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A definitive test of discrete scale (relativity, numerology)
I have decided to take two hours away from dead island and do something
equally productive with my time: test the numerology of Robert Oldershaw. Exhibit "A" is the VizieR catalog "J/A+A/352/555/table1", sourced from the Hipparcos catalog [1]. I am sure Robert is working on this very thing, as he has had more than a decade to find this publication, and about 2 weeks to find this catalog after I have first given it to him. I have only given myself two hours because dead island is fun but I only need a break rather than a sabattical. The quality of the mass data is rather impressive. Statistics: 12,090 stars to 10% or better. 3028 stars to 5% or better. 185 stars are known 1% or better. 30 with zero error :P 641 stars known to 0.02 M_sun or better. Procedu * Yoink the data off VizieR. * Export to ascii with the only meaningful/relevant parameters being mass and the uncertainty of the measurement. * Grep out invalid entires. I, personally, find zero error unlikely. Maybe they are good, maybe not. Don't feel like making big effort for 0.17% of the catalog. * Take the mass of each star, and subtract off the nearest integer multiple of 0.145. A computational step, not a physics step. What survives is the absolute difference from the predicted quantization, and the error in the measurement. I tried plotting out the 1% data with gnuplot, and even that small block of quality data is visually _meaningless_. Overlapping error bars make it impossible to get anything of value out of the data that way. So might as well go all the way and use all the stars where the masses are determined to 5% or better. Five percent or so, is about as large as you can go because the average mass of a star in this catalog is 1.3 M_sun and I'd like the results to be useful to at least one standard deviation. Results of DOES_STUFF.PL [2]: 3028 stars with masses determined to 5% or better: 37 exactly as predicted 1938 within 1 standard deviations 630 off by 1 to 2 standard deviations 215 off by 2 to 3 standard deviations 115 off by 3 to 4 standard deviations 40 off by 4 to 5 standard deviations 53 off by more than 5+ standard deviations Average standard deviation per star: 1.09 Average mass of star: 1.43 solar masses Seeing that about half the data is wrong by a standard deviation or more isn't all that telling given that the binning being tested for is only twice the average error of the sample. Let's go deeper. On a related note, Robert has frequently complained that rand("he can't find", "there doesn't exist", "he hasn't bothered to look for") a dataset that has masses determined to 1% or better. Negative twelve years after making that complaint, science has finally come through for him! Re-run with $cutoff = 0.01: 185 stars with masses determined to 1% or better: 3 exactly as predicted 42 within 1 standard deviations 36 off by 1 to 2 standard deviations 28 off by 2 to 3 standard deviations 23 off by 3 to 4 standard deviations 16 off by 4 to 5 standard deviations 37 off by more than 5+ standard deviations Average standard deviation per star: 2.97 Average mass of star: 1.35 solar masses Now all the subtleties of averaging a sample of data aside, I think it is pretty safe to say Robert's numerology is wrong when 75% of a precisely determined sample of relevant data is more than one standard deviation different from the predicted binning. Also when the _average_ standard deviation per sample is 3. Sure, there could be a corner case or twenty in there that could be jacking up the average by a large amount but I argue that tearing apart the data is an exercise for the reader at this point. None of this includes other data sets such as the various smaller sets such as ones that include eclipsing binaries, which are determined to fractions of a percent, or the Sun which has already been determined to not agree with Robert's numerology by 100 standard deviations and change. The catalog seems to cut out at about 0.88 solar masses on the low side, so small mass stars aren't really present here. There are other catalogs of low mass stars that Robert can ignore, so it isn't really an issue. Note that only 37 out of 3,028 stars are binned exactly as predicted. I doubt I will hear a sarcastic rendition of my mathematical point of the gauranteed existence of some matches, with such a small number. I figure taking 2 hours, wasting half of it re-learning gnuplot and realizing that there's no meaningful way to visualize this data, and spending the other half doing something useful with it, is a better investment of my time than repeatedly saying "HEY! HERE'S THE DATA! STOP CRYING AND BE THE SCIENTIST YOU CLAIM TO BE." I am going back to dead island so I can light undead minorities and Australians on fire. [3] ---- [1] : "Fundamental parameters of nearby stars from the comparison with evolutionary calculations: masses, radii and effective temperatures.", Allende Prieto C., Lambert D.L., Astron. Astrophys. 