Alex:
As you say in your P.S., you HAVE to have a functional model
describing your data for parametric least squares. So your proposed numerical example is incomplete. The assumption, of course, is that the model is a good approximation of the actual physical or mathematical relationship between the observations and the parameters to be estimated. The observations are assumed to have unknown random errors.
Let me illustrate with how GPS positioning works with redundant (more than 4) observations where we must estimate the receiver clock bias, which can be taken to be the same for all simultaneous observations.
Simplified model:
P = rho + c * dT
where P = measured pseudorange (includes a random error component)
rho = geometric range (distance between satellite and receiver)
c = speed of light
dT = receiver clock offset
rho is a non-linear function of the satellite coordinates (known) and the receiver coordinates
(unknown) as is dT. So, we have four unknowns: x, y, z, dT. With five or more simultaneous observations, we can estimate these parameters using least squares to get the "best" values.
Using vector/matrix representation, let X be the vector of unknowns, A is the matrix of partial derivatives of the observations with respect to the parameters (the "design matrix") and P is the vector of observations. Then
delta-X = A^-1 delta-P
where delta-X is the estimated increment to a starting value for X (X_0, from previous observations or some other knowledge or guesstimate) and delta-P are the differences between observed and computed values of the observables.
Then,
X = X_0 + delta-X.
Since this is a non-linear problem, iterations may be necessary until delta-X becomes sufficiently small. One can also compute a covariance matrix for the estimates, X, which comes from a propagation of the random errors in the measurements
into the estimated parameter values.
I'm fairly sure you could set up a parametric model for sextant observations that includes dip, which would allow you to estimate it (or refine a guess). And since you raised it in a subsequent posting, perhaps you could also (or alternatively) include a time error of the observations and estimate that. Both models would assume that the two nuisance parameters, the dip and the clock error, were constant for the suite of observations used.
I believe someone from USNO a few years ago wrote a paper that was published in the ION's journal Navigation on processing sextant observations using least squares (I haven't done it myself but I should). I'll try to dig it up.
-- Richard
P.S. I teach a course on introductory adjustment calculus (least squares) and another one that includes fundamental astronomy where the students use a T2 theodolite to reduce sun shots rather than a sextant.
On
10-Apr-12, at 10:29 AM, Alexandre E Eremenko wrote:
> Dear Richard,
> Sorry I do not understand your idea.
> I said "NO statistical method can eliminate such bias"
> Perhaps I am wrong. But then please explain me on a numerical example.
> Suppose you have a series of numbers,
> say observations taken at times 1,2,3,4,
> and the readings are 5,6,7,8.
> Please tell me from these data,
> what was the true quantity and what was the error.
>
> Alex.
>
> P.S. Parametric estimation in statistics must use some mathematical
> model of the data, the model involving parameters. From observation
> one can estimate these parameters, IF THE MODEL IS CORRECT.
> Unfortunately we do not have an appropriate mathematical model
> of the anomalous dip, expecially how it varies with time.
>
> On Tue, 10 Apr 2012, Richard B. Langley wrote:
>
>>
>> Thanks, Alex, but I was not talking about ordinary averaging but the
>> use of parametric least squares, which is able to estimate the value
>> of a bias along with the parameters of interest. So, if we have a
>> series of observations for which we can assume that the bias was
>> reasonably constant, then by simultaneously processing the complete
>> set, one should be able to get a single estimate of position and the
>> value of the bias (dip).
>> -- Richard
>>
>> On 10-Apr-12, at 9:49 AM, Alexandre E Eremenko wrote:
>>
>>> Dear Richard,
>>>
>>> Unfortunately, no statistical method, including least squares
>>> can help with dip. The reason is that dip can deviate from its
>>> normal value for relatively long periods.
>>> For example, if our much discussed
observation with Bill B on lake
>>> Michigan is explained by the dip (which a majority on the list seems
>>> to believe), this anomalous dip persisted for several hours,
>>> and was almost constant. (This is an extreme example of course).
>>> What averaging (or least square) helps to eliminate is a
>>> SUM of MANY small INDEPENDENT errors.
>>> The error of the dip is not a "random" error but a "systematic" one.
>>> And the only way to eliminate it is the use of some dip-meter device.
>>>
>>> However, we know that dip-meters were rarely used.
>>> (Western manuals almost never mention the device,
>>> Soviet ones do mention, and recommend, and it was a standard
>>> equipment,
>>> but the same manuals recognize that "people do not use it").
>>>
>>> This only shows that
navigators did not care about anomalous dip.
>>> That high accuracy in celestial navigation was not needed,
>>> and that large variations of the dip are probably rare.
>>>
>>> Alex.
>>>
>>> On Tue, 10 Apr 2012, Richard B. Langley wrote:
>>>
>>>>
>>>> Warning: academic exercise follows ;-)
>>>>
>>>> Perhaps if one has sufficient redundant observations and uses least
>>>> squares to estimate position, one could include dip as an additional
>>>> quantity estimated simultaneously from the (biased) observations. The
>>>> same procedure is used to process GPS measurements where one of the
>>>> "nuisance" parameters is the offset of the receiver's clock from GPS
>>>> System Time, which is generally unknown.
>>>>
>>>> -- Richard Langley
>>>>
>>>> On 10-Apr-12, at 1:31 AM, Antoine Couëtte wrote:
>>>>
>>>>> Still, your observations once again point out that DIP is definitely
>>>>> one "weak link" in the accuracy computation chain, since even under
>>>>> (quite) good conditions, dip standard deviation was already close to
>>>>> 0.15/0.20 arc minute.
>>>>>
>>>>
>>>> -----------------------------------------------------------------------------
>>>> | Richard B. Langley E-mail:
>>>> lang---ca |
>>>> | Geodetic Research Laboratory Web:
http://www.unb.ca/GGE/>>>> |
>>>> | Dept. of Geodesy and Geomatics Engineering Phone: +1 506
>>>> 453-5142 |
>>>> | University of New Brunswick Fax: +1 506
>>>> 453-4943 |
>>>> | Fredericton, N.B., Canada E3B
>>>> 5A3 |
>>>> | Fredericton? Where's that? See: http://
>>>> www.fredericton.ca/ |
>>>> -----------------------------------------------------------------------------
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> :
http://fer3.com/arc/m2.aspx?i=118883>>>>
>>>>
>>
>> -----------------------------------------------------------------------------
>> | Richard B. Langley E-mail:
>> lang---ca |
>> | Geodetic Research Laboratory Web:
http://www.unb.ca/GGE/>> |
>> | Dept. of Geodesy and Geomatics Engineering Phone: +1 506
>> 453-5142 |
>> | University of New
Brunswick Fax: +1 506
>> 453-4943 |
>> | Fredericton, N.B., Canada E3B
>> 5A3 |
>> | Fredericton? Where's that? See: http://
>> www.fredericton.ca/ |
>> -----------------------------------------------------------------------------
>>
>>
>>
>>
>>
>> :
http://fer3.com/arc/m2.aspx?i=118885>>
>>
-----------------------------------------------------------------------------
| Richard B. Langley
E-mail:
lang@unb.ca |
| Geodetic Research Laboratory Web:
http://www.unb.ca/GGE/ |
| Dept. of Geodesy and Geomatics Engineering Phone: +1 506 453-5142 |
| University of New Brunswick Fax: +1 506 453-4943 |
| Fredericton, N.B., Canada E3B 5A3 |
| Fredericton? Where's that? See:
http://www.fredericton.ca/ |
-----------------------------------------------------------------------------
