Table Of Contents
- Error Ellipses
- Derivation From Rotations
- Derivation From Eigen Values And Vectors
Error Ellipses
Introduction To Error Ellipses
-
Error ellipses are derived from the covariance
matrix, and provide a graphical means of viewing the results of
a network adjustment. Error ellipses can show:
-
Generally the standard deviation in x, y, and z
is used as a guide to the precision of a point, the error ellipse
gives more detailed information - the maximum and minimum standard
deviations, and their associated directions. The orientation of
the ellipse is basically given by the correlation term.
- The error ellipse is an approximation of the pedal curve, which
is the true shape of the standard error in all directions.
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Standard Error Ellipses and Confidence Regions
-
A standard error ellipse shows the region,
if centred at the true point position, where the least squares
estimate falls with a confidence of 39.4%. Since the truth
is rarely known, the error ellipse is taken to represent the 39.4%
confidence region, and when drawn is centred at the least squares
estimate of the position (rather than the true position).
To obtain different confidence regions the length of the ellipse
axis is just multiplied by an appropriate factor:
| Confidence region |
39.4% |
86.5% |
95.0% |
98.9% |
| Factor |
1.000 |
2.000 |
2.447 |
3.000 |
These multiplicative factors are determined from
the c2 distribution. Generally
the 95% confidence region ellipses are plotted (ie. TDVC's error
ellipses).
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Absolute and Relative Error Ellipses
-
Absolute error ellipses show the effects of the
datum - for example, points further away from the datum point in
a survey network generally have larger error ellipses than points
close to the datum point
-
Relative error ellipses are not influenced by the
choice of datum. Relative error ellipses are still derived from
the covariance matrix, only they involve 2 points, and an exercise
in propagation of variance.
- 2 methods for computing the error ellipse are introduced from a
theoretical point of view and with an example:
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Derivation From Rotations
Theory
We start with the covariance matrix of a 2D point based
on the local [x, y] coordinate system:
There is another coordinate system [u, v], that we
can rotate into using:
| u |
= |
sin(q) |
cos(q) |
x |
| v |
-cos(q) |
sin(q) |
y |
The corresponding covariance matrix for the [u, v]
system is gained via propagation of variance:
| Cuv = |
s2u |
suv |
= |
sin(q) |
cos(q) |
Cxy |
sin(q) |
-cos(q) |
| |
s2v |
-cos(q) |
sin(q) |
cos(q) |
sin(q) |
We are only really interested in su
AND sv which evaluate to:
s2u = s2xsin2(q)
+ 2sxy sin(q)
cos(q) + s2ycos2(q)
s2v = s2xcos2(q)
+ 2sxy sin(q)
cos(q) + s2ysin2(q)
If these equations are plotted for 0 < q
< 360 the pedal curve mentioned previously will be obtained.
The maximum and minimum values of su
and sv can be found by setting
the derivative (w.r.t. q ) of either of
the above equations to zero and solving for q
(note these maxima and minima will correspond to the directions of the
major and minor axes of the error ellipse):
| f(s2u) |
= 2s2x
sin(q) cos(q)
- 2s2y sin(q)
cos(q) - 2s2xy
sin2(q) + 2s2xy
cos2(q) |
|
| fq |
| |
= 2 sin(q) cos(q)
{s2x - s2y}
+ s2xy {cos2(q)
- sin2(q)} |
| |
= sin(2q) {s2x
- s2y} + 2sxy
cos(2q) = 0 |
| q = |
1 |
a tan ( |
- 2sxy |
) |
|
|
| 2 |
s2x
- s2y |
to which there are 2 solutions, 90 degrees apart. The
selection of rotation matrix means that q
is computed as a bearing. The values of q
are then substituted back into the equation for s2u
to determine the maximum and minimum values, and which axis they correspond
to.
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Example
As an example take:
Find q:
| q = |
1 |
a tan ( |
- 2sxy |
) = ~30 and 120 degrees
(as a bearing) |
|
|
| 2 |
s2x
- s2y |
Evaluating these angles gives:
su(30) = 4.00 and su(120)
= 1.93
which are the major and minor axis lengths and orientations
- all that is required to plot the error ellipse. Note that these axis
lengths correspond to the standard error ellipse. To plot the
95% confidence region the axis lengths are increased by a factor of
2.447:
su95%(30) = 9.79 and su95%(120)
= 4.72
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Derivation From Eigen Values And Vectors
Theory
It can be shown that the square root of the eigen values
of Cxy correspond to the error ellipse axis lengths,
and the eigen vectors that correspond to the eigen values define the
error ellipse axis directions. Start by finding the eigen values:
|Cxy - l I| = 0
 |
s2x - l |
sxy |
|
= 0 |
| syx |
s2y - l |
(s2x - l)(s2y
- l) - s2xy
= 0
(s2x - l)(s2y
- l) - s2xy
= 0
l2 + l(-s2x
- s2y) + s2xs2y
- s2xy = 0
Solving the quadratic in l
(gives the 2 eigen values):
| l = |
(s2x
+ s2y) +
 (s2x
+ s2y)2
- 4(s2xs2y
- s2xy) |
|
| 2 |
| l = |
(s2x
+ s2y) +
(s2x
+ s2y)2
+ 4s2xy |
|
| 2 |
The 2 corresponding eigen vectors e1
and e2 = [x2 y2]
can then be found using non-trivial solutions of:
| s2x
- l1 |
sxy |
x1 |
= |
0 |
| sxy |
s2y
- l1 |
y1 |
0 |
| s2x
- l2 |
sxy |
x2 |
= |
0 |
| sxy |
s2y
- l2 |
y2 |
0 |
These eigen vectors specify the direction of the ellipse
axis (and should be at right-angles). The lengths of the axes are given,
for the standard ellipse, by respectively l1
and l2.
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Example
Using the previous example:
| l = |
(s2x
+ s2y) +
(s2x
+ s2y)2
+ 4s2xy |
= |
19.743 + 12.255 |
|
|
| 2 |
2 |
l1 = 16
l2 = 3.744
(Standard axis lengths 4.00 and 1.93)
Eigen vector for l1:
| 6.822 - 16 |
5.315 |
x1 |
= |
0 |
| 5.315 |
12.921 - 16 |
y1 |
0 |
| y1 = 1.727x1, i.e. |
x1 |
= |
1 |
giving a direction (bearing) 30.0 degrees |
| y1 |
1.727 |
Eigen vector for l2:
| 6.822 - 3.744 |
5.315 |
x2 |
= |
0 |
| 5.315 |
12.921 - 3.744 |
y2 |
0 |
| x2 = 1.727y2, i.e. |
x2 |
= |
-1.727 |
giving a direction (bearing) 60.0 degrees |
| y2 |
1 |
Which all agrees with the previous method.
This algorithm can be used to obtain a 3D error ellipse
for a point. In fact the same approach is extendible to produce an n
dimensional hyper-ellipsoid, although these may bit a little difficult
to conceptualise, let alone draw.
- Extracting covariance data from TDVC (for major prac), use -lv
flag.
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