2.3 Total Least-Squared Estimation

A generalization of the LSE estimation technique discussed in Section 2.2 allows for the existence of noise in not only the measurements x _k in (2.3), but also for errors in the observation matrix H _k. When there is no noise in H _k, and the noise represented by n _k ~ N(0, ? ²) (this notation means that the noise is normal, or Gaussian, with zero mean and variance ? ²), then the LSE solution is the same as the maximum likelihood estimate. However, when noise is present in H _k, (2.17) with W _k = I is no longer optimal. It exhibits bias and increased covariance. To determine the optimal LSE estimate in this case, the method of TLSE estimation was developed [4]. The subscript k will be dropped here for notational convenience.

As described in Section 2.2, the LSE estimate is obtained by finding by calculating

(2.38)

where ? z ? ₂ is the L ₂ norm of z. The solution to (2.38) is given by

(2.39)

where H is the pseudoinverse of H _{M P}. ^[1] It is typically assumed that M > P and that H has full rank, so that H = (H ^*H) ^-1 H ^* and

(2.40)

The measurement vector and observation matrix can be expressed as

(2.41)

where ? x