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RE 1 - Linear-regression-other-tools-TonyOz-oct-2018-g43065

RE 2 - Linear-regression-other-tools-Couëtte-oct-2018-g43078

RE 3 - Linear-regression-other-tools-TonyOz-oct-2018-g43083

In RE 1, Tony, you gave us a very interesting example of 2nd order regression over noiseless data during a 1 hour time span.

Perhaps we might infer from this example that 2nd order regressions work equally well over such time span under most (if not all) circumstances, at least for noiseless data.

In RE2, I invited you to slightly modify your example with higher altitudes. In RE 3 you very kindly replied that you would "certainly re-do the simulations" as per my suggestion.

Meanwhile I have been extremely impressed by the beautiful performance of your 2nd order regressions over such a significant 1 hour time period. And I have decided to run these suggested simulations on my own (Noiseless data again).

I have also been extremely careful to compute accurate 2nd order regressions, working from offset data in order to reduce their magnitudes. I am using the HP41 STAT PACK by Hewlett Packard and trust that all 6 published digits are significant. I also do humbly acknowlege that publishing so many digits is definitely an over-kill. Hope I am absolved here ! Nonetheless, this may help interested readers with crosschecking their own software.

Tony, exactly like you proceeded in RE 1, the data in red serve as a benchmark for the 2nd order regressions results. Accordingly they have not been included as part of the date feeding such 2nd order regressions.

And as a slight refinement to your own method, Tony, I am using the "known true position" to compute the various intercepts from the 2nd order regressions derived heights.

I then rated all results as a fonction of the indicated "Rating list", with the smaller the intercepts, the better the rates. I have verified that under all cases, the LOP's angles exceed 45°, and - since these are noiseless data - their effects of the dilution of position are greatly minimized, if not totally wiped out.

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Here are my findings about the performance of the 2nd regressions. Interested readers can directly cross-check them from the enclosed document.

1 - Cases # 1a & 1b

Fom 60°N for bodies far from the Observer's Meridian with computed regression height at the exact time center of the observations set.

We are getting excellent intercepts. Hence 2nd order regressions do perform beautifully well over one hour time-span.

2 - Cases # 2a and 2c

From 60°N for bodies close to the Observer's Meridian with computed regression heights at the exact time center of the observations set with one height exceeding 72° or even 85° and the other one below 27°.

We are still getting either excellent or very good intercepts. 2nd order regressions still perform extremely well during one hour time-span under such environment.

2 - Cases # 2b and 2d

From 50°N for bodies close to the Observer's Meridian with computed regression heights at the exact time center of the observations set with one height exceeding 72° or even 85° and the other height not exceeding 22° .

We start getting either good or fair intercepts. 2nd order regressions still perform adequately well under such environment during a 1 hour time span.

3 - Case # 3a

From the Equator with both heights between 70° and 82°and the computed regression height still at the exact time center of the observations set.

We are getting an unsafe intercept. Then avoid the use of 2nd order regression over such a 1 hour time-span.

4 - Cases # 3b and 3c

From the Equator, with both heights between 70° and 82°and the computed regression heights at one end of the observations set.

We are getting totally unsafe intercepts. Then absolutely reject the use of 2nd regression over such a 1 hour time-span.

Lessons learnt

1 - The initial Examples 1a and 1b environment covers absolutely ideal conditions:

1.1 - High Latitude (60°). It is good to remember that the closer from a Pole the smaller the heights variations over a given time span. And:

1.2 - The samples are equally distributed over time. It has the effect of greatly minimizing odd order terms, which is excellent fwhen dealing with an even (2nd order) regression. And:

1.3 - The heights computed from the 2nd order regression are significantly constrained on both sides because they lie at the exact time center of the data set.

1.4 - The bodies are far from the Observer's meridian, hence the second order terms are minimized too. And:

1.5 - The bodies culmination heights - not observed here - stay reasonable.

2 - The Examples 2a and 2c still cover good conditions:

Altough one height of the data set reaches 75° while the other one does not exceed 27° the environmental factors 1.1, 1.2 and 1.3 listed immediately here-above do prevail. 

3 - The Examples 2b and 2d still cover good to fair conditions:

We are now working from 50°N with one height exceeding 72° or 85°. Then the cumulated effects contrary to 1.1, 1.4 and 1.5 start playing. However 1.2 and 1.3 become quite strong and we are still getting rather reliable 2nd order regressions. 

4 - The Examples 3a, 3b, and 3c : all unsafe id not very unsafe.

Environmental condition 1.3 here-above keeps limiting the damages in 3a.

But when condition 1.3 no longer applies like in 3b, and 3c, even with evenly spaced samples, the results are terrible.

As  a summary:

The closer from the poles, the better a 1 hour time-span 2nd regression performs.

Computing a 2nd order regression for a time close to the center of time-span greatly helps, and getting equally spaced time-samples also helps in a lesser way.

For cases 1a, 1b, 2a and 2c, I did not compute a linear regression for a time at one end of the 1 hour time-span. I leave it to the interested readers. Given the strenght of environmental condition 1.3, very likely the 2nd order regressions will perform not so well as published here.

Therefore, we have to be careful about blindly using any kind of regression without knowling its limitations. The examples here-above can give a good a priori feeling about their use for given environmental observations conditions.

Comments welcome.

Antoine M. "Kermit" Couëtte