Search Results for: eureqa

Equation search, part 2

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Some further thoughts on the Eureqa program which implements the curve-fitting method of Michael Schmidt and Hod Lipson:

The program kept running indefinitely, so I stopped it in the morning, at which point I noticed that the output didn’t quite make sense, and I went back and realized that I’d messed up when trying to delete some extra data in the file. So I re-ran with the actual data. The program functioned as before but moved much quicker to a set of nearly-perfect fits (R-squared = 99.9997%, and no, that’s not a typo). Here’s what the program came up with:

models.png

The model at the very bottom of the list is pretty pointless, but in general I like the idea of including “scaffolding” (those simple models that we construct on the way toward building something that fits better) so I can’t really complain.

It’s hard to fault the program for not finding y^2 = x1^2 + x2^2, given that it already had such a success with the models that it did find. Continue reading →

Equation search, part 1

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For some reason, Aleks doesn’t blog anymore; he just sends me things to blog. I guess he’s decided his time is more usefully spent doing other things. I, however, continue to enjoy blogging as an alternative to real work and am a sucker for all the links Aleks sends me (except for the videos, which I never watch).

Aleks’s latest find is a program called Eureqa that implements a high-tech curve-fitting method of Michael Schmidt and Hod Lipson. (And, unlike Clarence Thomas, I mean “high-tech” in a good way.) Schmidt and Lipson describe their algorithm as “distilling free-form natural laws from experimental data,” which seems a bit over the top, but the basic idea seems sound: Instead of simply running a linear regression, the program searches through a larger space of functional forms, building models like Tinker Toys by continually adding components until the fit stops improving:

eureqa.png

I have some thoughts on the limitations of this approach (see below), but to get things started I wanted to try out an example where I suspected this new approach would work well where more traditional statistical methods would fail.

The example I chose was a homework assignment that I included in a couple of my books. Here’s the description:

I give students the following twenty data points and ask them to fit y as a function of x1 and x2.

y x1 x2
15.68 6.87 14.09
6.18 4.4 4.35
18.1 0.43 18.09
9.07 2.73 8.65
17.97 3.25 17.68
10.04 5.3 8.53
20.74 7.08 19.5
9.76 9.73 0.72
8.23 4.51 6.88
6.52 6.4 1.26
15.69 5.72 14.62
15.51 6.28 14.18
20.61 6.14 19.68
19.58 8.26 17.75
9.72 9.41 2.44
16.36 2.88 16.1
18.3 5.74 17.37
13.26 0.45 13.25
12.1 3.74 11.51
18.15 5.03 17.44
16.8 9.67 13.74
16.55 3.62 16.15
18.79 2.54 18.62
15.68 9.15 12.74
4.08 0.69 4.02
15.45 7.97 13.24
13.44 2.49 13.21
20.86 9.81 18.41
16.05 7.56 14.16
6 0.98 5.92
3.29 0.65 3.22
9.41 9 2.74
10.76 7.83 7.39
5.98 0.26 5.97
19.23 3.64 18.89
15.67 9.28 12.63
7.04 5.66 4.18
21.63 9.71 19.32
17.84 9.36 15.19
7.49 0.88 7.43

[If you want to play along, try to fit the data before going on.]

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