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    Re: Linear Regression In Reverse
    From: Peter Fogg
    Date: 2005 Jun 6, 04:30 +1000

    George says:
    
    > I'm still in a bit of difficulty over Peter Fogg's proposal for
    > determining
    > altitude of a body at the exact moment it's on the observer's prime
    > vertical (his East-West line).
    > How is that precalculation done, I ask?
    
    Anyway you like. I'm not an expert on the alternatives.
    
    > What's the expression that's used
    > to provide the moment when the body is exactly East or West?
    
    90 or 270 degrees of azimuth. You're the expert on determining azimuth
    George, remember.
    
    > I can see that
    > it could be determined by plugging an azimuth of 90 degrees into a
    > standard
    > navigational triangle, but what's the resulting trig expression?
    
    A practical approach (when all at sea) that dispenses with trig expressions
    is to observe the prime vertical well enough to find the exact azimuth later
    by reiteration. I use my nav calculator, plugging in the next day's date
    when I plan to make the observation. I should not be surprised to learn that
    a trig formula could be devised.
    
    > Presumably the precalculation is based on an assumed position. How
    > sensitive is the result to errors in that assumed position, and does that
    > matter?
    
    The same issues are involved as with any timed sight that, in the absence of
    a known position, relies on the best alternative, a DR or assumed position.
    
    > Does the method provide for reiteration if the observation shows up
    > significant errors in that assumed position?
    
    The method, such as it is, involves making a timed sight (via a series over
    five minutes) at a predetermined moment. I can only answer your question by
    speculating. I can't see any problem with using reiteration to advance an
    erroneous DR closer to the LOP. Would that closer DR then give a more
    accurate intercept? This is the assumption behind reiteration.
    
    > How important is it, anyway, to know the exact moment when the Sun is on
    > the prime vertical? As I see it, it's a very undemanding requirement.
    
    The requirement is to achieve an LOP that runs due north/south. Its up to
    you to determine what constitutes your acceptable level of sloppiness. Once
    the moment of prime verticality is known the azimuth has, by definition,
    been determined. What is left to do is to match that moment with its
    corresponding observed altitude, using Reverse Linear Regression, then use
    sight reduction to find the intercept. The moment is fleeting, compared to a
    noon sight, since the apparent movement is greater. On the other hand, if
    the divisions on the graph paper represent 10 seconds then that limits the
    precision achievable. All of this looks more demanding in prose than it is
    in practice. This is why I encourage anyone interested to use the technique
    - its an easier way to understand it.
    
    > Is this a precalculated slope? If so, how is it calculated? What's the
    > expression used to give the slope?
    
    Yes. Its a function of azimuth and latitude. For example, at 50 degrees of
    latitude the slope at prime vertical is about 48' and rising if east,
    descending if west. At 34 degrees of latitude the slope is about 62',
    ' indicating minutes of arc, measured on the axis of altitude. The question
    is: by how many minutes of arc does the body rise or fall over five minutes
    of time. It could be established by observation. The last time the issue got
    a good thrashing here, when most contributors were focused on 'averaging',
    someone volunteered a formula for determination of slope. It should be
    findable in the archives; look for subject lines including 'averaging'.
    I use the calculator in George Bennett's book.
    
    > What's the slope used for?
    
    It expresses graphically the movement over time of the apparent rise or fall
    of the observed body.
    
    > If the
    > calculated slope doesn't correspond with the observed slope of the sights,
    > around the precalculated moment, what happens next? Is something then
    > adjusted to fit?
    
    Its the slope (in practice, a line parallel to the slope) that is adjusted
    to best fit the pattern of sights, thus the opposite of linear regression
    which uses the pattern of sights to devise a slope that best fits them.
    
    Linear regression is used in a variety of non-nav situations where a number
    of data points is available. It devises a slope from them, enabling an
    analysis of their commonality. Like many statistical techniques, it seeks to
    extract useful information from raw data. It is not the best tool for this
    nav purpose since what it seeks to determine is a fact easily established -
    the slope. We work in the opposite direction, from the slope towards the
    perfect line of sights that matches it.
    
    > To clarify matters to me, let me ask: If you DID manage to make that
    > altitude observation at exactly that precalculated moment, would any of
    > this "slope" business be required?
    
    You ask good questions, George. No, if the sight is made at the right
    instant then none of this slope business is necessary.
    
    Except, of course, to indicate a better result than any of the individual
    sights, and to provide a picture (literally) of just how accurate the
    individual sights are. The slope corrects and analyses the pattern of
    sights.
    
    > Peter, please don't hesitate to say more, in response to these requests,
    > to
    > explain just what you do and how you do it. In those circumstances, nobody
    > will accuse you of "just banging on about his idee fixe".
    
    Thanks for that. Again, its easier to do than to understand or write about.
    I'm happy to respond to specific requests.
    
    > I don't see how you can expect me to "provide my own example", when your
    > methods remain unclear to me (perhaps because of my own naivety). Only you
    > can do that, to illustrate those very methods.
    
    Tell me your latitude and I'll give you the slope. Its the only thing you're
    missing to get started.
    
    Studying statistics was rather depressing since I would sit through entire
    lectures and understand almost nothing. That came later, slowly, while
    working through exercises until I could do them. Understanding the
    mathematical logic behind them is something I am still struggling with. The
    point is: the direction of approach taken towards knowledge is important.
    You don't have to take the hard road if an alternative is available.
    
    > Well, I wasn't asking about the general topic of linear regression
    > (reversed or otherwise) but about the specific details of how Peter
    > himself
    > carries out his observation and analysis, questions that I have spelled
    > out
    > in detail above. Bennett's book provides no answers to those questions.
    
    If you have 'Bennett's book' then you have the best resource for this topic
    I know of.
    
    
    

       
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