Least Squares and Weighted Least Squares Estimates - Understanding GPS: Principles and Applications - page 664

and then setting this quantity equal to zero to obtain:

( )

− 1

$

T T

xHHHy = (A.5)

where it is assumed that the matrix inverse involved exists (i.e., that H T H is not sin-
gular).
The estimate described by (A.5) is referred to as a least squares estimate , since,
as shown in (A.3), it results in the minimum square error between the measurement
vector y and Hx , where the latter is the expected measurement vector based upon the
estimate of x .
Next, consider the more general case where the measurement errors are still
Gaussian distributed with zero-mean but are not necessarily identically distributed
or independent of each other. In this case, the maximum likelihood estimate can be
expressed as

1
2

1

( )

T − 1

( ) − yHx R yHx − −

$

n

x = arg max e

()

12

N 2

x

(A.6)

2 π R

n

( )

T

( )

− 1

= arg min yHxR −

n yHx

−

x

where R n is the covariance matrix associated with the measurement errors and | R n |is
its determinant.
Proceeding as before, (A.6) can be solved to yield:

( )

− 1

$

xHRHHRy
n

=

T − 1 T − 1

(A.7)

n

The estimate in (A.7) is referred to as a WLS solution.

Reference

[1] Stark, H., and J. W. Woods, Probability, Random Processes, and Estimation Theory for

Engineers , Englewood Cliffs, NJ: Prentice-Hall, 1986.