 •Introduction to adjustment calculus
 •Introduction to adjustment calculus (Third Corrected Edition)
 •Introduction
 •2. Fundamentals of the mathematical theory of probability
 •If d'cd; then p (d1) £ lf
 •Is called the mean (average) of the actual sample. We can show that m equals also to:
 •3.1.4 Variance of a Sample
 •Is called the variance (dispersion) the actual sample. The square root 2
 •In the interval [6,10] is nine. This number
 •VVII?I 00878'
 •In this case, the new histogram of the sample £ is shown in Figure 3.5.
 •Is usually called the rth moment of the pdf (random variable); more precisely; the rth moment of the pdf about zero. On the other hand, the rth central moment of the pdf is given by:
 •3.2.4 Basic Postulate (Hypothesis) of Statistics, Testing
 •3.3.4 Covariance and VarianceCovariance Matrix
 •X and X of a multivariate X as
 •It is not difficult to see that the variancecovariance matrix can also be written in terms of the mathematical expectation as follows:
 •3.3.6 Mean and VarianceCovariance Matrix of a Multisample The mean of a multisample (3.48) is defined as
 •4.2 Random (Accidental) Errors
 •It should be noted that the term иrandom error" is used rather freely in practice.
 •In order to be able to use the tables of the standard normal
 •X, we first have to standardize X, I.E. To transform X to t using
 •Is a normally distributed random
 •4.10 Other Measures of Dispersion
 •The average or mean error a of the sample l is defined as
 •5. Leastsquares principle
 •5.2 The Sample Mean as "The Maximum Probability Estimator"
 •5.4 LeastSqaures Principle for Random Multivariate
 •In very much the same way as we postulated
 •The relationship between e and e for a mathematical model
 •6.4.4 Variance Covariance Matrix of the Mean of a Multisample
 •Itself and can be interpreted as a measure of confidence we have in the correctness of the mean £. Evidently, our confidence increases with the number of observations.
 •6.4.6 Parametric Adjustment
 •In this section, we are going to deal with the adjustment of the linear model (6.67), I.E.
 •It can be easily linearized by Taylor's series expansion, I.E.
 •In which we neglect the higher order terms. Putting ax for XX , al for
 •The system of normal equations (6.76) has a solution X
 •In sections 6.4.2 and 6.4.3. In this case, the observation equations will be
 •In matrix form we can write
 •In metres.
 •6.4.7 VarianceCovariance Matrix of the Parametric Adjustment Solution Vector, Variance Factor and Weight Coefficient Matrix
 •I.E. We know the relative variances and covariances of the observations only. This means that we have to work with the weight matrix к£ 1
 •If we develop the quadratic form V pv 3) considering the observations l to be influenced by random errors only, we get an estimate к for the assumed factor к given by
 •Variance factor к plays. It can be regarded as the variance of unit
 •In metres,
 •Is satisfied. This can be verified by writing
 •Into a . О
 •6.U.10 Conditional Adjustment
 •In this section we are going to deal with the adjustment of the linear model (6.68), I.E.
 •For the adjustment, the above model is reformulated as:
 •Is not as straightforward, as it is in the parametric case (section 6.4.6)
 •VeRn VeRn
 •Into the above vector we get 0.0
 •0.0 In metres .
 •In metres.
 •Areas under the standard normal curve from 0 to t
 •Van der Waerden, b.L., 1969: Mathematical Statistics, SpringerVerlag.
Into a . О
The theoretical relation between the a priori and a posterior variance factors allows us to test statistically the validity of our hypothesis. However, this particular topic is going to be dealt with elsewhere. Let us just comment here on the results of the adjustment of the levelling network discussed in Examples 6.17 and 6.19. In computing the weight matrix P, we assumed = 1. After the adjustment we
obtained a^{2} = 0.00067. Thus the ratio a^{2}/a^{2} equals to 1500 which is con o о о
siderably different from 1. This suggests that the variancecovariance matrix E was postulated too "pessimistically" and that the actual variances
Li
