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Re: Finding The Symmedian
From: Peter Fogg
Date: 2010 Dec 29, 04:31 +1100
From: Peter Fogg
Date: 2010 Dec 29, 04:31 +1100
Gary LaPook wrote:
Thanks for putting together links to these relevant posts, Gary. Could they be taken collectively as saying that the Symmedian point is the 'least squares' centre?
According to Wiki:
"The Gauss–Markov theorem. In a linear model in which the errors have expectation zero conditional on the independent variables, are uncorrelated and have equal variances, the best linear unbiased estimator of any linear combination of the observations, is its least-squares estimator."
http://en.wikipedia.org/wiki/Least_squares
In other words, 'least squares' assumes non-correlated, unbiased, or random error.
The problem, it seems to me, is with the assumption that error, in our application, is entirely random. How can this assumption be supported? I don't see how it can be, and to date nobody has been able to do so here.
In our application of position lines derived from reduced sights that fail to meet at a point there is likely to be a mix of random and non-random error, and it is not possible after the LOPs create a triangle (or other shape, if there are more than 3) to separate random from non-random error.
There is a method to solve for non-random error, and that is to bisect the angles formed by intersecting position lines, and find the fix position where these bisector lines meet. If the spread in azimuths is less than 180d then the bisector lines will meet outside the triangle, and if the spread in the azimuths of observed azimuths is greater than 180d then the bisector lines will meet at the centre of the triangle - although a different centre than the Symmedian point.
Evidently, this method of resolving error assumes that the error is entirely non-random, although in practice, and especially if no method has been employed to deal with random error, there is likely to be unknown quantities of both present.
Therefore a rational approach to deal with both random and non-random error may consist of dealing with random error via, for example, an analysis of multiple sights of the same body over a short period of time - such as the slope method.
Then, having dealt with random error as best as can be practically achieved, it may be reasonable to then address non-random error via bisecting those angles and deriving a fix free (again; as best as can be practically achieved) of both random and non-random error.
In this case establishing the Symmedian point as the centre of the triangle would be incorrect, since the 'least squares' approach assumes random error only.
Incidentally, the '25% chance of the fix being within the triangle' contention relies on the same unsupportable assumption - that error is only random.
According to Wiki:
"The Gauss–Markov theorem. In a linear model in which the errors have expectation zero conditional on the independent variables, are uncorrelated and have equal variances, the best linear unbiased estimator of any linear combination of the observations, is its least-squares estimator."
http://en.wikipedia.org/wiki/Least_squares
In other words, 'least squares' assumes non-correlated, unbiased, or random error.
The problem, it seems to me, is with the assumption that error, in our application, is entirely random. How can this assumption be supported? I don't see how it can be, and to date nobody has been able to do so here.
In our application of position lines derived from reduced sights that fail to meet at a point there is likely to be a mix of random and non-random error, and it is not possible after the LOPs create a triangle (or other shape, if there are more than 3) to separate random from non-random error.
There is a method to solve for non-random error, and that is to bisect the angles formed by intersecting position lines, and find the fix position where these bisector lines meet. If the spread in azimuths is less than 180d then the bisector lines will meet outside the triangle, and if the spread in the azimuths of observed azimuths is greater than 180d then the bisector lines will meet at the centre of the triangle - although a different centre than the Symmedian point.
Evidently, this method of resolving error assumes that the error is entirely non-random, although in practice, and especially if no method has been employed to deal with random error, there is likely to be unknown quantities of both present.
Therefore a rational approach to deal with both random and non-random error may consist of dealing with random error via, for example, an analysis of multiple sights of the same body over a short period of time - such as the slope method.
Then, having dealt with random error as best as can be practically achieved, it may be reasonable to then address non-random error via bisecting those angles and deriving a fix free (again; as best as can be practically achieved) of both random and non-random error.
In this case establishing the Symmedian point as the centre of the triangle would be incorrect, since the 'least squares' approach assumes random error only.
Incidentally, the '25% chance of the fix being within the triangle' contention relies on the same unsupportable assumption - that error is only random.
