Description Usage Arguments Details Value Author(s) See Also Examples

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.

1 |

`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`

, `corner`

,
`plot.regvec3d`

, `pointOnLine`

,
`regvec3d`

, `vectors3d`

,
`vectors`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
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)
# orthogonal complements
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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