352, 555 (1999) [2] : #!/usr/bin/perl use warnings; use strict; use Math::Round ':all'; open (DATA, "stars2.txt"); my $cutoff = 0.01; my $percent = 100 * $cutoff; my $zero = 0; my $one = 0; my $two = 0; my $three = 0; my $four = 0; my $five = 0; my $morethanfive = 0; my $totaldeviations; my $count; my $totalmass = 0; while (my $line = DATA) { my @data = split(' ', $line); my $mass = $data[2]; my $mass_error = $data[3]; my $multiple = round( nearest(0.145, $mass) / 0.145 ); my $residual = $mass - ($multiple * 0.145); my $truncated_residual = substr($residual, 0, 5); my $deviations = substr( abs($residual / $mass_error), 0, 4); if ($mass_error / $mass le $cutoff) { $totaldeviations += $deviations; $count++; $totalmass += $mass; if ($deviations eq 0) { $zero++; next; }; if ($deviations le 1 && $deviations ne 0) { $one++; next; }; if ($deviations le 2 && $deviations gt 1) { $two++; next; }; if ($deviations le 3 && $deviations gt 2) { $three++; next; }; if ($deviations le 4 && $deviations gt 3) { $four++; next; }; if ($deviations le 5 && $deviations gt 4) { $five++; next; }; if ($deviations gt 5) { $morethanfive++; next; }; } } my $average_deviations = substr($totaldeviations / $count, 0, 4); my $average_mass = substr($totalmass / $count, 0, 4); print "$count stars with masses determined to $percent% or better: $zero exactly as predicted $one within 1 standard deviations $two off by 1 to 2 standard deviations $three off by 2 to 3 standard deviations $four off by 3 to 4 standard deviations $five off by 4 to 5 standard deviations $morethanfive off by more than 5+ standard deviations Average standard deviation per star: $average_deviations Average mass of star: $average_mass solar masses "; [3] : Fire is the cleanser. |
#2
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A definitive test of discrete scale (relativity, numerology)
In article ,
eric gisse wrote: I have decided to take two hours away from dead island and do something equally productive with my time: test the numerology of Robert Oldershaw. To help people doing this sort of work: there's a standard way of saying whether a given number of standard deviations away from a model is significant, the chi^2 test. Eric's code almost does this already, but below[1] is a slightly modified version which computes the chi^2 statistic, which is simply the sum of the squares of the deviations over the errors. The higher chi^2 is, the worse the model agrees with the data. The advantage of doing that is that the probability of obtaining a value of chi^2 as extreme, or more extreme than the one that you actually see under the 'null hypothesis' that the data are actually consistent with the model can be computed: if this probability p is low, then we say that the model is ruled out at the (1-p) confidence level. Various online calculators or the Perl module Statistics:istributions can be used to do this[2]. Thus it's possible to show, for example, that taking the stars with 5% or better errors on masses, the model is ruled out at the 99.9999999999% confidence level. Similar tests could be done with other databases. [1] #!/usr/bin/perl use warnings; use strict; use Math::Round ':all'; use Statistics:istributions; open (DATA, "stars2.txt"); my $chi2=0; my $count=0; my $cutoff=0.05; my $confidence=1e-12; while (my $line = DATA) { # $line just contains mass and error my @data = split('\s+', $line); my $mass = $data[0]; my $mass_error = $data[1]; if ($mass_error/$mass=$cutoff) { my $multiple = round( nearest(0.145, $mass) / 0.145 ); my $residual = $mass - ($multiple * 0.145); my $deviation = $residual / $mass_error; $chi2+=$deviation**2; $count++; } } print("Chi^2 is $chi2 for $count stars.\n"); printf("Probability under null hypothesis: %g\n",Statistics:istributions::chisqrprob($count ,$chi2)); printf("(chi^2 required to rule out null hypothesis at (1-%g)\n confidence level is %g)\n",$confidence,Statistics:istributions::chis qrdistr($count,$confidence)); [2] I haven't used this module before and can only say that the numbers it produces look entirely reasonable. -- Martin Hardcastle School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK Please replace the xxx.xxx.xxx in the header with herts.ac.uk to mail me |
#3
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A definitive test of discrete scale (relativity, numerology)
The problem is that all those stars could have invisible companions.
