A package which calls itself "an industry-leading scientific graphing and data analysis software" suggests breaking the samples into real and imaginary parts, and fitting curves to each. Hmmmph. I guess it is not a common task that they could be bothered coding. Now surely, one can just set up the Vandermonde matrix, where the elements are the sums of x, x squared, x cubed et cetera. Or with complex data, would it end up as Hermitian (upper triangle is conjugate of lower triangle)?
polynomial fitting for COMPLEX data
Started by ●November 11, 2016
Reply by ●November 12, 20162016-11-12
<mbjorn@y7mail.com> wrote:>A package which calls itself >"an industry-leading scientific graphing and data analysis software" >suggests breaking the samples into real and imaginary parts, and >fitting curves to each. Hmmmph. I guess it is not a common task >that they could be bothered coding.Are the abscissae complex, or are they real? If real, I think doing this is the appropriate answer even if the range is complex. If complex, you can still do a Lagrangian interpolation but it would take some working out of equations. There are possibly also nuances depending on whether the abscissae are uniformaly spaced. Steve
Reply by ●November 13, 20162016-11-13
On Sat, 12 Nov 2016 09:03:59 +0000, Steve Pope wrote:> <mbjorn@y7mail.com> wrote: > >>A package which calls itself "an industry-leading scientific graphing >>and data analysis software" >>suggests breaking the samples into real and imaginary parts, and fitting >>curves to each. Hmmmph. I guess it is not a common task that they could >>be bothered coding. > > Are the abscissae complex, or are they real? > > If real, I think doing this is the appropriate answer even if the range > is complex.Yes, at least if you're looking for a least-squares fit. If you're looking for a fit that minimizes some nonlinear cost criteria then the real and complex solutions may well interact.> If complex, you can still do a Lagrangian interpolation but it would > take some working out of equations. > > There are possibly also nuances depending on whether the abscissae are > uniformaly spaced. > > Steve-- Tim Wescott Wescott Design Services http://www.wescottdesign.com I'm looking for work -- see my website!
Reply by ●November 14, 20162016-11-14
On 14/11/2016 01:46, Tim Wescott wrote:> On Sat, 12 Nov 2016 09:03:59 +0000, Steve Pope wrote: > >> <mbjorn@y7mail.com> wrote: >> >>> A package which calls itself "an industry-leading scientific graphing >>> and data analysis software" >>> suggests breaking the samples into real and imaginary parts, and fitting >>> curves to each. Hmmmph. I guess it is not a common task that they could >>> be bothered coding.Indeed, Origin is very cool for the sort of problems where you don't have to write your own code.>> Are the abscissae complex, or are they real? >> >> If real, I think doing this is the appropriate answer even if the range >> is complex. > > Yes, at least if you're looking for a least-squares fit. If you're > looking for a fit that minimizes some nonlinear cost criteria then the > real and complex solutions may well interact. >If I get it correctly, in case of Gaussian noise the least-squares criterion is the same thing as maximizing the likelihood function. Incidentally, makes me feel a pity that while finding the global maximum of the likelihood function is a standard method in communications, in scientific packages fits work as some kinds of iterative processes, so if you want to search for the global maximum (useful when the problem is ill-defined) you have to write the code instead of just clicking a couple of buttons in a standard package. ;) Gene
Reply by ●November 14, 20162016-11-14
On Mon, 14 Nov 2016 16:07:32 +0300, Evgeny Filatov wrote:> On 14/11/2016 01:46, Tim Wescott wrote: >> On Sat, 12 Nov 2016 09:03:59 +0000, Steve Pope wrote: >> >>> <mbjorn@y7mail.com> wrote: >>> >>>> A package which calls itself "an industry-leading scientific graphing >>>> and data analysis software" >>>> suggests breaking the samples into real and imaginary parts, and >>>> fitting curves to each. Hmmmph. I guess it is not a common task that >>>> they could be bothered coding. > > Indeed, Origin is very cool for the sort of problems where you don't > have to write your own code. > >>> Are the abscissae complex, or are they real? >>> >>> If real, I think doing this is the appropriate answer even