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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 HCs on 2018-10-13:

    • 12:00UT, 60°00'N;30°00'E: HCDeneb = 43°19'12"; HCCapella = 18°22'10"
    • 12:10UT, 60°02'N;30°02'E : HCDeneb = 44°32'18"; HCCapella = 17°59'26"
    • 12:20UT, 60°04'N;30°04'E : HCDeneb = 45°45'59"; HCCapella = 17°39'02"
    • 12:40UT, 60°08'N;30°08'E : HCDeneb = 48°14'50"; HCCapella = 17°05'17"
    • 12:50UT, 60°10'N;30°10'E : HCDeneb = 49°29'51"; HCCapella = 16°51'59"
    • 13:00UT, 60°12'N;30°12'E : HCDeneb = 50°45'07"; HCCapella = 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 HC: 0.13199529·x2 + 4.154611098·x -25.42931898
    • for Capella's HC: 0.709664195·x2 - 19.42687497·x + 149.3005809

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

    • Deneb's HCat 12:30 is 47°00'12"
    • Capella's HC at 12:30 is 17°20'59"

    Now he looks-up GHAAries 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 HCs 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

    PS

    GHAAries 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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