That is why I looked at Alpha centauri system, since I thought that in such a close star that problem would be solved... As I see it, this prediction is impossible to verify really. As far as the data that eric shows, the answer is negative: there is no quanta of matter when star systems are measured. Is that the final truth? We will know when we know for sure the exact characteristics of thousands of systems. (And I mean "exact" i.e. when we can rule out any unseen companions, we know the masses of all the planets, etc) |
#4
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A definitive test of discrete scale (relativity, numerology)
On Sep 18, 5:43*am, Martin Hardcastle
wrote: print("Chi^2 is $chi2 for $count stars.\n"); printf("Probability under null hypothesis: %g\n",Statistics:istributions::chisqrprob($count ,$chi2)); printf("(chi^2 required to rule out null hypothesis at (1-%g)\n * * confidence level is %g)\n",$confidence,Statistics:istributions::chis qrdistr($count,$confidenc*e)); [2] I haven't used this module before and can only say that the numbers it produces look entirely reasonable. -------------------------------------------------------------------------------------------------------------- I am currently looking at the paper "Accurate masses and radii of normal stars: Modern results and applications" by Torres, Andersen and Gimenez. This catalog is available at VizieR, and published Astronomy & Astrophysics Review, vol. 18, 2010. It is also available at arxiv.org (search Torres, 2009, astro-ph) This sample is heavily weighted toward "massive" stars, but it might yield some interesting results in the 1 to 4 solar mass range for total system masses. An independent analysis of the data would be most welcome. RLO Fractal Cosmology |
#5
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A definitive test of discrete scale (relativity, numerology)
Le 18/09/11 18:53, jacob navia a écrit :
The problem is that all those stars could have invisible companions. And the problem could be also that they have LOST visible companions. A couple of stars can be destroyed in gravitational interactions when the couple is born, both of them becoming lone stars and leaving no trace of their common origin... See: http://www.sciencedaily.com/releases...0915083715.htm or the scientific article: http://arxiv.org/abs/1109.2896 I think that proving or disproving Robert's hypothesis could be VERY difficult. The only way to know if he is right would be to weight star systems when they are just born... |
#6
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A definitive test of discrete scale (relativity, numerology)
Martin Hardcastle wrote in
: In article , eric gisse wrote: I have decided to take two hours away from dead island and do something equally productive with my time: test the numerology of Robert Oldershaw. To help people doing this sort of work: there's a standard way of saying whether a given number of standard deviations away from a model is significant, the chi^2 test. Eric's code almost does this already, but below[1] is a slightly modified version which computes the chi^2 statistic, which is simply the sum of the squares of the deviations over the errors. [...stuff...] Irritating perl bug: The value of $residual is going to be negative a significant portion of the time, but squaring the negative quantity triggers some sort of overflow/insanity. Might need to do some reporting/squashing after this. After calculating the chi squared of the sample, I noticed I was getting values north of 10^22. I found that to be odd. It turns out that perl does not square a negative number correctly. So I have to add yet another hackish/goofy computational protection in the form of abs($residual) so the squaring doesn't go nutbar. Now for the chi squared test, let's do that right. Given most of my experience is with ye olde standard deviation and associated distribution widgetry, my knowledge of the test has been limited to "what I learned in school and forgot, and the residual notion that a large chi squared means the result is crap". Re-reading my statistical analysis textbook (Taylor, haven't opened it in about 2 years) and the discussion of this, it turns out that this is a _far_ better test of a distribution hypothesis rather than what I was doing. I was merely generalizing my previous behavior of 'take the latest example, and testing whether it works' which is fine for the small scale but does not generalize. Now, you don't want just the regular chi-squared value. That isn't useful. What is wanted is the reduced chi-squared value, which is chi- squared divided by the number of degrees of freedom. For this, the hypothesis is a binning of 0.145 solar masses. This requires figuring out the amount of bins, and can't be done beforehand because the data set covers a wide range of masses with a wider range of confidence. Now this may be a point of contention, but I argue the degrees of freedom is not the amount of stars themselves but rather the amount of bins of 0.145 M_sun required to cover the mass range. This is actually more generous to the testing, given that more degrees of freedom makes it more likely that the hypothesis is true. Turns out to not matter much, since the answer is "zero" for meaningful chunks of the data. Knowing the probability that the resultant reduced chi squared is going to be larger than chi squared for a certain amount of degrees of freedom requires a computation using a variation