3. Least-Squares Estimation
If the preceding functional and stochastic models are correct, a least-squares method yields the adjusted parameters/measurements that are the best linear unbiased estimates (Koch, 1999).
3.1 Parameter Corrections and Measurement Residuals
To begin with, the nonlinear equations (1-4) have to be linearized to form a system of error equations, of which the expansion point is at the available measurements and parameter approximations:
where the n*n coefficient matrix B contains the partial derivatives of Eqs. (1-4) with respect to the n*1 measurement vector
(...,r
i,t
j,...). The n*1 vector v stands for the measurement residuals
(...,
Vr
i,
Vt
j,...).
Analogously, the n*u design matrix A contains the partial derivatives with respect to the u unknown parameters.
The u*1 vector x represents the parameter corrections
(dt,dM
b,da
k,db
k,dc
c) with k = 0, 1, 2, 3.
The n*1 vector l is the reduced observation vector.
The measurement error covariance matrix is denoted by
s02, where
s02Q is the a priori unit weight reference variance. A least-squares method, which requires a minimization of the quadratic form
v
TQ
-1v, produces the parameter corrections and the measurement residuals, as follows:
where the u*u scaled covariance matrix
Q
x and the n*n scaled covariance matrix Q
v refer to the parameter correction vector x and the measurement residual vector v, respectively. The covariance matrices can be obtained by using the law of error propagation. Both Leick (1995) and Mikhail (1976) detailed the derivation of
Q
x and Q
v so that their explicit expressions are not repeated here.
The v-vector quadratic form can lead to the a posteriori estimate
s02 of a unit weight reference variance. The same v-quadratic form also leads to a chi-square
(
c2) test statistic:
where n-u represents the degree of freedom.
a is a chosen significance level, e. g. at 5%,
used to create the lower and upper bounds of the inequality equation (9),
in the course of a global model hypothesis testing.
If this testing fails, an analyst can state that, with a 1-
a confidence level, the functional and stochastic models (1-4) are not in order.
3.2 Optimal Parameter Selection
Another statistical testing can be utilized to validate parametric significance when trajectory polynomial modelings, such as described in Eq. (4), are involved. For any element x of a parameter correction vector x (7a), an F-distribution test statistic can be given as the term on the left-hand side of the following inequality equation (Zhong, 1997):
where q
x denotes the scaled variance of x. The F-distribution has (1, n-u) degrees of freedom.
Upon choosing a significance level
a, the upper critical value
F
1-a;1,n-u is read from an F-distribution look-up table.
If the test quantity
fulfills the inequality relationship (10), this parameter element x is considered to be insignificant. After its deletion, the new parameter set will have u-1 elements. The measurement vector and its error covariance matrix remain unchanged.
A repeated least-squares adjustment is performed by using the algorithmic equations (6-7). New u-1 parameter corrections and new n measurement residuals are estimated. Their acceptance is based on the required global model test, as indicated in Eq. (9). The following minimum criteria, all related to the quadratic form of the estimated measurement residuals (Zhong, 1997), serve as the optimization indices in order to distinguish between the old/previous and new/current estimation results:
where Eq. (11a) is identical to Eq. (8). If the new model has a better performance than the old one, the search for another possible insignificant parameter continues. This means that the significance testing, Eq. (10), will be invoked. If, on the other hand, the previous model produces more optimal results than the current model does, the old/previous functional model represents the sought-after model solution.
