De Rerum Natura

I am writing a Mathematica program to produce the precision error signal and reconstruction matrix for an arbitrary number of models. The maximum number I had tried before was ten models because it corresponded to the number of maps we have for the 29 Palms dataset.

My first try consisted of systematically creating all possible permutations of the precision error equations, squaring them, and then storing the coefficients. The program would then systematically look at the equations and augment an independent set every time it found an equation that could not be written as linear combinations of the previous ones in the set.

This worked okay for ten or so models, but I want to produce the full covariance matrix for twenty questions in a multiple-choice exam. No problem, I was making some grilled lamb for Easter dinner yesterday, so I put the computer to work and walked away. Three hours later, the computer was still trying to finish the list of all possible equation permutations! I confess that I have not worked out the combinatorics for the equations yet so perhaps it is of order $20!$. This would be 2,432,902,008,176,640,000 combinations. Compare this to $10!=3628800$ and you can see why the computation got hard quickly.

So my second incantation of the program was to do the computation randomly. Two integers $P$ and $Q$ are picked randomly such that $1\le P\le m$ and $1\le Q\le m$ where $m$ is the number of models. These random numbers are then used to randomly sample the model variables and construct a precision error difference equation. If the equation is independent from the set currently at hand, it is kept, otherwise discarded.

This second version is taking about ten minutes to produce a result. This made me think about how we perceive randomness as haphazardly: “Oh, you are just randomly trying to guess the right answer.” We perceive randomly as wasteful or misguided. The case presented here is just another example of how random is sometimes faster than systematic.