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Least Squares Adjustment
1. Mixed Model
f ( x, l )  0
f is made of m equations, x of u unknowns and l of n observations.
This model is also known as implicit model.
1.1 Functional Model
The model must be linearized around x 0 approximate values for the unknowns and l
observed values for the observations.
f ( xˆ, lˆ)  f ( x 0 , l ) 
f
f
( xˆ  x 0 ) 
x x 0 , l
l
(lˆ  l )    0
x0 , l
  v  w  0
With
w  f ( x0 , l )
f

x x 0 , l
  xˆ  x 0
f

l x 0 , l
v  lˆ  l
misclosure vector
first design matrix
unknown vector (for the linearized problem)
second design matrix
residual vector
1.2 Stochastic Model
The observations are characterized by their variance-covariance matrix Cl . Because
sometimes the exact covariance matrix is not known, a scale factor is introduced and two
new matrices are derived from the covariance matrix: the cofactor matrix Ql and the
weight matrix P.
C
Cofactor matrix Ql  2l
0
Weight matrix P   02 Cl1
 02 is called variance factor or variance of unit weight or apriori variance factor.
1.3 Least Squares Solution
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Find δ such as   v  w  0 and v t v minimum.
Use Lagrange multipliers k and variation function: ( , v, k )  vt v  2k t (  v  w)

 2v t   2 k t   0
v

 2k t   0


 2 t t  2v t t  2wt  0
k
By dividing by 2 and transposing
  t

 0
 0 t

0


0 
 v   0  0 
 k    w   0 
     
   0  0
By eliminating v
  1t

t
 
  k  w 0
   
0     0  0
By eliminating k
t ( 1t ) 1   t ( 1t ) 1 w  0
Giving the unknowns   (t (1t )1 )1 t ( 1t )1 w
Or     1u with   t ( 1t )1  and u  t (1t )1 w
Or again     1u with   t  1 and u  t  1w assuming   1t
Replacing δ into   1t k    w  0 gives k
k  ( 1t )1 (   w)   1 (   w)
Replacing k into v  t k  0 gives v
v  1t (1t )1 (  w)  1t  1 (  w)
The adjusted quantities are now simply
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xˆ  x 0  
lˆ  l  v
The cofactor matrices of all the adjusted quantities can be derived starting with the
cofactor matrix of the misclosure vector w  f ( x0 , l ) . Applying the general law of
variance-covariance propagation and noting that the jacobian of w is simply the second
design matrix leads to:
Qw   Ql t    1 t  
The cofactor matrix of the linearized unknowns  is obtained by applying the law of
variance-covariance propagation to the re-written formula:
  (t ( 1t ) 1 ) 1 t ( 1t )1 w  (t  1) 1 t  1w
Q  ( t  1) 1 t  1Qw ( 1( t  1) 1 )  ( t  1) 1
Because xˆ  x 0   , the cofactor matrix of the adjusted parameters is equal to the cofactor
matrix of the linearized unknowns:
Qxˆ  Q
Similarly, the cofactor matrix of the residuals can be computed through the law of
variance-covariance propagation to:
v  1t (1t )1 (  w)  (1t  1( t  1)1 t  1  1t  1 )w
Qv  Ql t  1 Ql  Ql t  1 1t  1 Ql  Ql t  1 (   1t  1 )  Ql
Computing the cofactor matrix for the adjusted observation lˆ  l  v requires the
derivation of the jacobian of lˆ and some more matrix multiplications to obtain:
Qlˆ  Ql  Qv
Finally, the computation of the minimum of the weighted residuals v t v needs to be
performed to complete the least squares analysis.
Recalling that v  (1t  1(t  1)1 t  1  1t  1 ) w leads to:
vt v  wt ( 1   1( t  1) 1 t  1 ) w
The expected value of this quantity is:
 (vt v)   (Tr (vt v))   (Tr ( wt ( 1   1( t  1)1 t  1 ) w)) 
 (Tr (( 1   1( t  1)1 t  1 ) wwt ))  Tr ( ( 1   1( t  1)1 t  1 ) ( wwt ))
By definition of the variance covariance matrix and by noting that  (w)  
 ( wwt )  Cw  ( w) ( w)t   02 Qw   t t
So
 (v t v)   02 Tr (    1( t  1) 1 t )   02 (m  u )
The difference m-u is called the degree of freedom and is equal to the number of
redundant equations in the model. To be more exact, the degree of freedom is m-rank(A)
but this will be covered later. The a posteriori variance of unit weight is:
vt v
̂ 02 
mu
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For the situation of  02 unknown, ˆ 02 is used to rescale the covariance matrices for
statistical testing purposes.
2. Parametric Model
A more common model is given by l  f (x)
It is also called observation equation model as each observation has its own equation.
2.1 Least Squares Solution
This model is a simplified version of the mixed model with   
So using the same notation as above:
v    w
Qw  Ql   1
w  f ( x0 )  l
  (t )1 t w Q  ( t ) 1
Generally the term normal matrix and the notation N are used leading to the well known
formula:
    1t w   1u
Qxˆ  Q   1
Qv  Ql   1t
Qlˆ  Ql  Qv   1t
It is useful to rewrite the formula for the residuals as follows:
v    1t  w  w  (  1t   1 ) w  Qv  w
The matrix Qv  is symmetrical and idempotent. Due to the last property, the trace of the
matrix is equal to its rank, so the redundancy of the adjustment problem can be written
as:
r  trace (Qv )
The diagonal element i of this matrix Qv  is noted ri and represents the contribution of
the observation i to the overall system redundancy.
2.2 Statistical Tests
Until now, there was no assumption on the distribution of the observations. Now, we will
assume that they follow a normal distribution and we will introduce the concept of
standardized residuals.
v
The standardized residual of the observation li is yi  i where vi is the regular residual
v
i
and  vi is the square root of the i diagonal element of the variance covariance matrix
th
Cv   02 Qv . Because of the assumption on normality for the observation li , yi should
follow a normal distribution of zero mean and one standard deviation.
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The test to verify this assumption is the  2 goodness of fit test.
The standardized residuals are pooled together and an histogram is plotted. The statistics
b
(a  e ) 2
y   i i follows a chi2 distribution of degree b-1,
ei
i 1
where b is the number of histogram classes, ai is the observed frequency in class i and
ei is the expected frequency in class i.
Failure of the chi2 goodness of fit test is an indication that the observations are not
normally distributed and/or that there is a problem in the model. Another test for the
assessment of the observations and the model is the test of the weighted quadratic form of
vt v r ˆ 2
the residuals. The statistics y  2  20 follows a chi2 distribution of degree r, where
0
0
r is the degree of freedom of the adjustment.
2.3 Sequential Solution
Let’s assume that the observations are gathered into two groups and that both groups are
uncorrelated. The two linearized systems can be written as follows:
v1  1  w1
v2   2  w2
with their own weight matrices 1 and 2 but they can also be grouped together as:
 v1   1 
 w1 
v      w 
 2  2
 2
 0 
with the total weight matrix    1

 0 2 
The normalization of the system is:
 0   1 
  t   1t t2  1
 1t 1  t22



 0 2  2 
 0   w1 
u  t w   1t  t2  1
 1t 1w1  t22 w2



 0 2  w2 




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