Weights of Observations

Introduction

• Weights can be assigned to observations according to their relative quality

• Example: Interior angles of a traverse are measured – 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

Relation to Covariance Matrix

With correlated observations, weights are related to the inverse of the covariance matrix, Σ.

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.

20

ijijq

Recall, Covariance Matrix

2

2

2

11

2212

1211

nnn

n

n

xxxxx

xxxxx

xxxxx

For independent observations, the off-diagonal terms are all zero.

Cofactor Matrix

20

1

Q

We can also define a cofactor matrix which is related to the covariance matrix.

The weight matrix is then:

120

1 QW

Weight Matrix for Independent 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

20

iiw

If the weight, wi = 1, then

is the variance of an observation of unit weight (reference variance)

220 i

20

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

iiw

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 y

Weighted Mean Formula

n

ii

n

iii

w

zwz

1

1

Weighted Mean – Example 2A 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 σ1 = 0.010 m D2 = 1097.241 m σ2 = 0.005 m

Solution: First, compute the weights.

222

22

222

11

m000,40)m005.0(

11

m000,10)m010.0(

11

w

w

Example - Continued

Now, compute the weighted mean.

m243.1097

40,000m10,000m1097.241m40,000m1097.253m10,000m

22

22

D

D

Notice that the value is much closer to the more precise observation.

Standard Deviations – Weighted Case• When computing a weighted mean, you want an

indication 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

Standard Deviation Formulas

11

2

0

n

vwS

n

iii

)1(1

2

nw

vwS

i

n

iii

i

Standard deviation of unit weight

Standard deviation of observation, i

Standard deviation of the weighted mean

n

ii

n

iii

M

wn

vwS

1

1

2

)1(

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 to consider entire lines of levels rather than individual setups. Weights are:– Inversely proportional to line length– Inversely proportional to number of setups

Angle Example 9.2

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.

Differential Leveling ExampleFour different routes were taken to determine the elevation difference between two benchmarks (see table). Computed the weighted mean elevation difference.

Example - ContinuedWeights: (note that weights are multiplied by 12 to produce integers, but this is not necessary)

Compute weighted mean:

ft366.2524

78.60824612

30.25238.25441.25635.2512

M

M

What about significant figures?

Example - Continued

ft090.01

1

2

0

n

vwS

n

iii

Compute residuals

Compute standard deviation of unit weight

Compute standard deviation of the mean

ft018.0)1(

1

1

2

n

ii

n

iii

M

wn

vwS

Example - ContinuedStandard deviations of weighted observations:

Summary

• Weighting allows us to consider different precisions of individual observations

• So far, the examples have been with simple means

• Soon, we will look at least squares adjustment with weights

• In adjustments involving observations of different types (e.g. angles and distances) it is essential to use weights