Least Squares

  • More generally, linear least-squares problems have the form:

f(t)=c1f1(t)++cnfn(t)

Where the function fi are all known functions.

  • The fit will only be approximate, with residuals yif(ti).

  • The least squares approach minimizes:

R(c1,,cn)=mi=1[yif(ti)]2 - This can be made into a matrix problem:

r=[y1y2ym1ym][f1(t1)f2(t1)fn(t1)f1(t2)f2(t2)fn(t2)f1(tm1)f2(tm1)fn(tm1)f1(tm)f2(tm)fn(tm)][c1c2cn]=bAx

  • The linear least squares problem is then to minimize R=rTr or more generally:

3.1.1 Defintion {-} 3.1.3:

Given ARm×n and bRm, with m>n, find:

argminxRn