The swp function “sweeps” a matrix on the rows and columns given in index to produce a new matrix
with those rows and columns “partialled out” by orthogonalization. This was defined as a fundamental statistical operation in
multivariate methods by Beaton (1964) and expanded by Dempster (1969). It is closely related to orthogonal projection,
but applied to a cross-products or covariance matrix, rather than to data.
swp(M, index)a numeric matrix
a numeric vector indicating the rows/columns to be swept. The entries must be less than or equal
to the number or rows or columns in M. If missing, the function sweeps on all rows/columns 1:min(dim(M)).
the matrix M with rows and columns in indices swept.
If M is the partitioned matrix
$$\left[ \begin{array}{cc} \mathbf {R} & \mathbf {S} \\ \mathbf {T} & \mathbf {U} \end{array} \right]$$
where \(R\) is \(q \times q\) then swp(M, 1:q) gives
$$\left[ \begin{array}{cc} \mathbf {R}^{-1} & \mathbf {R}^{-1}\mathbf {S} \\ -\mathbf {TR}^{-1} & \mathbf {U}-\mathbf {TR}^{-1}\mathbf {S} \\ \end{array} \right]$$
Beaton, A. E. (1964), The Use of Special Matrix Operations in Statistical Calculus, Princeton, NJ: Educational Testing Service.
Dempster, A. P. (1969) Elements of Continuous Multivariate Analysis. Addison-Wesley, Reading, Mass.
data(therapy)
mod3 <- lm(therapy ~ perstest + IE + sex, data=therapy)
X <- model.matrix(mod3)
XY <- cbind(X, therapy=therapy$therapy)
XY
#> (Intercept) perstest IE sexM therapy
#> John 1 26 3 1 32
#> Susan 1 24 4 0 40
#> Mary 1 22 8 0 44
#> Paul 1 33 4 1 44
#> Jenny 1 27 6 0 48
#> Rick 1 36 4 1 52
#> Cathy 1 30 10 0 56
#> Robert 1 38 4 1 56
#> Lisa 1 30 12 0 60
#> Tina 1 34 15 0 68
M <- crossprod(XY)
swp(M, 1)
#> (Intercept) perstest IE sexM therapy
#> (Intercept) -0.1 30 7 0.4 50
#> perstest 30.0 250 10 13.0 300
#> IE 7.0 10 152 -13.0 304
#> sexM 0.4 13 -13 2.4 -16
#> therapy 50.0 300 304 -16.0 1000
swp(M, 1:2)
#> (Intercept) perstest IE sexM therapy
#> (Intercept) -3.70 0.120 5.80 -1.160 14.0
#> perstest 0.12 -0.004 0.04 0.052 1.2
#> IE 5.80 0.040 151.60 -13.520 292.0
#> sexM -1.16 0.052 -13.52 1.724 -31.6
#> therapy 14.00 1.200 292.00 -31.600 640.0