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[REBOL] Re: GPS: minimizing sum of squares algorithm

From: jjmmes:y:ahoo:es at: 11-Nov-2002 21:21

Hi Tom, Thanks for helping me out. I think the general approach to solving this is with Jan's code, specially when you want to acount for 4+ data points. It turns out that with 3 data points I only have 2 sets of independent equations, u: f(x,y) and v: f'(x,y), so I only have to solve one set of 3 equations with 3 unknowns and can get the unknows expressed as a function of u,v . Regards, Jose --- Tom Conlin <[tomc--darkwing--uoregon--edu]> escribió:
> Hi Jose, > > Hmmm > > I recall "root mean square" from Calculus not Linear > and it was the sqrt of the area under f(x^2) between > a couple of points > if this is the correct interpretation I am not > seeing how it will help you > find your constants. > > I think you are closer to your mark in the subject > line > and may be looking for a "linear-least-squares" fit. > (or if you are less lucky a > "non-linear-least-squares" fit) > > disclaimer: I have only been through standard > undergraduate classes and > have not used this stuff in several years > (winter 99, and never in the real world). > > if I were you I would try to view what I needed > as 2 problems each with 3 unknowns since your > x and y equations seem independent and the amount of > work will > normally square with the number of unknowns, > that is 2*3^2 < 6^2 > > looking at just the x half, > a few gps samples you could have in something > similar to your notation > > 1*a + lat1*b + lon1*c = x1 > 1*a + lat2*b + lon2*c = x2 > 1*a + lat3*b + lon3*c = x3 > > or in matrix form > > | 1 lat1 lon1 | |a| |x1| > | 1 lat2 lon2 | * |b| = |x2| > | 1 lat3 lon3 | |c| |x3| > > in a more standard notation this is typically > represented as > > A x = b > > the linear-least-squares solution 'x' to Ax = b > is > > x = (A_t * A)^-1 * A_t * b > > where A_t is the transpose of A > (transpose means the indices for a row and column > are swapped > or that a matrix is reflected about its diagonal) > > and the (A_t A)^-1 means the inverse of the matrix > that results from > multiplying the transpose of A by A (order matters) > > more specifically it is the matrix you would have > to multiply (A_t * A) > by to get the "Identity matrix" where the identity > matrix is > a square matrix with 1's from the top-left to > lower-right and 0's > elsewhere > > a matrix must be square to have an inverse > There is no guarantee a square matrix has an > inverse. > > I am not going into inverting a matrix because there > are alternatives. > > all in all it is pretty expensive to go this way > but at least it is straight forward if you want to > plow through it. > > If you are going to do allot of these there is a > shortcut which > stems from the rearrangement > > A_t A x = A_t b > > where the two sides of the equation can be expressed > as > "Augmented Matrices" > > going back to something that resembles your notation > translated to matrix > form we get > > || 1 1 1| |1 lat1 lon1| : a| | 1 > 1 1 : x1| > ||lat1 lat2 lat3| * |1 lat2 lon2| : b| = |lat1 > lat2 lat3 : x2| > ||lon1 lon2 lon3| |1 lat3 lon3| : c| |lon1 > lon2 lon3 : x3| > > where the [a b c] & [x1 x2 x3] vectors are > augmenting the square matrices > > these augmented matrices (representing your > simultaneous equations) > may be solved by adding multiples of the rows in a > matrix to one another. > to end up with 1s, 0s, and solutions (if there are > any) > there are methods by Gauss, and an improvement > called "Gauss-Jordan" > which have been around for hundreds of years. > > in the end you end up with identity matrices > augmented with the solutions > > |1 0 0 : i*a| |1 0 0 : p| > |0 1 0 : j*b| = |0 1 0 : q| > |0 0 1 : k*c] |0 0 1 : r] > > so a = p/i > b = q/j > c = r/k > > there is a function "naive-gauss" in code I wrote > to double check > the longhand results required in my homework. > (the professor would not even allow a four function > calc in the class) > and was not finished beyond the point where it > served my immediate > purpose > > http://darkwing.uoregon.edu/~tomc/core/math/matrix.r > > that should work to reduce your augmented > equations. > > there is another "gauss-jordan" which would be > better if it did not have > a bug I never got around to tracking down > > note: > the "comp" function in the file was derived from > Eric Long's compare > > best of luck > > On Sat, 9 Nov 2002, [iso-8859-1] jose wrote: > > > a,b,c,d,e,f are constants. > > > > I basically want to solve a set of 6 equations > with > > six unknowns, using one of the existing linear > algebra > > algorithms. The algorithm implementation I'm > looking > > for is also known as RMS, root mean squares but > > couldn't find a good algorithm description in the > web > > > > Regards > > Jose > > --- Jan Skibinski <[jan--skibinski--sympatico--ca]> > > escribió: > Jose, > > > I could certainly help you with this but I would > > > need to know more about the shape of the > functions b > > > c e f > > > in order to model them appriopriately. For > example, > > > depending on what those functions represent a > > > polynomial > > > approximation might be of no use or it might be > > > quite good actually. > > > A reference to existing georeference algorithms > > > could > > > be a good starting point. > > > Regards, > > > Jan > > > > > > > > > jose wrote: > > > > > > > Has anybody written this algorithm in Rebol ? > > > > > > > > I need this to georeference gps data, i.e. > find a, > > > b, > > > > c, d, e, f below given three calibration > points >
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