if the >>> range is complex. >> >> Yes, at least if you're looking for a least-squares fit. If you're >> looking for a fit that minimizes some nonlinear cost criteria then the >> real and complex solutions may well interact. >> >> > If I get it correctly, in case of Gaussian noise the least-squares > criterion is the same thing as maximizing the likelihood function. > > Incidentally, makes me feel a pity that while finding the global maximum > of the likelihood function is a standard method in communications, in > scientific packages fits work as some kinds of iterative processes, so > if you want to search for the global maximum (useful when the problem is > ill-defined) you have to write the code instead of just clicking a > couple of buttons in a standard package. ;)That's because of the way the problem is structured. In comms, if the noise isn't Gaussian then the maximum of the likelihood function is not found in a least-squares fit. For my Master's thesis I worked on a system that operated at 400kHz or so, where the noise primarily comes from electrostatic discharge and where the best noise models all had infinite variance. Under those circumstances, you basically threw out all the tidy math that comes from the Gaussian assumption. -- Tim Wescott Control systems, embedded software and circuit design I'm looking for work! See my website if you're interested http://www.wescottdesign.com
Reply by ●November 14, 20162016-11-14
On 14/11/2016 20:33, Tim Wescott wrote:> That's because of the way the problem is structured. In comms, if the > noise isn't Gaussian then the maximum of the likelihood function is not > found in a least-squares fit. For my Master's thesis I worked on a > system that operated at 400kHz or so, where the noise primarily comes > from electrostatic discharge and where the best noise models all had > infinite variance. Under those circumstances, you basically threw out > all the tidy math that comes from the Gaussian assumption. >From the sound of it, looks like you might have used some erasure codes like Reed-Solomon? Gene
Reply by ●November 14, 20162016-11-14
On Mon, 14 Nov 2016 21:17:39 +0300, Evgeny Filatov wrote:> On 14/11/2016 20:33, Tim Wescott wrote: >> That's because of the way the problem is structured. In comms, if the >> noise isn't Gaussian then the maximum of the likelihood function is not >> found in a least-squares fit. For my Master's thesis I worked on a >> system that operated at 400kHz or so, where the noise primarily comes >> from electrostatic discharge and where the best noise models all had >> infinite variance. Under those circumstances, you basically threw out >> all the tidy math that comes from the Gaussian assumption. >> >> > From the sound of it, looks like you might have used some erasure codes > like Reed-Solomon? > > GeneI built a receiver that delivered bit-slice integrator levels to the decoding algorithm. It was accompanied by a section in my thesis explaining how those levels should be interpreted (IIRC, +/- 64 indicated highest likelihood, with the likelihood dropping both above and below that absolute value). It was when the Coast Guard was first developing their differential GPS service; my Thesis advisor had a bunch of graduate students who were working on developing the codes. -- Tim Wescott Control systems, embedded software and circuit design I'm looking for work! See my website if you're interested http://www.wescottdesign.com
Reply by ●November 14, 20162016-11-14
On 14.11.2016 21:33, Tim Wescott wrote:> On Mon, 14 Nov 2016 21:17:39 +0300, Evgeny Filatov wrote: > >> On 14/11/2016 20:33, Tim Wescott wrote: >>> That's because of the way the problem is structured. In comms, if the >>> noise isn't Gaussian then the maximum of the likelihood function is not >>> found in a least-squares fit. For my Master's thesis I worked on a >>> system that operated at 400kHz or so, where the noise primarily comes >>> from electrostatic discharge and where the best noise models all had >>> infinite variance. Under those circumstances, you basically threw out >>> all the tidy math that comes from the Gaussian assumption. >>> >>> >> From the sound of it, looks like you might have used some erasure codes >> like Reed-Solomon? >> >> Gene > > I built a receiver that delivered bit-slice integrator levels to the > decoding algorithm. It was accompanied by a section in my thesis > explaining how