project-methods: Projections

In mosaic: Project MOSAIC Statistics and Mathematics Teaching Utilities

Projections

Description

Compute projections onto the span of a vector or a model space, dot products, and vector lengths in Euclidean space.

Usage

project(x, ...)

## S4 method for signature 'formula'
project(x, u = NULL, data = parent.frame(2), coefficients = TRUE, ...)

## S4 method for signature 'numeric'
project(x, u = rep(1, length(x)), type = c("vector", "length", "coef"), ...)

## S4 method for signature 'matrix'
project(x, u, data = parent.frame())

vlength(x, ...)

dot(u, v)

Arguments

x	a numeric vector (all functions) or a formula (only for project). Left-hand sides of formulas should be a single quantity
...	additional arguments
u	a numeric vector
data	a data frame.
coefficients	For project(y ~ x) indicates whether the projection coefficents should be returned or the projection vector.
type	one of "length" or "vector" determining the type of the returned value
v	a numeric vector

Details

project (preferably pronounced "pro-JECT" as in "projection") does either of two related things: (1) Given two vectors as arguments, it will project the first onto the second, returning the point in the subspace of the second that is as close as possible to the first vector. (2) Given a formula as an argument, will work very much like lm(), constructing a model matrix from the right-hand side of the formula and projecting the vector on the left-hand side onto the subspace of that model matrix.

In (2), rather than returning the projected vector, project() returns the coefficients on each of the vectors in the model matrix. UNLIKE lm(), the intercept vector is NOT included by default. If you want an intercept vector, include +1 in your formula.

Value

project returns the projection of x onto u (or its length if u and v are numeric vectors and type == "length")

vlength returns the length of the vector (i.e., the square root of the sum of the squares of the components)

dot returns the dot product of u and v

See Also

link{project}

Examples

x1 <- c(1,0,0); x2 <- c(1,2,3); y1 <- c(3,4,5); y2 <- rnorm(3)
# projection onto the 1 vector gives the mean vector
mean(y2)            
project(y2, 1)
# return the length of the vector, rather than the vector itself
project(y2, 1, type='length')
project(y1 ~ x1 + x2) -> pr; pr
# recover the projected vector 
cbind(x1,x2) %*% pr -> v; v
project( y1 ~ x1 + x2, coefficients=FALSE )
dot( y1 - v, v ) # left over should be orthogonal to projection, so this should be ~ 0
if (require(mosaicData)) {
project(width~length+sex, data=KidsFeet)
}
vlength(rep(1,4))
if (require(mosaicData)) {
m <- lm( length ~ width, data=KidsFeet )
# These should be the same
vlength( m$effects )  
vlength( KidsFeet$length)
# So should these
vlength( tail(m$effects, -2) )
sqrt(sum(resid(m)^2))
}
v <- c(1,1,1); w <- c(1,2,3)
u <- v / vlength(v)  # make a unit vector
# The following should be the same:
project(w,v, type="coef") * v 
project(w,v)
# The following are equivalent
abs(dot( w, u ))
vlength( project( w, u) )
vlength( project( w, v) )
project( w, v, type='length' )

mosaic documentation built on May 29, 2024, 5:27 a.m.