- •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 0-0878'
- •In this case, the new histogram of the sample £ is shown in Figure 3.5.
- •Is usually called the r-th moment of the pdf (random variable); more precisely; the r-th moment of the pdf about zero. On the other hand, the r-th central moment of the pdf is given by:
- •3.2.4 Basic Postulate (Hypothesis) of Statistics, Testing
- •3.3.4 Covariance and Variance-Covariance Matrix
- •X and X of a multivariate X as
- •It is not difficult to see that the variance-covariance matrix can also be written in terms of the mathematical expectation as follows:
- •3.3.6 Mean and Variance-Covariance 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. Least-squares principle
- •5.2 The Sample Mean as "The Maximum Probability Estimator"
- •5.4 Least-Sqaures 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 X-X , 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 Variance-Covariance 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, Springer-Verlag.
In metres.
The computations can be checked by deriving the heights of
points b, с and d from using the adjusted hu . The resulting values
must not differ from the adjusted values EL , H and H_{Л}.
bed
6.4.7 Variance-Covariance Matrix of the Parametric Adjustment Solution Vector, Variance Factor and Weight Coefficient Matrix
The parametric adjustment solution vector X is given by equation (6.77), i.e.
T -1 T - -1 T
X = (A PA) A PL = (N A P) L
This can he written as:
X = В L
(6.81:
■where
-1 T В = N A P •
(6.82)
The variance-covariance matrix E* of the solution vector X can be easily
x
deduced by applying the covariance law (equation (6.15)) on (6.8l)j we get:
^{E}X ^{=} ^{BE}L (6.83)
From equation (6.66), we have
P = кЕг-""*^{1}
Ju
and inverting both sides we obtain
= к P""^{1} . (6.Qk)
Substituting (6.82) and (6.81+) into (6.83) we get
E* = (N^{wl}A^{T}P) к P™^{1} (N^P)*. (6.85)
л.
Both P and N are symmetric matrices, so that we can write:
T Т т m i
P = P, N = N and (1 ) = N .
Substituting this into (6.85) we get
-1 T -1 -1
= к N A P P P. A N
X
= к N""^{1} (A^{T}PA) if^{1} = к N~^{1} N if^{1} = к N""^{1} ,
that is
E~ = к jf^{1} = ic (А^АГ^{1}
(6.86)
On the other hand, by putting P •= кЕ-"^{-1} in (6.86) we get
1»
^{J}X
z,: = - к (a^{t} г-Л)"^{1} = (a^{t}e_{f}*^{1}a)*^{1} ,ic L L '
(6.87:
which shows that does not depend on the choice of the factor к. In fact,
this statement is valid only if we know the correct values of the elements
of Е- . Unfortunately_{5} however, £- is often known only up to a scale factor. L li
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
without knowing the actual value of the factor к. Therefore cannot be
jL
computed from equation (6.87).
^{Л}Т ^{л}
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
V^{T}pv = (n - u) к . (6.88)
The multiplier in the right-hand side is nothing else but the difference between the number of independent observations and the number of unknown parameters, i.e. the number of redundant observations, which is sometimes denoted by'cif^and called the number of degrees of freedom, i.e.
df = n - u . (6.89)
df must be greater than zero in order to be able to perform a least-squares adjustment. Hence equation (6.88) becomes
(6.90)
Usuallyj in the literature, к is known as the a priori variance factor and к is called the least-squares estimate of the variance factor^or,simply» estimated or a posteriori variance factor. ФЬе estimated variance factor can be now used instead of the a priori one, yielding an estimate of E^.
-1 а Т -1 = к N = к(A PA)
X
(6.91)
which is known as the estimated variance-covariance matrix of X.
To discuss the influence of the chosen variance factor к in
the weight matrix P = к Е-^{1} on E", as defined by (6.91), we take another
L X
factor, say к^{1}. We obtian P^{1} = к'Е- ^{1} = yP. Substituting in equation
L
(6.91) we get:
^ _{=} Y (V^{T}PV) 1 _{(A}T_{pA)}-l _{=}
^{h}X df у X "
The above result indicates that 2> given by equation (6.91) is independent
of the choice of the a priori variance factor к. We recall that the same
a
holds true for the estimated solution vector X (equation 6.79).
It often happens in the adjustment calculus, that we have to use
the estimated parameters X in subsequent adjustments as "observations".
Then we have to take into account their respective weights. We know that
the weight matrix of an observation vector must be proportional to the
inverse of its variance-covariance matrix (equation 6.66). Thus, we can
weight matrix of the vector X, since the inverse N is proportional to the variance-covariance matrix E^. Accordingly, the matrix N ^{1} is known also as the weight coefficient matrix and the square roots of its diagonal
see that the matrix of normal equations, N, can be immediately used as the-1
will Ъе uncorrelated if L is uncorrelated, i.e. P is diagonal, and if the design matrix A is orthogonal. On the other hand, N may he diagonal even for some other general matrices P, A.
Let us now turn once more hack to the "adjustment" of the sample mean (see 6.4.3). It is left to the student to show that the normal equations degenerate into a single- equation, namely equation (6.40) On the other hand, using eq. (6.91), we obtain the estimated variance of the mean x as
^{л}гп ^{л} ^{л}ф ^{л}
^{А} р V PV V PV / , \
2
Evidently the estimated variance of x differs from the variance
(see eq. б.бо) in the denominator. By analogy we define a new statis-
"2
tical quantity, the estimated variance S of a sample L
л p 1 — P 1
SL = —-—E (£.-£)=——- T) n - 1 . , 1 n - 1 .
n n _{A} _{p}E v. i=l " i=l
(6.93)
(compare with eq. 3.6) which is used in statistics wherever the mean I of the sample L is also being determined. It is again left to the student to show that using the estimated variances for the grouped observations (see 6Л.2) the formula(6.92) (instead of 6.60) can be derived using the argument at ion of 6.4.2 and 6.4.3.
The estimated variances of the sample L and its mean £ can be
also computed using non-normalized weights,.i.e. weights p^ for which
n
£ p. ф 1 (see 6.4.6). It can be shown that the appropriate formulae are i«l ^{1}
: 2 1 ? - 2
and
^{s}l ^{=}7-—^ A ^{p}i ^{v}i ' ^{(6}-^{95)}
(n-1) _{E} p 1=1 i=l
To conclude this section, let us try to interpret the meaning of the variance factor к, introduced for the first time in б. U • 5 • Let us take, for simplicity, an experiment yielding a unit matrix of normal equations, i.e. N = I. What would he the variance-covariance matrix of the solution vector X? It will he a diagonal matrix
= diag (к", k,*.._{5}k). (6.96)
2 ^{л}
This implies that all the variances Sa of the components of X equal to
x. 1
к. Since the square roots of the diagonal elements of N (all equal to l) can be considered as the weights P^ of the components x^ of X we can also
write:
P, S ^{2} = P_{0}S_{0}^{2} = . . . = P S ^{2} = к . (6.97)
1 1 2 2 n n
Comparison with equation (6.59) gives some insight into the role the
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