A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Linear Regression In Reverse
From: Peter Fogg
Date: 2005 Jun 6, 04:30 +1000
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.