Satisfying the
“Condition of Least Squares”
Land Surveyors’ Association of Washington
2012 Annual Conference Presentation
March 6, 2014
by Jon B. Purnell, PLS © 2012 Alidade Consulting
What is “Least Squares?”
• A method for computing “the most
likely value” from a set of
observations
• A method for reconciling (adjusting)
observations such that they conform
to “constraints”
How is the “Condition of Least
Squares” satisfied?
• When the sum of the squared
residuals is minimized…
• …the condition of least squares is
said to be “satisfied”
Compute the “most likely value”
given the following observations
• 4.0, 5.0, 5.0, 5.2, 6.3, 7.9, 9.9
• Possible approaches:
– Use the most frequent value (mode)
– Use the middle value (median)
– Use the average value (mean)
• Which of these approaches best satisfies
“the condition of least squares?”
Using Mode
Using Mode as the Measure of Central Tendency
Obs #
value
-mode
1
2
3
4
5
6
7
4.0
5.0
5.0
5.2
6.3
7.9
9.9
-5.0
-5.0
-5.0
-5.0
-5.0
-5.0
-5.0
mode=
5.0
Squared
Residuals Residuals
-1.0
0.0
0.0
0.2
1.3
2.9
4.9
1.00
0.00
0.00
0.04
1.69
8.41
24.01
35.15
2.42
0.91
35.15
σs =
6
= Sum of Squared Residuals
= Standard Deviation of the Mean
= Standard Error of the Mean
Using Median
Using Median as the Measure of Central Tendency
Obs #
value
1
2
3
4
5
6
7
4.0
5.0
5.0
5.2
6.3
7.9
9.9
median=
5.2
Squared
-median Residuals Residuals
-5.2
-5.2
-5.2
-5.2
-5.2
-5.2
-5.2
-1.2
-0.2
-0.2
0.0
1.1
2.7
4.7
1.44
0.04
0.04
0.00
1.21
7.29
22.09
32.11
2.31
0.87
32.11
σs =
6
= Sum of Squared Residuals
= Standard Deviation of the Mean
= Standard Error of the Mean
Using Mean
Using Mean as the Measure of Central Tendency
Obs #
value
-mean
1
2
3
4
5
6
7
4.0
5.0
5.0
5.2
6.3
7.9
9.9
-6.2
-6.2
-6.2
-6.2
-6.2
-6.2
-6.2
mean =
6.2
Squared
Residuals Residuals
-2.2
-1.2
-1.2
-1.0
0.1
1.7
3.7
4.78
1.41
1.41
0.97
0.01
2.94
13.80
25.31
2.05
0.78
25.31
σs =
6
= Sum of Squared Residuals
= Standard Deviation of the Mean
= Standard Error of the Mean
What does “Least Squares” do
to my data?
• Keeps your data as close as possible
to the original observations
• When fitting observations to a
“constraint,” LSQ adjusts the
observations by the smallest amount
possible
Given, distance AB and weights:
• AB = 244.43’ (measured 5 times)
• AB = 244.25’ (measured 50 times)
• Simple mean = 244.34
• Weighted mean = 244.27
• Which “mean” will best “satisfy the
condition of least squares?”
All observations “adjusted” by a
similar amount…
(this is what compass rule does!)
Using Simple Mean
Using Simple Mean as the Measure of Central Tendency
Squared
Squared
Residuals * Freq.
Obs #
value
Freq
Residuals
Residuals
1
2
244.43
244.25
5.0
50.0
0.09
-0.09
0.0081
0.0081
Sum=
55.0
0.0405
0.405
Simple mean = 244.34
Sum of Squared Residuals =
Standard Deviation of the Mean =
“Adjusted” value
Standard Error of the Mean =
0.44550
0.67
0.47
Observations adjusted
according to weights,
resulting in smaller
adjustments overall
throughout the data set!
Using Weighted Mean
Using Weighted Mean as the Measure of Central Tendency
Obs #
value
Freq
Residuals
Squared
Residuals
1
2
244.43
244.25
5.0
50.0
0.16
-0.02
0.0256
0.0004
Sum=
55.0
Squared
Residuals * Freq.
0.128
0.02
Weighted Mean= 244.27
Sum of Squared Residuals =
“Adjusted” value
Standard Deviation of the Mean =
Standard Error of the Mean =
0.14800
0.05
0.01
What should you look for in
LSQ software?
• Look for software that allows you to
control (globally and to individual obs)
–
–
–
–
instrument centering errors
target centering errors
instrument pointing errors
EDM constant and scalar errors (according to
the manufacturer’s specifications)
– HI and HT errors
– Zenith angle errors
What should you look for in
LSQ software?
• Look for software that allows you to
see
– Observations, adjusted values, and
residuals (adjustments)
– Standard errors (weights) for each
measurement
– Standardized residuals for each
measurement (ratios of Residuals to
Standard Errors)
Be wary of LSQ software that
• Applies the same standard error to
entire classes of measurements
– Standard errors for distances should
vary by the lengths of the lines
– Standard errors for angles should vary
according the FS/BS lengths and by the
magnitudes of the angles