of the observations are much lower.
6.U.10 Conditional Adjustment
In this section we are going to deal with the adjustment of the linear model (6.68), I.E.
В L = С , (r < n), (6.102)
which represents a set of ^{ff}r" independent linear conditions between n observations L« Note that С is an r by 1 vector of constant values arising from the conditions.
For the adjustment, the above model is reformulated as:
В (L + V)  С = 0 or, as we usually write it
В V + W =0 ^{4}) j (6.103)
where:
W = В L
(6.10k)
The system of equations (6.103) is known as the condition equations, in which В is the coefficient matrix, V is the vector of discrepencies and W is the vector of constant values. We recall that "n" is the number of observations and "r" is the number of independent
conditions. It should also be noted that no unknown parameters,.i.e. vector X, appear in the condition equations. The discrepencies V are the only unknowns
We wish again to get such estimate V of V that would minimize
T ol
the quadratic form V PV, where P = 2> is the assumed weight matrix
T
of the observations L. The formualtion of this condition, i.e. min V PV,
Is not as straightforward, as it is in the parametric case (section 6.4.6)
This is due to the fact that V in equation (6.103) can not be easily
expressed as an explicit function of В and W. However, the problem can
be solved by introducing the vector К of r unknowns, called Lagrange^{1}s
r,l
multipliers or correlates^{5}). We can write:
min V^{T}PV = min [V^{T}PV + 2K^{T} (BV + W)] , (6.105)
VeRn VeRn
since the second term on the right hand side equals to zero. Let us denote
ф = V^{T}PV + 2K^{T} (BV + W) .
To minimize the above function, we differentiate with respect to V and
equate the derivatives to zero. We get:
Эф T T
r^{1} = 2V P + 2K В = 0
a
T
which, when transposed, gives PV + В К = 0.
1 T V = ~P В К
The last equation can be solved for V and we obtain:(6.106)
This system of equations is known as the correlate equations.
Substituting equation (6.106) into (6.103), we eliminate V: 1 T
B(P В К) + W = 0 ,
or
(BP""^{X}B^{T}) К = W . j (6.107)
This is the system of normal equations for conditional adjustment. It is usually written in the following abbreviated form:
М К = W, (6108)
where
м = в р"*^{1} в^{т}, (6.109)
The solution of the above normal equations system for К yields: к = m"*"^{1} ¥ = (bp""^{2}^)"^{1} W. (6.110)
Once we get the correlates K, we can compute the estimated residual vector V from the correlate equations (6.106). finally the adjusted observations L are computed from
L = L + V . (6.111)
In fact, if we are not interested in the intermediate steps, the formula for the adjusted observations L can be written in terms of the original matrices в and p and the vectors L and C. We get
L = L + V
1 T = L  P В К
= L  p""^{1}b^{t}(bp"^{1}b^{t})""^{1} (BLC) . (6.112)
It can also be written in the following form:
L = (I  T) L + HC ,
(6.113)
where: ^{1} is the identity matrix _{%}
T = p"^{1} [в^{т}(вр~^{1}в^{т})""^{1} B],
н = p"^{1} b^{t}(bp"^{x} bV^{1}
(6.11U)
Example 6.20: Let us solve example 6.l6 again, using this time the conditional method of adjustment. We have only one condition equation between the observed height differences h^, i = 1,2,3> and we thus note that the number of degrees of freedom
is the same as in example 6.l6. Denoting H  H ЬуДН, the existing condition can he written as:
Eh. = AH.
1
After reformulation we get:
y_{±} + v_{2} + v_{3} + (h + h_{2} + h_{3}  ЛН) = 0,
which can he easily written in the matrix form as
Б V + W = 0 1,3 3,1 1,1
where
В = [l, 1, l], V = v.
and
W = h + h_{2} + h^  ДН
= E h. ~ ДН i ^{1}
The weight' matrix of the observations is given by (see
example 6.l6):
P = diag (j, ~~> 3,3 1 2 ^{a}3
and
P"^{1} = diag (d_{1}, d_{2}, d_{3}). 3,3
The system of normal equations for the correlates К is given
by equation (6.108) as
.M К = W 1,1 1,1 1,1
where
M = BP""^{1} B^{T} » (d_{1} + d_{2} + d_{3}) = Ed_{i<}
The solution for К is
К = M"^{1} ¥ = yp (Eh  ДН). i i i
The estimated residuals are then computed from equation (6.IO6) as:
^{d}l ^{0}
1 1 1
1 T у = P В К
d_{2} 0
0 0
?h.  AH
^{ )}
11
Г 1 

