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 yi−f(ti).
The least squares approach minimizes:
R(c1,…,cn)=m∑i=1[yi−f(ti)]2 - This can be made into a matrix problem:
r=[y1y2⋮ym−1ym]−[f1(t1)f2(t1)⋯fn(t1)f1(t2)f2(t2)⋯fn(t2)⋮f1(tm−1)f2(tm−1)⋯fn(tm−1)f1(tm)f2(tm)⋯fn(tm)][c1c2⋮cn]=b−Ax
- The linear least squares problem is then to minimize R=rTr or more generally: