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    Re: Taking four stars for checking accuracy of fix - and "Cocked Hats"
    From: Peter Fogg
    Date: 2008 Aug 3, 19:37 +1000

    Geoffrey Kolbe wrote:
    >
    > Since, in this discussion, my name has been associated with a fix taken
    > with four bodies at the cardinal points, I have a few observations to make.
    
    Go Geoffrey.
    
    > Why can't you correct for systematic error first? The point about
    > systematic errors is that for any particular instrument, they can be
    > identified and measured and thus accounted for in a measurement on that
    > instrument.
    
    Sorry, Geoffrey, I thought I explained this:
    "You CAN'T 'correct systematic errors
    first' since their correction leads to a single point - a fix
    position.  How are you then going to go about correcting for your ...
    whatever you like to call the other, erratic variety?"
    
    And as George has pointed out, the constant error could be time-based,
    it does not need to be related to any particular angle-measuring
    instrument.
    >
    > I have used this example before, but as it seems appropriate I make no
    > apologies in using it again.
    >
    > Look at http://www.pisces-press.com/C-Nav/nav-plots/14%20March%20big.jpg
    > and you will see a plot of an actual fix in the Egyptian Western desert,
    > worked up with observations of bodies at (or close to) the cardinal points.
    > There are a number of points to note:
    >
    > First, each of those LOPs is actually the result of five measurements on
    > the body itself. Averages were taken of the five altitudes and times of
    > observation - usually a minute apart - and then an Hc and Zn was calculated
    > for the average time and altitude. Peter's method is to plot altitude
    > against time and draw a straight line through the results.
    
    No.  This is back to front. Peter's method (I certainly didn't invent
    it - and David Burch has a nice explanation and example on a Starpath
    page) is to establish a FACT (albeit a simplification),  the ACTUAL
    line of apparent rise/fall, against which to compare the round of
    sights.  This is not a fine distinction, it is a most vital one.  It
    is necessary to understand that this works in the opposite direction
    to the way averaging normally functions.
    
    Usually, in situations where averaging can be helpful, your data
    points, imperfect as they may be, are all you have to work with.  Then
    averaging can usefully reduce erratic error (only), and derive
    (assuming the data is affected purely randomly) an improved result
    from those data points.  Usually this is the best that can be done,
    without other knowledge of what the result should be.
    
    But this situation is different.  We don't need to derive the line
    from the sights, its the other way around: the line is a FACT.  The
    sights HAVE TO conform to the line, not the converse.  Your 'other way
    around' constitutes a regression analysis.  THIS IS INHERENTLY
    INFERIOR.
    
    > This amounts to
    > much the same thing and - if the straight line is drawn correctly - my
    > average should sit on a straight line fit through the results.
    
    My advice is to forget about 'averaging'.  This is something else.
    Averaging is inherently inferior since it accords equal weight to all
    data points.  Why would you want to do this when the FACT of the
    correct line, the one your sights would match in a perfect world, is
    there in black and white in front of you?  You are talking about
    deriving a line from the sights, but what you need to do is see how
    well your sights agree with the ACTUAL line, not your derived one,
    which may rely on one or more bad bits of information.  If I remember
    correctly, the Starpath page has an example that illustrates this
    nicely - does anyone have that link to hand?
    
    Deriving a line from your data is misleading if its the wrong line.
    It will then lead to the wrong LOP.  Why not start with the right
    line?
    
    > As George has mentioned, taking averages of data to minimize random error
    > in data does not get rid of the random error, as Peter seems to imply.
    
    Forget about averaging.  It is only confusing you.  And forget about
    'implying', I am trying to explain very clearly and openly how
    comparing the sights taken with the actual line leads to an improved
    LOP, freed to a useful extent of random error.
    
    > I would also caution against throwing out data. In my experience, I have
    > ended up with a better Hc by keeping all the altitudes - even the "weirdo"
    > ones. It is fairly easy to see if a succession of sights on the same body
    > are giving good results. Given that the time between sights will be fairly
    > constant, the difference in measured altitude should also be fairly
    > constant. If I see it is wandering about for some reason, I take more
    > sights and let statistics take care of whatever is getting in the way of me
    > taking accurate sights.
    
