Confidence ellipse - Java·Applied·Geodesy·3D

Hello,

I've got a few questions about the confidence ellipses I get in JAG3D. When adjusting 2D network I get a (major semi axis) and c (minor semi axis) with gamma as a rotation angle of the confidence ellipse.
But once I take the same project (and add a height data) and do 3D network adjustment I get the same results when it comes to the coordinates, but I also get b (middle semi axis) and two more rotation angles that I didn't have in 2D network. Now when I look at the numbers I see that b and c are very close in values that I had as a and c before. What is the meaning of major semi axis in this case then?
And none of the rotation angles are close or similar to one I had before in 2D network adjustment. So how would I draw one for example in CAD software when having 3 rotation angles? What do they represent?

Thanks,
Marko

Hello Mark0,

I've got a few questions about the confidence ellipses I get in JAG3D. When adjusting 2D network I get a (major semi axis) and c (minor semi axis) with gamma as a rotation angle of the confidence ellipse.

Yes, that is right.

But once I take the same project (and add a height data) and do 3D network adjustment I get the same results when it comes to the coordinates, but I also get b (middle semi axis) and two more rotation angles that I didn't have in 2D network. Now when I look at the numbers I see that b and c are very close in values that I had as a and c before. What is the meaning of major semi axis in this case then?
And none of the rotation angles are close or similar to one I had before in 2D network adjustment. So how would I draw one for example in CAD software when having 3 rotation angles? What do they represent?

Each dispersion matrix (covariance matrix) is diagonalizable using, for instance, the eigendecomposition. Appliying the eigendecomposition results in a diagonal matrix, which contains the so-called eigenvalues, and an orthogonal matrix, which contains the corresponding eigenvectors. In case of a spatial network, a 3x3 sub-matrix of the total dispersion matrix refers to each estimated point. Decomposing a 3x3 dispersion yields three eigenvalues and three eigenvectors forming a 3x3 orthogonal matrix. The eigenvalues relate to the semi-axis of the confidence region, which is represented by an ellipsoid in case of a spatial network. (In a horizontal network, the confidence region forms an ellipse. In case of a levelling network, the confidence region is just an interval.) The eigenvectors are the direction vectors of these semi-axes. The eigenvectors forms a 3x3 orthogonal matrix. If the determinate of this orthogonal matrix is +1, it is also a rotation matrix. In case of a horizontal network, a single Euler angle represents the rotation. However, in case of a spatial network, three Euler-angles are required to describe a full rotation sequence. For more details, please take a look to the documentation.

All the best
Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

Tags:
Confidence region, Confidence ellipse, Confidence ellipsoid

Thanks for the reply and the sources.

If I understood this correctly, I should rotate around x-axis first using alpha, then around y-axis using beta, and finally around z-axis using gamma angle. And I did that, and it appears to be quite similar to the ellipse I get in graphic tab of the project after the adjustment. So that seems fine.

Now, I have another question regarding about standard deviations of x, y and z for each point. If I draw them in CAD and overlap over my ellipses that I got as described above, I find them to be much showing much smaller area. Reading more in the documentation I saw that those standard deviations are about 20% of confidence in case of 3D points.
After a shorter googling I have found this

https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-statistics/h-how-directional-distribution-standard-deviationa.htm

And if I scale my standard deviations by the value of 5.20, I do get closer to the ellipse.

But it doesn't make much sense to me seeing that scale factor of 5.20 is for 3 sigma, and reading in the documentation it says that Šidak correction scales the deviations by a quantil from χ2_n distributon. So, I am wondering how do I find that scale factor exactly in order to have for example 95% confidence level? And do my standard deviations for x, y and z represent only 20% confidence level?

I am missing something here, but unsure what.

Hi,

So, I am wondering how do I find that scale factor exactly in order to have for example 95% confidence level?

The (adapted) α can be taken from the following table. The last column refers to the quantile used to scale the confidence interval to the desired probability.
[image]

And do my standard deviations for x, y and z represent only 20% confidence level?

Each single standard deviation represents a confidence level of about 68 %. However, the following plot shows two confidence regions as ellipses having identical standard deviations.
[image]

Obviously, both ellipses are different, so they cannot have an identical confidence level.

/Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

Ah, okay, this does make sense now. So standard deviations are 68% confidence level, but drawing an ellipsoid using those values would "degrade" my confidence level to about 20%? So I need to scale it for about 4.21 (or some close value I got in my project) in order to be at 95%.

I've mixed up confidence level of a single standard deviation with confidence level of an ellipsoid.

Thanks

Hello Marc0,

So standard deviations are 68% confidence level, but drawing an ellipsoid using those values would "degrade" my confidence level to about 20%?

Assuming a normal distribution, the standard deviation of each parameter considered in isolation relates to a confidence level of about 68 %. However, the confidence level is reduced if more than one parameter is considered at the same time. Thus, the standard deviation of an isolated considered x-component relates to confidence level of about 68 %, and the standard deviation of an isolated considered y-component relates to confidence level of about 68 % and so on. However, the question is how large is the conditional probability that both, x and y, lie within a confidence region. Obviously, there are four cases (with different probabilities of occurrence)

both are outside
x is inside a region, but not y
y is inside a region, but not x
both are inside a region

Let us assume that a single parameter (x or y) lies within the 68 % confidence interval, then in three out of four cases there is a confidence region in which both parameters do not lie simultaneously - the conditional confidence level must be less than 68 %.

So I need to scale it in order to be at 95%.

Yes.

Just a question: Do you implement a specific method to transfer/import the adjustment results to a CAD software? What is your strategy (do you use the values from the HTML-report)?

Kind regards
Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

Thank you for all the explanation, it is much clearer now.

I am just playing around in CAD to figure out the angles and other values that have geometric meaning. I usually use html report and copy paste values as I have only 4 points. I know there are export options, although I haven't found any that enables rotation angles to be exported. Is there any option that can export to CAD software that I missed?

Thanks for all the replies.
Marko

Hello,

Is there any option that can export to CAD software that I missed?

No, not yet. However, the formatted HTML report bases on a freemarker-template. A DXF file for CAD application would be possible with a customized template. To get a better picture, this simple template generates a text file with coordinates as a coordinate list. So, in general it is possible...

/Micha

--
applied-geodesy.org - OpenSource Least-Squares Adjustment Software for Geodetic Sciences

Okay, maybe I'll check that once I get a lot more points, but for now this works.

Still, thanks for you replies, you helped a lot.

Marko

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