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Re: Averaging
From: Herbert Prinz
Date: 2004 Nov 3, 11:19 -0500
From: Herbert Prinz
Date: 2004 Nov 3, 11:19 -0500
Alexandre Eremenko wrote: > As I understand from your last messages on > the topics, these authors only discuss a Least Square algorithm, > but DO NOT discuss why the simple averaging of the altitudes > can sometimes be wrong, that is they do not discuss non-linearity > of the altitude changes? Dear Alex, That's correct. Why would they? Simple averaging of the altitudes is always wrong. I mention the papers in that paragraph because they are examples for the rigorous method of "averaging". Linearity of of the altitude does not come into play here at all. You seem to be arguing that because the altitude function is linear in reasonable intervals, averaging the altitudes is permissible. Not so. Consider an overdetermined system of linear equations. The equations are linear, by definition! Ok? You cannot solve this system by any kind of simple averaging. M * x = a The correct method is to multiply with the transposed matrix M_t and solve the system that results from that. (M_t * M) * x = M_t * a The algorithm in the N.A. that I mentioned several times is an exact equivalent to this method. The reason why the N.A. procedure is iterative (just like the original St. Hilaire intercept method!) is that the LOPs aren't straight lines. Whether an iteration is necessary in actual practice depends on how good an approximation our linear system was. So you see, non linear equations are perfectly acceptable. They are NOT the source of the problem. All I am saying in my message of 2004-10-09, third paragraph, is that if you have more than two observations for a fix, you treat the problem as an overdetermined system. In my message of 2004-10-19, esp. paragraphs starting with "Second,...", and "Third, ...", I elaborated in more detail on the reasons for this. I say it again: One problem with averaging groups of sights individually before reducing them is that you minimize separately the residuals within each group of observations, instead of their total sum. I asked you whether you can justify this statistically and you did not address the problem. I find it all the more surprising that since then, you have claimed twice to have refuted my "exaggerated claim" that averaging was a thing of the past. I have not seen a refutation, yet. Best regards Herbert Prinz