LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 129

x T x

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= minimum (4.214)

This property can be used to obtain a geometric interpretation of the inner constraints.
For example, it can be shown that the approximate parameters x 0 and the adjusted
parameters x P can be related by a similarity transformation whose least-squares
estimates of translation and rotation are zero. For inner constraint solutions, the
standard ellipses show the geometry of the network and are not affected by the
definition of the coordinate system. It can also be shown that the trace of Q P is the
smallest compared to the trace of the other cofactor matrices. All minimal constraint
solutions yield the same adjusted observations, a posteriori variance of unit weight,
covariance matrices for residuals, and the same values for estimable functions of the
parameters and their variances. The next section presents a further explanation of
quantities invariant with respect to changes in minimal constraints.

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4.9 STATISTICS IN LEAST-SQUARES ADJUSTMENT

Lin
—
- ——
No
PgE

Statistics completes the theory of adjustments, because it allows one to make objec-
tiv e statements about the data. The basic requirements, however, are that the mathe-
m atical model and the stochastic model be correct and that the observations have a
m ultivariate normal distribution. Statistics cannot guarantee the right decision, but it
ca n be helpful in gaining deeper insight into often unconscious motives that lead to
ce rtain decisions.

[12

4.9.1 Multivariate Normal Distribution

Th is section contains a brief introduction to multivariate normal distribution. A few
th eorems are given that will be helpful in subsequent derivations. The multivariate
no rmal distribution is especially pleasing, because the marginal distributions derived
fro m multivariate normal distributions are also normally distributed. An extensive
tre atment of this distribution is found in the standard statistical literature. To simplify
no tation, the tilde is not used to identify random variables. The random nature of
va riables can be readily deduced from the context.
Let x be a vector with n random components with a mean of

E( x ) = µ (4.215)

an d a covariance matrix of

E ( x ) T = n Σ n

− µ )( x − µ (4.216)

If x has a multivariate normal distribution, then the multivariate density function is

f x 1 ,...,x n =

1

1 / 2 e − ( x − µ ) T

Σ − 1 ( x − µ )/ 2

(4.217)

( 2 π ) n/ 2 | Σ |