

Let the n \times n covariance matrix R (origCov) be partitioned into non-empty, disjoint sets p and q. Let R_{ij} denote the covariance matrix between the p and q variables where the subscripts denote the variable subsets (e.g. R_{pq}). Let column vectors μ_p and μ_q contain the means of p and q variables, respectively. We wish to compute the conditional covariances of the variables in q for a subset of the population where R_{pp} and μ_p are known (or partially known)—that is, we wish to condition the covariances and means of q on those of p. Let V_{pp} (newCov) be an arbitrary covariance matrix of the same dimension as R_{pp}. If we replace R_{pp} by V_{pp} then the mean of q (origMean) is transformed as

μ_q \to μ_q + R_{qp} R_{pp}^{-1} μ_p

and the covariance of p and q are transformed as

≤ft[ \begin{array}{c|c} R_{pp} & R_{pq} \\ \hline R_{qp} & R_{qq} \end{array} \right] \to ≤ft[ \begin{array}{c|c} V_{pp} & V_{pp}R_{pp}^{-1}R_{pq} \\ \hline R_{qp}R_{pp}^{-1}V_{pp} & R_{qq}-R_{qp} (R_{pp}^{-1} - R_{pp}^{-1} V_{pp} R_{pp}^{-1}) R_{pq} \end{array} \right]

References

Aitken, A. (1935). Note on selection from a multivariate normal population. Proceedings of the Edinburgh Mathematical Society (Series 2), 4(2), 106-110. doi:10.1017/S0013091500008063

Examples

library(OpenMx)

m1 <- mxModel(
  'selectionTest',
  mxMatrix('Full', 10, 10, values=rWishart(1, 20, toeplitz((10:1)/10))[,,1],
           dimnames=list(paste0('c',1:10),paste0('c',1:10)), name="m1"),
  mxMatrix('Full', 2, 2, values=diag(2),
           dimnames=list(paste0('c',1:2),paste0('c',1:2)), name="m2"),
  mxMatrix('Full', 10, 1, values=runif(10),
           dimnames=list(paste0('c',1:10),c('v')), name="u1"),
  mxAlgebra(mxPearsonSelCov(m1, m2), name="c1"),
  mxAlgebra(mxPearsonSelMean(m1, m2, u1), name="u2")
)

m1 <- mxRun(m1)