8
¥Solving the Linear System : Y = Xb + e
_the basic goal of 3dDeconvolve
_Least Square Estimate (LSE): making sum of squares of residual (unknown/unexplained) error eÕe minimal ˆ Normal equation: (XÕX)
b = XÕY
_When X is of full rank (all columns are independent), b^ = (XÕX)-1XÕY
¥Geometric Interpretation:
_project vector Y onto a space spanned by the regressors (the column
vectors of design matrix X)
_find shortest distance from Y to X-space
Y
e
X-Space
r1
r2
Xb
0