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Turing Equation Explains how Leopard Spots Develop
BilZ0r writes "A slight modification of an equation developed by Alan Turing in 1952 has been used to show how the patterns of big cats change from kitten to adult markings. Sy-Sang Liaw of National Chung-Hsing University in Taichung, Taiwan, and colleagues set out to replicate these patterns using Turing's equations. But they found they had to do more than just tweak the parameters of the reaction-diffusion equation. Instead they had to assume two stages of spot growth with different rules: the first to get the baby cats their spots, and the second to create the final configurations. It took them a year to find a final solution."
Some researchers dicking around with orange molds accidentally discovered this little thing called PENICLLIN. Some Swiss mountain hiker got irritated with little seeds that kept sticking to his clothes, which upon further inspection led to the invention of VELCRO.
On the other hand, researchers trying to solve a critical rubber shortage during World War II came up with an earth-shattering invention: SILLY PUTTY.
Point is, you just never know. ;)
All the techniques ever used to make men moral have been themselves thoroughly immoral... (Nietzsche)
Here we witness the micro through the macro, through all scales of physical dimension, in an interplay of force, energy and motion, with the final result happening both all at once and forever spread over time. Incredible.
No, not really."
If you find something as mundane as a mathematical model of how spots deveop on leopards to be "incredible", I think the wonder is all in you and not in the thing itself. Setting aside the wonder that is life itself, leopard spots are pretty boring -- roughly the equivalent to modeling how freckles develop on redheads.
In machine learning (really statistical modelization), people are interested in developing methods of representing relations. Expressive models can "learn" complex mathematical functions (like the distribution of spots). Because learning uses finite (and sometime noisy) data, it is necessary to limit the expressiveness of the models so that chance occurrences in the data are smoothed out of the representation and only meaninful correlations are retained. Also the model is "objectively" measured. One way to do it is to set aside some of the data and use it to validate the model. That is, try to predict unseen data. If the model performs well on unseen data then it has "learned" a meaningful representation of the function governing the creation of the data. Even then, the model doesn't have explanative power (Correlation doesn't imply causation), just predictive power.
To link back to the parent: If all you want is description, you rely on the predictive power of the model and there are standard ways to make sure the model is accurate and conservative with respect to the data used to build it. If you want to explain the underlying biological process then: though luck.
Finally, (I'm officially ranting now) you're asking a lot at the same time. It's like discarding Newton's mechanics because he wasn't aware of all the factors (atoms, quarks, quantum mechanics) even if the model fits reality well. At some point you have to fence the known and unknown and structure what you know into a coherent system of axioms. Then you bite at the unknown a little bit more and try to fit/extend/simplify your system of axioms. I mean, to this day, even if we can predict gravitational phenomena to incredible precision, nobody would be foolish enough to think that they know what gravitation really is; they can predict how it behaves but they can't explain it.
The objective of scientific research is to find ways to make useful predictions (almost by definition; if a field of study cannot make predictions or is not useful, it's not science - it's philosophy or art something like that). If you can generate a sufficiently accurate prediction, then the method by which you attain it is immaterial. 'Understanding' the processes is one highly effective way to discover such things, but it's not the only way. These modelling techniques are a good way to bridge gaps in understanding and make accurate predictions. Understanding things is more important for 'pure research' goals - extending the foundations on which you can construct new predictions. There's no reason for concern so long as you keep these things clear and don't confuse them; they have nothing in common except that they're both ways of constructing a prediction.
However, I am unable to think of a reason why predicting the patterns of leopard spots could be useful, unless you're trying to engage in some form of leopard topiary or need to compress a large number of images of leopards.
They aren't supposed to define anything. Attempting to do so would be confusing 'modelling' with 'understanding'. For an analogy: you can get mince out of a mincing machine, but no matter how hard you look, you are not going to get any mincing machines out of the mince.
Those parameters are almost certainly related to some real-world process, but the relationship is very unlikely to be a direct mapping of parameters to factors - and no amount of staring at the equations will result in understanding the unknown expression that defines this relationship. Most likely they are an approximation of the result of the true expression, sufficiently accurate to describe all the real-world cases but otherwise uninformative.
And if you're an animation TD who has been assigned the task of creating a huge school of fish, each one of which should look different and yet still look like the right kind of fish, you'll be glad that someone has studied the problem of how to model animal markings.
No, this is not hypothetical. It's real, and it's done today.
sub f{($f)=@_;print"$f(q{$f});";}f(q{sub f{($f)=@_;print"$f(q{$f});";}f});