Projection of Vector y on columns of X

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.

Usage

Proj(y, X, list = FALSE)

Arguments

y

a vector, treated as a one-column matrix

X

a vector or matrix. Number of rows of y and X must match

list

logical; if FALSE, return just the projected vector; otherwise returns a list

Value

the projection of y on X (if list=FALSE) or a list with elements y and P

Details

The projection is defined as \(P y\) where \(P = X (X'X)^- X'\) and \(X^-\) is a generalized inverse.

See also

Other vector diagrams: arc(), arrows3d(), circle3d(), corner(), plot.regvec3d(), pointOnLine(), regvec3d(), vectors(), vectors3d()

Author

Michael Friendly

Examples

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
#> [1] 2.5 2.5 2.5 2.5
Proj(y, X[,2])
#> [1] -1 -1  1  1
Proj(y, X)
#> [1] 1.5 1.5 3.5 3.5

# 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))
#> [1] 1 0

# orthogonal complements
y <- 1:4
yp <-Proj(y, X, list=TRUE)
yp$y
#> [1] 1.5 1.5 3.5 3.5
P <- yp$P
IP <- diag(4) - P
yc <- c(IP %*% y)
crossprod(yp$y, yc)
#>      [,1]
#> [1,]    0

# P is idempotent:  P P = P
P %*% P
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.5  0.5  0.0  0.0
#> [2,]  0.5  0.5  0.0  0.0
#> [3,]  0.0  0.0  0.5  0.5
#> [4,]  0.0  0.0  0.5  0.5
all.equal(P, P %*% P)
#> [1] TRUE