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Re: Linear regression and other tools
From: Tony Oz
Date: 2018 Oct 13, 15:34 -0700

Hello!

Please consider a fictitious adventure. A navigator (apparently capable to see the stars in the daylight) obtains the folowing H_Cs on 2018-10-13:

12:00UT, 60°00'N;30°00'E: H_CDeneb = 43°19'12"; H_CCapella = 18°22'10"
12:10UT, 60°02'N;30°02'E : H_CDeneb = 44°32'18"; H_CCapella = 17°59'26"
12:20UT, 60°04'N;30°04'E : H_CDeneb = 45°45'59"; H_CCapella = 17°39'02"
12:40UT, 60°08'N;30°08'E : H_CDeneb = 48°14'50"; H_CCapella = 17°05'17"
12:50UT, 60°10'N;30°10'E : H_CDeneb = 49°29'51"; H_CCapella = 16°51'59"
13:00UT, 60°12'N;30°12'E : H_CDeneb = 50°45'07"; H_CCapella = 16°41'05"

When he takes the sights he does not know his position yet.

Our navigator also posesses a TI-83+ calculator using which with the above data he finds the following approximating functions (QuadReg function):

for Deneb's H_C: 0.13199529·x² + 4.154611098·x -25.42931898
for Capella's H_C: 0.709664195·x² - 19.42687497·x + 149.3005809

He then substitutes the 12:30 as x in both equations and gets:

Deneb's H_Cat 12:30 is 47°00'12"
Capella's H_C at 12:30 is 17°20'59"

Now he looks-up GHA_Aries for 12:30 (which is 209°31'20") and finds himself at 60°06'N;30°06'E using the two-simultaneous-altitudes SR. That fix is perfectly equal to his real position at that time.

If he choose just to average the H_Cs he would get somewhat different values: 47°01'05" and 17°26'30". One is almost there while the other is quite the way off.

Considering the inerval (one hour) - the quadratic regression did the great job to recover good approximation functions. Bi-quad or cubic are only marginally better.

This, I think, proves that interleaving sights of several bodies done even in rather longish time can give quite good fix.

I hope it is not unrealistic to fit 5 interleaved sights of two bodies into ~15~20 minutes.

Please comment.

Regards,

Tony

60°N 30°E

GHA_Aries was calculated with the same TI-83+ using the formulas from Hennig Umland's excellent "Short Guide to Celestial Navigation" (chaper 15, long-term almanac).

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