    Ah dear.  A true recalcitrant.  Normally, that is when you don't know
    which data points are good or bad, you certainly would not 'throw out
    data'.  But this is different.  You DO know which points are much more
    likely to be useful.  If most of the sights follow the line, more or
    less, and one obviously doesn't, then either all of the others are
    consistently wrong, and the errant one correct, OR it must be the
    other way around.  I can only see those two alternatives.
    >
    > Second, it will be seen that the LOPs in the picture form a box, with the
    > bodies observed always on the far side of the box. From this, I can deduce
    > that the dominant error in the data is now a systematic error, which in
    > this case is an index error due to the desert heat warping the sextant or
    > index arm slightly.
    
    Yes.  This is what I have been advocating.  Reduce erratic error at
    source then assume that what is left must be systematic, from
    wherever, and eliminate it by bisecting those crossing LOPS, thus
    deriving a fix at the centre of the shape.  A fix thus freed, to a
    useful extent, of both kinds of error.
    >
    > Given that the box is about 10 minutes of arc on a side, I can see that
    > there is some 5 minutes of  unaccounted for systematic error in the data.
    
    > Now look at
    > http://www.pisces-press.com/C-Nav/nav-plots/19%20March%20big.jpg for a fix
    > taken a few days later, at our next campsite a little further North. This
    > time I included the extra 5 minutes of systematic error which I found a few
    > days earlier and the result is pretty much the single point fix Peter was
    > talking about.
    >
    > So, to sum up, this is an example where I DID "correct systematic errors
    > first" and where their correction did lead to a "single point fix". (Of
    > course, where the pen is drawing a line which is about a minute wide, a
    > "single point fix" should be taken in context.)
    
    You got a single point because you will always get a single point.
    But unless you have analysed the random error present, then got rid of
    it to a useful extent, it will still be there.  And so we're back to
    having three times more chance of not being within the shape.  Wrong
    approach.
    >
    > It should be noted though, that the final fix was still a minute or so away
    > from the actual position. In this case, it is not possible to determine if
    > the residual errors are random or systematic. There are ALWAYS residual
    > errors in data.
    
    Yep.
    
    Where random and systematic errors have been reduced to the
    > precision of the instrument, it is not worth the effort to try and reduce
    > these errors still further. With this instrument (an A-12 bubble sextant)
    > an accuracy of 1 or 2 nautical miles on the fix is as good as it is going
    > to get.
    
    Sure.
    >
    > On the subject of "COCKED HATS".
    >
    > In the past, I have always agreed with George that where there is no
    > systematic error in the measured altitudes, the probability of the actual
    > position being outside the cocked hat is three times higher than being
    > inside. But my position has shifted slightly on this one. I wonder if I can
    > induce George to come along with me. (In the following analysis, assume
    > that there is no systematic error of any kind).
    >
    > First, consider a measured altitude on Polaris. On reducing the sight, I
    > will end up with an LOP which will run East-West across the chart. What can
    > I say about my actual position relative to this line? I can say that the
    > probability of my position being on or near one point of the line is equal
    > to that for any other point on the line. (In other words, I have no
    > information regarding my longitude). I can say that the probability of my
    > actual position being North of the line is 0.5, exactly the same as the
    > probability of being South of the line.
    >
    > Now, suppose I take a sight on the star Canopus when it is on the meridian
    > to the South. I reduce the sight and draw another LOP on the chart. This
    > LOP also runs East-West across the chart, but I find that it is about 2
    > nautical miles (say) to the South of the Polaris LOP.
    >
    > What can I say about my position relative to this new Canopus LOP? Since it
    > runs parallel to the Polaris line, I still have no new information about my
    > longitude. The chance of me being on or near one point of this line is just
    > the same as for any other point. What can I say about my position North or
    > South of the line? Is the probability of my actual position being to the
    > North of the Canopus LOP still 0.5, exactly the same as it being South of
    > the line? I would now say no.
    >
    > Because the Polaris LOP sits to the North of the Canopus LOP, I have an
    > independent piece of data which indicates that my position is actually more
    > likely to be North of the Canopus LOP than South of it. Statistically
    > speaking, this is a so-called "Bayesian" approach to statistical analysis.
    > I have some a priori information regarding my actual position, which can
    > inform my estimation of my position with respect to the Canopus LOP.
    >
    > Actually, there is no reason why I should have taken the Polaris sighting
    > first. I could just have easily have taken the Canopus sighting first. That
    > being the case, I can say by symmetry that the Canopus LOP can similarly
    > inform, regarding my position relative to the Polaris LOP. Now, I can say
    > that the probability of my position being South of the Polaris LOP is
    > higher than the probability of being North of it.
    >
    > I will stop there as ...
    
    Looks good to me, at first glance.
    
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