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8/3/2019 Weights of Observations
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Weights of Observations
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Introduction
Weights can be assigned to observations
according to their relative quality
Example: Interior angles of a traverse aremeasured half of them by an inexperienced
operator and the other half by the best
instrument person. Relative weight should be
applied.
Weight is inversely proportional to variance
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Relation to Covariance Matrix
With correlated observations, weights are related to the
inverse of the covariance matrix,.
For convenience, we introduce the concept of a
cofactor. The cofactor is related to its associated
covariance element by a scale factor which is the
inverse of the reference variance.
2
0W
W ijijq !
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Recall, Covariance Matrix
-
!7
2
2
2
11
2212
1211
nnn
n
n
xxxxx
xxxxx
xxxxx
WWW
WWW
WWW
.
/1//
.
.
For independent observations, the off-diagonal terms
are all zero.
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Cofactor Matrix
7!2
0
1
WQ
We can also define a cofactor matrix which is related to the
covariance matrix.
The weight matrix is then:
12
0
1 7!! WQW
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Weight Matrix forIndependent
Observations Covariance matrix is diagonal
Inverse is also diagonal, where each diagonal term is
the reciprocal of the corresponding variance element
Therefore, the weight for observation i is:
2
2
0
i
iwW
W!
If the weight, wi = 1, then
is the variance of an observation of unit weight (reference variance)
22
0 iWW !
2
0W
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Reference Variance
It is an arbitrary scale factor (a priori)
A convenient value is 1 (one)
In that case the weight of an independent observation
is the reciprocal of its variance
2
1
i
iwW
!
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Simple Weighted Mean Example
3.1523
5.1525.1529.151!
!
y
A distance is measured three times, giving values of 151.9, 152.5,
and 152.5. Compute the mean.
Same answer by weighted mean. The value 152.5 appears twice
so it can be given a relative weight of 2.
3.1523
5.15229.151!
v!y
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Weighted Mean Formula
!
!!
n
ii
n
i
ii
w
zw
z
1
1
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Weighted Mean Example 2
A line was measured twice, using two different total stations. The
distance observations are listed below along with the computed
standard deviations based on the instrument specifications. Compute
the weighted mean.
D1 = 1097.253 m 1 = 0.010 m
D2 = 1097.241 m 2 = 0.005 m
Solution: First, compute the weights.
2
22
2
2
2
22
1
1
m000,40
)m005.0(
11
m000,10)m010.0(
11
!!!
!!!
W
W
w
w
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Example - Continued
Now, compute the weighted mean.
m243.1097
40,000m10,000m
1097.241m40,000m1097.253m10,000m22
22
!
vv!
D
D
Notice that the value is much closer to the more
precise observation.
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Standard Deviations Weighted Case
When computing a weighted mean, you want anindication of standard deviation of observations.
Since there are different weights, there will be
different standard deviations
A single representative value is the standard
deviation of an observation of unit weight
We can also compute standard deviation for a
particular observation
And compute the standard deviation of the
weighted mean
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Standard Deviation Formulas
11
2
0
!
!
n
vw
S
n
i
ii
)1(1
2
!
!
nw
vw
Si
n
i
ii
i
Standard deviation
of unit weight
Standard deviation
of observation, i
Standard deviation
of the weighted
mean
!
!
!n
i
i
n
i
ii
M
wn
vw
S
1
1
2
)1(
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Weights for Angles and Leveling
If all other conditions are equal, angle weights
are directly proportional to the number of turns
For differential leveling it is conventional toconsider entire lines of levels rather than
individual setups. Weights are:
Inversely proportional to line length
Inversely proportional to number of setups
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Angle Example
This example asks for an adjustment and uses the
concept of a correction factor which has not been
described at this point. We will skip this type of
problem until we get to the topic of least squares
adjustment.
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Differential Leveling Example
Four different routes were taken to determine the elevation
difference between two benchmarks (see table). Computed the
weighted mean elevation difference.
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Example - ContinuedWeights: (note that weights are multiplied by 12 to produce
integers, but this is not necessary)
Compute weighted mean:
ft366.2524
78.608
24612
30.25238.25441.25635.2512
!!
vvvv!
M
M
What about significant figures?
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Example - Continued
ft090.01
1
2
0 !
!
!
n
vw
S
n
i
ii
Compute residuals
Compute standard deviation of unit weight
Compute standard deviation of the mean
ft018.0
)1(1
1
2
!
!
!
!
n
i
i
n
i
ii
M
wn
vw
S
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Example - Continued
Standard deviations of weighted observations:
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Summary
Weighting allows us to consider differentprecisions of individual observations
So far, the examples have been with simplemeans
Soon, we will look at least squares adjustmentwith weights
In adjustments involving observations ofdifferent types (e.g. angles and distances) it isessential to use weights
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