those levels should be interpreted (IIRC, +/- 64 indicated > highest likelihood, with the likelihood dropping both above and below > that absolute value). > > It was when the Coast Guard was first developing their differential GPS > service; my Thesis advisor had a bunch of graduate students who were > working on developing the codes. >Thanks; now I better appreciate your contribution. I looked up Wikipedia on "differential GPS", and seen that "The United States Coast Guard and Canadian Coast Guard each run such systems in the U.S. and Canada on the longwave radio frequencies between 285 kHz and 325 kHz near major waterways and harbors." About the same frequencies you've mentioned. Kinda cool! Gene
Reply by ●November 14, 20162016-11-14
On Tue, 15 Nov 2016 01:20:23 +0300, Evgeny Filatov wrote:> On 14.11.2016 21:33, Tim Wescott wrote: >> On Mon, 14 Nov 2016 21:17:39 +0300, Evgeny Filatov wrote: >> >>> On 14/11/2016 20:33, Tim Wescott wrote: >>>> That's because of the way the problem is structured. In comms, if >>>> the noise isn't Gaussian then the maximum of the likelihood function >>>> is not found in a least-squares fit. For my Master's thesis I worked >>>> on a system that operated at 400kHz or so, where the noise primarily >>>> comes from electrostatic discharge and where the best noise models >>>> all had infinite variance. Under those circumstances, you basically >>>> threw out all the tidy math that comes from the Gaussian assumption. >>>> >>>> >>> From the sound of it, looks like you might have used some erasure >>> codes >>> like Reed-Solomon? >>> >>> Gene >> >> I built a receiver that delivered bit-slice integrator levels to the >> decoding algorithm. It was accompanied by a section in my thesis >> explaining how those levels should be interpreted (IIRC, +/- 64 >> indicated highest likelihood, with the likelihood dropping both above >> and below that absolute value). >> >> It was when the Coast Guard was first developing their differential GPS >> service; my Thesis advisor had a bunch of graduate students who were >> working on developing the codes. >> >> > Thanks; now I better appreciate your contribution. > > I looked up Wikipedia on "differential GPS", and seen that "The United > States Coast Guard and Canadian Coast Guard each run such systems in the > U.S. and Canada on the longwave radio frequencies between 285 kHz and > 325 kHz near major waterways and harbors." About the same frequencies > you've mentioned. Kinda cool! > > GeneActually, I think those frequencies are correct and my memory was faulty. I just fired up the receiver to check (my thesis advisor gave it back to me about 10 years ago), but some of the LED segments are out -- but as far as I can tell that's the frequency range it supports. The USCG transmitters were originally piggy-backed on their radio direction finding beacons (I don't know if they even support that any more); originally there was some thought that it would be nice to also be capable of using aviation RDF beacons as well; those are at higher frequencies, and are probably why I was remembering 400-ish instead of 300-ish. According to my thesis advisor it was the second-ever design to work in that service, which I take with a glow of pride and a grain of salt, because I expect that SOMEONE must have been working on it. (Radio direction finding, BTW, is the reason for those round antennas on top of old airplanes: they were designed so that you'd get a null in reception when the hole of the donut was pointed along a line to the transmitter. Plot a few of those on a map, and you'd know where you were.) -- Tim Wescott Wescott Design Services http://www.wescottdesign.com I'm looking for work -- see my website!
Reply by ●November 15, 20162016-11-15
On 15/11/2016 03:01, Tim Wescott wrote:> > (Radio direction finding, BTW, is the reason for those round antennas on > top of old airplanes: they were designed so that you'd get a null in > reception when the hole of the donut was pointed along a line to the > transmitter. Plot a few of those on a map, and you'd know where you > were.) >Satellite navigation has by now evolved to centimeter-precision Real Time Kinematic. I'm a complete ignoramus, but made me wondering whether anyone has bothered to upgrade Radio direction finding with modern techniques, such as spread spectrum for multipath mitigation or perhaps more accurate beamforming with MIMO... Gene
