Proj | R Documentation |
Fitting a linear model, lm(y ~ X)
, by least squares can be thought of geometrically as the orthogonal projection of
y
on the column space of X
. This function is designed to allow exploration of projections
and orthogonality.
Proj(y, X, list = FALSE)
y |
a vector, treated as a one-column matrix |
X |
a vector or matrix. Number of rows of |
list |
logical; if FALSE, return just the projected vector; otherwise returns a list |
The projection is defined as P y where P = X (X'X)^- X' and X^- is a generalized inverse.
the projection of y
on X
(if list=FALSE
) or a list with elements y
and P
Michael Friendly
Other vector diagrams:
arc()
,
arrows3d()
,
circle3d()
,
corner()
,
plot.regvec3d()
,
pointOnLine()
,
regvec3d()
,
vectors3d()
,
vectors()
X <- matrix( c(1, 1, 1, 1, 1, -1, 1, -1), 4,2, byrow=TRUE) y <- 1:4 Proj(y, X[,1]) # project y on unit vector Proj(y, X[,2]) Proj(y, X) # project unit vector on line between two points y <- c(1,1) p1 <- c(0,0) p2 <- c(1,0) Proj(y, cbind(p1, p2)) # orthogonal complements y <- 1:4 yp <-Proj(y, X, list=TRUE) yp$y P <- yp$P IP <- diag(4) - P yc <- c(IP %*% y) crossprod(yp$y, yc) # P is idempotent: P P = P P %*% P all.equal(P, P %*% P)
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