Internal Reliability computation

[Manu Cledat]

Dear developers of the SALSA software,

 

Congratulation for your OpenSource Software and its very detailed user manual.

I have a question about some equations in the user manual.

https://wwwext.arlut.utexas.edu/salsa/pdfs/SalsaUserManual_1150.pdf

 

In the equation (B.3) page 164

Line 1 of the equation B.3 is valid only if MCov is diagonal

Line 2 of the equation B.3 is always valid even if MCov is not diagonal

 

 

In the equation (B.4) page 164

Line 1 of the equation B.3 is always valid even if MCov is not diagonal

Line 2 of the equation B.3 is valid only if MCov is diagonal

 

In the equation (B.21) page 173

Line 2 is equal to Line 3 only if MCov is diagonal

Line 3 is always equal to Line 4 even if MCov is not diagonal

Line 4 is equal to Line 5 only if MCov is diagonal

Line 5 is always equal to Line 6 even if MCov is not diagonal

Line 2 is equal to Line 5 & 6 only if MCov is diagonal

 

 

My question is the following:

What is the best formulae to compute the Internal Reliability?

I think the best formulae is the equation (B.21) line 2, but I wonder if you agree with me.

 

Thanks a lot in advance for your answer!

 

Emmanuel Cledat, Professor of Photogrammetry at IGN (French Mapping Agency)

[Matthew Graham]

Professor Cledat,

Thank you for reaching out and asking such a detailed question. We really appreciate getting these kinds of questions about our software and documentation. 

To answer very briefly, yes we agree that line 2 of B.21 is the best formula to use to compute internal reliability. 

I've also provided a little more explanation and discussion below that I think may provide some helpful context. 

You've correctly noted that some of the equations in Appendix B explicitly require a diagonal covariance matrix. These formulae follow from similar derivations in Ghilani's textbook (Adjustment Computations) and Leick's textbook (GPS Satellite Surveying), which both explicitly require a diagonal covariance matrix. 

The diagonal covariance assumption is made because even in cases where the measurement covariance is not diagonal, one can always transform the original measurements using a whitening transformation that will yield a diagonal covariance matrix. For example, in the SALSA code, we use the Cholesky decomposition (MCov = LL^T) as the whitening transformation before computing the least-squares solution.

We are currently working on revising and updating our SALSA documentation, specifically Appendix B of the user manual. As part of those revisions we will be adding additional text to explicitly note the diagonal covariance assumption that applies to the reliability formulae. We will also be providing more details about how SALSA applies whitening transformations when solving the least squares problems. 

I hope that answered your question. If you have any additional questions or follow-ups, please feel free to reach out. 

Matthew Graham 

[Manu Cledat]

Dear Matthew Graham,

 

 

Thanks a lot for your quick, accurate, clear and sharp answer! Thus, I will only consider the line 2 of B.21 of your user manual to compute internal reliability.

 

Thanks also for the reference to [Ghilani 2006] (I’ve found the Fourth Edition but I haven’t found the Fifth one) and for the reference to [Leick 2004].

 

 

Line 2 of equation (B.21) page 173 of SALSA userManual corresponds to

-the left side of equation (4.358) of [Leick 2004]

-equation (21.19) of [Ghilani 2006]

It seems to be the best way to compute internal reliability. However, I have to confess that I don’t really understand the demonstration of this equation, nor even the notation of equations (11.10) & (11.12) of the very famous [Baarda 1968].

 

 

Line 6 of equation (B.21) page 173 of SALSA userManual corresponds to

-the right side of equation (4.358) of [Leick 2004]

-equation (4.363) of [Leick 2004]

 

 

I will be glad to see any further updates either in your software or in this very complete user-manual!

 

Thanks again for your answer.

 

Emmanuel Cledat

 

 

[Ghilani 2006] Charles D. Ghilani, Paul R. Wolf, « ADJUSTMENT COMPUTATIONS Spatial Data Analysis » John Wiley & Sons, Inc, Fourth Edition 2006

[Leick 2004] A. Leick, “GPS Satellite Surveying”. John Wiley & Sons, Inc., 3rd ed., 2004.

[Baarda 1968] “A testing procedure for use in geodetic Networks” NETHERLANDS GEODETIC COMMISSION PUBLICATIONS ON GEODESY NEW SERIES VOLUME 2 NUMBER 5, 1968