on the gamma function. Screw that, that is what CPAN is for. The suggested module does that! Putting all this together, i have DOESSTUFF_mod2.pl It turns out that the hypothesis is so wrong that it does not matter what data quality cutoff I use. BEEP BOOP...analyzing 17187 stars with masses determined to 100% or better Average standard deviation per star: 0.48 Average mass of star: 1.28 solar masses Mass range of sample: 0.88 to 7.77 solar masses Chi-squared of the expected binning hypothesis: 35097 Reduced chi-squared: 731.1875 The probability that the reduced chi-squared value of 731.1875 is larger than the value of 35097 for 48 degrees of freedom is 0. BEEP BOOP...analyzing 185 stars with masses determined to 1% or better Average standard deviation per star: 2.97 Average mass of star: 1.35 solar masses Mass range of sample: 1.00 to 4.63 solar masses Chi-squared of the expected binning hypothesis: 537 Reduced chi-squared: 21.48 The probability that the reduced chi-squared value of 21.48 is larger than the value of 537 for 25 degrees of freedom is 0. BEEP BOOP...analyzing 10 stars with masses determined to 0.6% or better Average standard deviation per star: 2.99 Average mass of star: 2.07 solar masses Mass range of sample: 1.71 to 4.63 solar masses Chi-squared of the expected binning hypothesis: 26 Reduced chi-squared: 1.3 The probability that the reduced chi-squared value of 1.3 is larger than the value of 26 for 20 degrees of freedom is 0.16581. The only remarkable thing here is that there is a 4.63 M_sun determined to 0.6%! Sorry Robert. The probability that your theory is right is competing with the limits of floating point math. -------------- #!/usr/bin/perl use warnings; use strict; use Math::Round ':all'; use Statistics:istributions; open (DATA, "stars2.txt"); my $cutoff = 1; my $percent = 100 * $cutoff; my $totaldeviations; my $count; my $totalmass = 0; my $chisq = 0; my @stars; while (my $line = DATA) { my @data = split(' ', $line); my $mass = $data[2]; my $mass_error = $data[3]; my $multiple = round( nearest(0.145, $mass) / 0.145 ); my $residual = $mass - ($multiple * 0.145); my $truncated_residual = substr($residual, 0, 5); my $deviations = substr( abs($residual / $mass_error), 0, 4); if ($mass_error / $mass le $cutoff) { $totaldeviations += $deviations; $count++; $totalmass += $mass; $chisq += abs($residual / $mass_error)^2; push (@stars, $mass); } } my @sorted = sort @stars; my $largest = $sorted[-1]; my $smallest = $sorted[0]; my $binning_largest = nearest_ceil(0.145, $largest); my $binning_smallest = nearest_floor(0.145, $smallest); my $degrees = ($binning_largest - $binning_smallest) / 0.145; my $reduced_chisq = $chisq / $degrees; my $probability = Statistics:istributions::chisqrprob ($degrees, $chisq); my $average_deviations = substr($totaldeviations / $count, 0, 4); my $average_mass = substr($totalmass / $count, 0, 4); print "BEEP BOOP...analyzing $count stars with masses determined to $percent% or better Average standard deviation per star: $average_deviations Average mass of star: $average_mass solar masses Mass range of sample: $smallest to $largest solar masses Chi-squared of the expected binning hypothesis: $chisq Reduced chi-squared: $reduced_chisq The probability that the reduced chi-squared value of $reduced_chisq is larger than the value of $chisq for $degrees degrees of freedom is $probability. "; |
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A definitive test of discrete scale (relativity, numerology)
"Robert L. Oldershaw" wrote in
: On Sep 18, 5:43*am, Martin Hardcastle wrote: print("Chi^2 is $chi2 for $count stars.\n"); printf("Probability under null hypothesis: %g\n",Statistics:istributions::chisqrprob($count ,$chi2)); printf("(chi^2 required to rule out null hypothesis at (1-%g)\n * * confidence level is %g)\n",$confidence,Statistics:istributions::chis qrdistr($count, $conf idenc*e)); [2] I haven't used this module before and can only say that the numbers it produces look entirely reasonable. ---------------------------------------------------------------------- - --------------------------------------- I am currently looking at the paper "Accurate masses and radii of normal stars: Modern results and applications" by Torres, Andersen and Gimenez. This catalog is available at VizieR, and published Astronomy & Astrophysics Review, vol. 18, 2010. It is also available at arxiv.org (search Torres, 2009, astro-ph) This sample is heavily weighted toward "massive" stars, but it might yield some interesting results in the 1 to 4 solar mass range for total system masses. Why would you think it'd tell you anything we don't already know? ~ 17,000 stars, the majority within that mass range, explicitly disproves your theory. Another hundred won't change anything. An independent analysis of the data would be most welcome. How fortunate for you that someone did your research for you. There's a program right there for you to analyze the data with. RLO Fractal Cosmology |
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A definitive test of discrete scale (relativity, numerology)
jacob navia wrote in news:mt2.0-20923-1316381120
