From: Ralph Hartley (hartley_at_aic.nrl.navy.mil)
Date: Mon Feb 12 2001 - 15:33:44 CET
Received: (from mdom_at_localhost) by karto.ethz.ch (8.9.3/8.9.3/SuSE Linux 8.9.3-0.1) id PAA08061 for cavexml-outgoing; Mon, 12 Feb 2001 15:30:31 +0100 Received: from sun0.aic.nrl.navy.mil (sun0.aic.nrl.navy.mil [132.250.84.10]) by karto.ethz.ch (8.9.3/8.9.3/SuSE Linux 8.9.3-0.1) with ESMTP id PAA08057 for <cavexml_at_cartography.ch>; Mon, 12 Feb 2001 15:30:31 +0100 Received: from aic.nrl.navy.mil (pc31.aic.nrl.navy.mil [132.250.84.181]) by sun0.aic.nrl.navy.mil (8.9.3+Sun/8.9.3) with ESMTP id JAA04963 for <cavexml_at_cartography.ch>; Mon, 12 Feb 2001 09:30:33 -0500 (EST) Message-ID: <3A87F448.8000501@aic.nrl.navy.mil> Date: Mon, 12 Feb 2001 09:33:44 -0500 From: Ralph Hartley <hartley_at_aic.nrl.navy.mil> User-Agent: Mozilla/5.0 (X11; U; Linux 2.2.16-22 i686; en-US; m18) Gecko/20010124 X-Accept-Language: en To: cavexml_at_cartography.ch Subject: Re: Other areas that haven't been discussed. References: <Pine.GSO.4.05.10102111518080.26377-100000_at_cor.oz.cc.utah.edu> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: owner-cavexml_at_karto.baug.ethz.ch Precedence: bulk Reply-To: cavexml_at_cartography.ch
John Halleck wrote:
>
>>
>> From: Ralph Hartley <hartley_at_aic.nrl.navy.mil>
>>
>>> > If a measurement is marked reliability="godgiven" any program should
>>> > treat it as fixed, and apply no corrections, adjustments, etc. to it.
>>> > Loop closures should consider its variance to be zero. It is the users
>>
>> > If you have more than one point with variance zero, then the least
>> > squares adjustment has a singular matrix, and the adjustment is
>> > therefore invalid.
>>
>> > I don't think you understand the ramifications of zero variance.
>>
>> I think I do. I don't think there are any problems unless there is at least one loop.
>
>
> Matrices with zero rows are singular.
> Inverting such a matrix is invalid.
> Zero weight shots give zero rows. (Whether or not they are in a loop, if they are in
> the adjustment.
Shots marked reliability="godgiven" would (if just plugged blindly into
a matrix, which is not a good idea) be infinitely weighted, not zero. Of
course, if one is not careful, that could be just as bad. If one insists
on just dropping everything into one big matrix, one could use an
arbitrary large factor instead. You would have to worry about numerical
stability problems, but you should be doing that anyway.
It's the reliability="error" shots that should be given zero weight. The
reasonable thing to do with them is to leave them out of the matrix
altogether, not to add zero rows.
While it is true that a singular matrix has no inverse, there are
actually a number of techniques for getting useful information in a
situation that calls for the inverse of a matrix, but where the matrix
may be singular. In fact, least squares itself is the oldest of those,
but there are others.
>
>> I should have said that loops specified as goodgiven should not be closed.
>
> If they are good given, why would they missclose to begin with?
They shouldn't, and that does make the feature a bit dangerous. What I
mean is that a program, when given such loops, is permitted to assume
that the loops close perfectly, and is not responsible for the
consequences if that assumption is wrong. Different programs could use
that information in different ways, or not at all. From the users point
of view, if you mark all the shots of a loop as godgiven, the data had
better really BE perfectly accurate (or at least consistent).
In practice, it would be very unusual for an entire loop to be marked
godgiven. It would be most useful for defining things like coordinate
systems.
>
>> You would use this, for example, to set the
>> coordinates of the datum to (0,0,0), which by definition is exact. It
>> should be used very sparingly.
>
> If you have a single control point, then the weighting of it is
> going to cancel out, and a numerically stable adjustment is not
> going to move it anyway.
This is true. But if you have more than one point who's coordinates are
given, one of them has to be singled out as fixing the coordinates,
while the others would be subject to adjustment.
> May I suggest that the details of LS adjustments is probably best
> discussed off line, or by reference to any good book on the topic.
> I recommend either Mikhail and Gracie's "Analysis and adjustment
> of survey measurements" or Wolf and Ghilani's "Adjustment Computations".
I agree, we are straying a little from the topic of this mailing list.
Reply to me only. Anything that turns out to be relevant can be reposted
to the list.
Ralph Hartley
PS Strictly speaking, adjustments to the weights given to measurements,
ether a priori or depending on residual errors etc. are not part of
least squares. They are methods often applied along with least squares,
or parts of algorithms that use least squares to compute other things.
Least squares itself takes the weights (equivalently variances) as
given, and is characterized as the algorithm that gives the best
estimate if the errors are independent, normally distributed, and have
the given variances. The other algorithms are useful in the (common)
cases where those assumptions need to be relaxed.
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