Just a couple of things:

• A Google search on "Titius-Bode" and "55 Cancri" reveals that other thread on the front page.
• They report an r² score like that's a normal thing to do, yet in the literature I've been scouring lately, the only other reference where I found something similar is in an obscure letter to the editor of the The American Statistician from 1969, (Vol. 23, No. 5 (Dec., 1969), pp. 52-53), penned by a C. Mitchell Dayton where he calculates a correlation coefficent for the Solar System applied to a linear TBL model. (BTW Poveda & Lara mistakenly refer to their R² as a "correlation coefficient" which is just plain R; R² is the "coefficient of determiniation".)
• The forumula they use is mathematically virtually identical to the one I constructed, yet it's like they changed it around to make it look different, sort of like using the capitalized R².

Well, as they say, imitation is the most sincere form of flattery; or was it that great minds are to be found in the same gutter? At least, they've given the idea the publicity it deserves. Indeed, the internet's practically on fire!

In any case, they didn't get into significance testing, so thus the meaning of their R² is open to interpretation, to say the least. Dayton, in his letter goes on to say that Remember, Dayton was writing in 1969, when computing power was slow and expensive. For example, Ephron (1971, Journal of the American Statistical Association Vol. 66: 552-559--it's in JSTOR) attempted a Monte Carlo simulation of sorts, but he could only do a measly 2,000 attempts, and his histogram is pathetically jerky-looking.

(BTW, there was an interesting flurry of interest in the Titius-Bode law among certain American statisticians in the very late 1960's and very early 1970's. I'm trying to put together a bibliography.)

But nowadays, a typical desktop computer could do 1 billion or more simulations in a single day if it wanted to. So I think I was on the right track by calculating a probability distribution for each specific scenario. However, reading through the statistical literature on the TBL has made me realize that I perhaps made too much of the actual distributions that I constructed. This is because the question of whether any proposed hypothesis is statistically significant is relative to an alternative null hypothesis that is rejected. Here is what Ephron has to say:
So in that other thread, I chose an uninteresting alternative null hypothesis, all right--just take the random planetary distances from a totally uniform distribution. The only problem is that it was too uninteresting to be interesting. It was a straw man, really, I must admit, guaranteed to produce the desired result. As Ephron's rival, I. J. Good wrote (1969, American Statistical Association Journal 64: 23-34, page 30):
BAM! Cut down to size by a single sentence by a man writing 40 years ago!

Quoting Good, Ephron added
So, noting that as one progresses out of the Solar System, the spacing between successive planets always increases, Ephron proposed his own uninteresting null hypothesis: the law of increasing differences.

Ephron's "law" of increasing differences may or may not seem absurd to you, but it has taken on some historical significance because it was addressed in some detail in Hayes and Tremaine's (1988) Icarus paper (preprint available at the ArXiv; the original at ScienceDirect). Now, Hayes and Tremaine's paper is one of the more opaque papers I've read in a long time (though they provide a complete discussion of possible radius exclusion principles). But their bottom line with regard to the law of increasing distances is that any system "that satisfies the law of increasing differences will often satisfy all but the most stringent exclusion laws." (p. 555) In other words, "the law of increasing differences is a much more restrictive assumption than radius-exclusion laws, and in the absence of any physical justification, it does not form a sound basis from which to judge the validity of Bode's law." (p. 555-556). That is, the law of increasing differences is an unfair null hypothesis because it's too strong.

On the other hand, if it could be shown that the law of increasing differences as a comparatively uninteresting, alternative, null hypothesis to the Titius-Bode law should be rejected, that would indeed warrant the assertion that the Titius-Bode spacing pattern of 55 Cancri is a real pattern.

So, naturally, I had to try it.

I used a simple broken stick algorithm to generate random systems that satisfy the law of increasing differences.

The first step is to break a stick of length 1 into 7 pieces--2 end pieces and 5 interior pieces. This is accomplished by drawing six numbers from the uniform distribution ([U[0, 1]). These numbers are then sorted in a nondescending order.

Then I calculated the length of the interior sticks by taking the absolute value of the differences between adjacent random numbers, and then the interior sticks themselves were sorted.

The next step to generating the random system was to reconstruct the semimajor axes: a₂, a₃, a₄, a₅. Note that the one exception to the law of increasing differences in our Solar System is Mercury, in that it is 0.4 AU from the Sun, whereas the distance to Venus is 0.7 AU, thus the difference is only 0.3 AU. Thus, a₁ and a₆ are already given because the two end pieces were never sorted.

The final step was to throw out the 5th planet, because of the empty orbital that Poveda and Lara's model assumes.

Once the random system is generated, then the r² can be computed in the ordinary matter, and stored into the histogram to construct the probability distribution necessary to conduct a significance test.

Once the histogram was constructed out of 100 million random solar systems (see attached chart), it was determined that the 95th percentile of r² fell out at right about 0.9967. Thus, for a 5% level of significance, 0.9967 would be the critical value. The r² reported by Poveda and Lara of 0.997 exceeds the critical value. (I reported an r² of 0.9975; but that was rounded up from 0.997488; if I had chosen three significant figures, I also would have had 0.997.) Therefore, we can say with 95% confidence that we can reject the null hypothesis of the law of increasing differences as a possible explanation for the spacing pattern an 55 Cancri.



Granted, a 95% confidence level is not nearly as high as the 99.9% confidence level I reported in that other thread, but there's no glory to be gained in showing that a straw man is false with a confidence level of 99.9%; it is much more satisfying to be 95% sure that one is warranted in rejecting the strongest null hypothesis proposed in the literature so far.

(Caveat: these are preliminary results, I need to double check everything and increase the number of random systems to 1 billion to smooth out the curve even more before I send the manuscript into Icarus!)

Attached Images

PROBDIST.png (142.7 KB, 7 views)

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"Ignorance more frequently begets confidence than does knowledge" -- Charles Darwin