@hydra.herts.ac.uk: Le 18/09/11 18:53, jacob navia a écrit : The problem is that all those stars could have invisible companions. And the problem could be also that they have LOST visible companions. A couple of stars can be destroyed in gravitational interactions when the couple is born, both of them becoming lone stars and leaving no trace of their common origin... See: http://www.sciencedaily.com/releases...0915083715.htm or the scientific article: http://arxiv.org/abs/1109.2896 I think that proving or disproving Robert's hypothesis could be VERY difficult. The only way to know if he is right would be to weight star systems when they are just born... The 12k star sample have their masses determined spectroscopically. Besides, Robert has argued that invididual stars are quantized in mass in addition to the systems themselves. Yes, really. Not my theory, I don't have to justify it. |
#9
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A definitive test of discrete scale (relativity, numerology)
In article ,
eric gisse wrote: Irritating perl bug: The value of $residual is going to be negative a significant portion of the time, but squaring the negative quantity triggers some sort of overflow/insanity. Might need to do some reporting/squashing after this. Not in the perl I use! (I did test the script I posted...) That would be a very serious and basic bug if it were really present! I think you are mistaking the operator ^ for the operator ** (see my code). ^ in perl is not 'raise to power' but 'bitwise xor'. ** is the exponentiation operator. Now, you don't want just the regular chi-squared value. That isn't useful. What is wanted is the reduced chi-squared value, which is chi- squared divided by the number of degrees of freedom Not so. The reduced chi^2 is a useful way of telling at first glance whether there is something wrong with a model: a reduced chi^2 much greater than 1 indicates a problem. However, the reduced chi^2 doesn't tell you everything that you want to know, quantitatively -- a reduced chi^2 of 2 for 2 degrees of freedom is very much less interesting than a reduced chi^2 of 2 for several thousand degrees of freedom. So in fact the standard thing to do is use the chi^2 value itself and calculate (or look up) the critical value for a given number of degrees of freedom. You can do the equivalent thing for a reduced chi^2, so it's not a big deal, but chi^2/d.o.f. is what people quote in the astrostatistics literature, so it's what I used. For this, the hypothesis is a binning of 0.145 solar masses. This requires figuring out the amount of bins, and can't be done beforehand because the data set covers a wide range of masses with a wider range of confidence. Now this may be a point of contention, but I argue the degrees of freedom is not the amount ouf stars themselves but rather the amount of bins of 0.145 M_sun required to cover the mass range. This is actually more generous to the testing, given that more degrees of freedom makes it more likely that the hypothesis is true. Turns out to not matter much, since the answer is "zero" for meaningful chunks of the data. I don't think this is right. The degrees of freedom is the number of data points, minus the number of free parameters of the model (none in this case): see e.g. http://en.wikipedia.org/wiki/Chi-square_statistic. So I don't think your numbers are correct (in particular, it would seem very bizarre if, as you suggest, including stars with larger errors caused the model to be ruled out more stringently... in fact, if you include all the stars with huge errors *and* calculate the chi^2 correctly, you should find an acceptable fit, but that's only because you'd be diluting the stars that can actually constrain the model with the many more that can't). However, the key point is that this test can be done, and, when it's done with stars with accurately measured masses, it is inconsistent with the proposed model at a very high confidence level, as I said earlier. Martin -- Martin Hardcastle School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK Please replace the xxx.xxx.xxx in the header with herts.ac.uk to mail me |
#10
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A definitive test of discrete scale (relativity, numerology)
In article ,
Robert L. Oldershaw wrote: This sample is heavily weighted toward "massive" stars, but it might yield some interesting results in the 1 to 4 solar mass range for total system masses. An independent analysis of the data would be most welcome. As these are non-contact binaries and the authors say they should have evolved independently, I did the chi^2 test as described in the earlier posting for the individual stars with errors less than 0.145 solar masses: chi^2 of 16085 for 172 degrees of freedom, null hypothesis ruled out at such a large confidence level that I can't calculate it offhand, but basically such that the model can't possibly be right. If I add up the two components and take only the systems where the combined error on mass is less than 0.145 solar masses, I get a chi^2 of 1154 for 82 d.o.f., again wildly inconsistent with the model. So, again, the data are not telling you what you would like them to tell you. It took me about ten minutes to find the data you referred to, get them into the right format, modify and run my code, and do the modifications needed to run it again on the sums of the masses. Testing models, when they make quantitative predictions, is easy, and it's a skill that any would-be-modeller ought to learn. The half-hour or so I've spent on this today is enough for me, though. -- Martin Hardcastle School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK Please replace the xxx.xxx.xxx in the header with herts.ac.uk to mail me |
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