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Star*net Least Squares question
Posted by rfc on May 10, 2016 at 2:50 pmTrying to write the formulas to answer this, but getting lost:
Let’s say you have a series of angle observations: Direct A1, A2, A3…A8, and reverse A9, A10, A11…A16.
then you repeat that series n number of times.
Is there any difference in the mean between adjusting the observations 16 at a time, and then meaning the n sets, vs. taking all 16*n observations and adjusting all at once? Will the answer be any different?rfc replied 8 years, 2 months ago 5 Members · 13 Replies 
13 Replies

If you don’t want to wrestle through the statistics, why not try it both ways in Star*Net?

Jim Frame, post: 371430, member: 10 wrote: If you don’t want to wrestle through the statistics, why not try it both ways in Star*Net?
Haven’t taken all the observations yet; still planning.

If you are only inputting angles taken from one setup point, then yes it should give the same final estimate.
When mixed with distances and other angles in a triangle, the nonlinear functions of angles create a theoretical difference, but if we are talking about arc seconds of std error, that is not significant.
Doing them all at once automates the process of examining goodness of fit and posteriori statistics.
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rfc, post: 371432, member: 8882 wrote: Haven’t taken all the observations yet; still planning.
It’s easy enough to create a dummy data set.

If you measure angle once, you enter it.
Twice, enter both. Thrice enter all 3. And so on.If you measure an angle 10x, and enter its mean as one angle, (not 10 angles), the std err of that angle is quite small, and not the global 23″ entered in options. So you would have to calc and enter the std err for that one angle.
Consider an angle measured by 4 sets and entered as one mean angle. And another angle by 8 sets and its mean value entered. The 8set angle is more reliable than the 4set angle, but Starnet will have only 1 occurrence for each, and each angle will therefore have identical weights. Starnet won’t know a 16set angle from a 1set angle, unless you independently enter a unique std err for each. (the 1set angle should be the global err entered in options.)
If you measure an angle 10 times, and enter it 10 times, then Starnet will take all 10 occurrences at global a priori std err for each, and the repetition will strengthen that angle by the statistics. And the 4set angle occurring 4 times as well. And the one set angle, and so on.
If you measure an angle 10x, and 7 of them are +/ 2″, and 3 of them are 45″, let Starnet determine the individual residuals of each occurrence.
But, that’s pencil and paper angle recording. A TS with a data collector may return a unique std err for each angle if measured repeatedly, which will over ride the global err setting in Starnet. A data collector may be doing a lot of statistics real time. You have to know what it is returning in the output.

Google: Theodolite Observations and Least Squares.
There’s a free pdf version from UNSW. 
Good article. And good points on entering both F1 angles and F2 angles letting the LSA software perform the statics. And the errors associated with the ‘grand mean’ of an angle are not the global errors, but may be unique.

And, if an angle is measured 10x, that set of angles may have a std dev of 2″. That same angle measured 20x may have have a std dev of 2″. (Std dev doesn’t decrease with population.) But the std err of a 20set angle would be much smaller than the std err of a 10angle. (Std error is inversely proportional to population.)
So beware of mistaking std dev for std err.
I recently ran a network with direction sets (paper and pencil recording) ranging from 4 to 20 sets, and I entered all sets (direction sets really) D and R separately. If I entered ‘mean values’ then the high repetition direction sets would have to have a lot spreadsheet error determination preadjustment to standardize the std err. Entering all data allowed for global std err to be used.

Scott Zelenak, post: 371456, member: 327 wrote: Google: Theodolite Observations and Least Squares.
There’s a free pdf version from UNSW.That’s a cool document. I’ve not read it before. It explains things somewhat differently than Ghilani and Wolf does.
In answer to Bill93’s question, yes, all the observations are from one point from the same backsight, but they’re astro shots, not fixed angles. I’m not using Star*net to reduce each set; I’m using Larry’s fabulous spreadsheet. His sheet works, as Star*net can’t, in that each observation in the set is different from the last. My plan is to simply start adding the meaned results from that sheet over time to a Star*net adjustment. As I add more and more, I would guess that the std errors would continue to get smaller and smaller as the number of sets increases.

Larry Scott, post: 371469, member: 8766 wrote: Std error is inversely proportional to population
Standard error is inversely proportional to the square root of the number of measurements from a given population.
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An Astro azimuth is a little different. The FS is moving. And the spreadsheet does generate a std dev per set. (I guess std error too would be good, but std error of small populations is sometimes not realistic.)
So, one set, (48 D, 48 R), returns one 1 azimuth. Enter it.
Day2, another set observing the same line, enter it too. Later in the day, another D/R set of the same line, enter it also.
Day 3 ….
With a single ‘grand mean’ of multiple days or sets, you’ll have to supply the error estimate. Take 5 sets, enter 5 azimuths with equal std err each. The mean of 5 sets, entered as 1 azimuth observation is a smaller std err.
However, if you are measuring the same line repeatedly, (i.e.; only one azimuth for the network) it won’t matter since there will be only one azimuth and the residuals will only reflect the averages of that azimuth. If you are observing azimuth multiple times on multiple lines, then take care in the std err of each. Your angles are are better than Astro az, so the azimuths should receive residual – best fit orientation.
But one line with many azimuth obs, Starnet will mean them out for you. And you’ll probably not have a big population like 12. But 46 maybe?

As for std err of Astro azimuths. If you accumulate sufficient azimuth data of one line, by independent sets of that same line, it would be best to independently determine the std err. ‘Text Book’ values of std err of Astro may not reflect your observed, empirical, data.

Larry Scott, post: 371481, member: 8766 wrote:
And you’ll probably not have a big population like 12. But 46 maybe?Depends on how many sunny days we get here in Vermont in the next 6 weeks:D
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