d„ 

1 
?h.  1 1 
d 

2 
? d. 1 1 
d_{0} 

L 3 

and we get:
ДН  Zh.
v. = —±± d., i = 1, _{2}; 3.
.d.
1 1
This is the same result as obtained in example 6.l6 when using the parametric adjustment (note that ЛН = H_{T}  E_{n}).
J la
in this particular problem, we notice that the adjustment
divide the misclosure, i.e. (ДН ~Е^1ь ), to the individual
observed height differences proportionally to the corresponding
lengths of levelling sections, i.e. inversely proportionally
to the individual MSE^{1} s. Tbe adjusted observations are given by equation (6.111) i.e.
L = L + V ,
or
This yields:
AH  Eh
h. = h. + — d, ,
^{1} ^{1} ?d.
1 1
Finally, the estimates of the unknown parameters, ,i.e, X = (H^5 H^) , are computed from the known elevations BL, H_{T} and the adjusted observations h, , as follows:
(id 1
H * H + h
1 G 1
^{я} ^{H}G ^{+} ^{h}l ^{+}
?d.
l l
d„(AH  Eh. )
i ^{1}
and
^{H}2 ^{=} ^{H}J ~ ^{h}3
 _{Hj}.h_{3}
(AH  lh_{±})
The results are again identical to the ones obtained from the parametric adjustment.
Example 6.21: Let us solve example 6.17 again, but this time using the
conditional adjustment. The configuration of the levelling network in question is illustrated again in Figure 6.8, for convenience.
From the above mentioned example we have: No. of observations, n = 6, No. of unknown parameters, u = 3.
Then df = 6  3 = 3, and we shall see that we can again formulate only 3 independent condition equations between the given observations.
By examining Figure 6.8, we see that there are к closed loops, namely: (a ~ с  d  a), (adЪа), (b  с  d  b) and (a  с  b  a).
This means that we can write k condition equations, one for each, closed loop. However, one can be deduced from the other 3, e.g. the last mentioned loop is the summation of the other three loops.
Let us ..for instance, choose the following three loops :
loop I=acba>
loop 11= acda>
loop III = a«dba. These loops give the condition equations as follows
^{(E}1 ^{+} ^{V}l^{)} " ^{(}^6 ^{+} V " ^{(}4 * V ^{=} °'
^1 ^{+} ^{V}l^ ^{+} ^3 ^{+} V " ^2 ^{+} ^{Y}2^ ^{=} °> (h_{2} + v_{2})  (h_{5} + v_{5})  (h^ + v^) = 0.
Then we get
^{v}i  v_{k}  v_{6} + (5_{1}  h_{k}  Eg) = 0 ,
^{V}l  ^{Y}2 ^{+} ^{V}3 ^{+} ^{(5}1 " ^{E}2 ^{+} V ^{=} °> ^{V}2 " ^{r}k ~ ^{v}5 ^{+} ^{(й}2 " \ ~ S^{)} ^{=} ° '
The above set of condition equations can be written in the matrix notation as
В V + ¥ = 0 , 3,6 6,1 ЗА
where:

1 
0 
0 
1 
0 
1 
В 
1 
1 
1 
0 
0 
0 
3,6 







0 
1 
0 
1 
1 
0 
f = (V ^{T}2' ^{Y}V V V V 1,6
and
¥ = 3,1
(h^  h_{2} 4 h^) ^(h_{2}  \  h_{5})j
Substituting the observed quantities h\ , i = 1, 2, 6,
W = 3,1
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