Ordinary least squares through the normal equations

Formula: (X'X) beta = X'y; beta = (X'X)^-1 X'y; Var(beta) = sigma2 (X'X)^-1; H = X(X'X)^-1 X'

Usage

Olsnormeq(X, y, add_intercept = TRUE)

Arguments

X

Design matrix, one record per row.

y

Response vector of length n.

add_intercept

Prepend a column of ones to X.

Value

List with beta, fitted, resid, rss, sigma2, se, leverage, n, p.

References

Montesinos Lopez, Montesinos Lopez and Crossa (2022), Multivariate Statistical Machine Learning Methods for Genomic Prediction, Springer, doi:10.1007/978-3-030-89010-0. Chapter 3, Sect. 3.2 pp. 72-73: setting the gradient of the residual sum of squares to zero gives the normal equations (X'X)beta = X'Y with unique solution beta = (X'X)^-1 X'y, variance sigma2 (X'X)^-1, and hat matrix H = X(X'X)^-1 X'. sigma2 is the unbiased residual variance RSS/(n - p). Read from the chapter PDF, not recalled.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Olsnormeq(V, V)
#> $beta
#> [1] 6.097555e-16 1.000000e+00
#> 
#> $fitted
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $resid
#> [1] -2.220446e-16  0.000000e+00  4.440892e-16  8.881784e-16  8.881784e-16
#> [6]  8.881784e-16  1.776357e-15  1.776357e-15
#> 
#> $rss
#> [1] 8.923989e-30
#> 
#> $sigma2
#> [1] 1.487331e-30
#> 
#> $se
#> [1] 9.502751e-16 1.881825e-16
#> 
#> $leverage
#> [1] 0.4166667 0.2738095 0.1785714 0.1309524 0.1309524 0.1785714 0.2738095
#> [8] 0.4166667
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 2
#> 
#> $method
#> [1] "OLS via the normal equations, MVSML Sect. 3.2"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"