When choosing models, one either needs to have a convincing mechanism OR one must test the model on data not used to choose the model. (Best is to have both.)

In my field, cross-validation is the standard, since we often have lots of data from past experiments, but collecting new data is slow and expensive. In cross validation, you do all your model building and model selection from part of the data, then check to see how well the model fits on the remaining data.

You have to be very careful when doing this, since you can only use the verification set once. If you go back and change your model, based on a failure on the verification set, then you’re using all the data for training, and have no independent verification.

If the model-building and selection process is automatic, you can do random splits of the data, and see the distribution of how well the data fits. This can give you a good idea whether you have enough data to fit the sort of model you are using.

What’s interesting to me is the (volume)*(height) term itself. Volume alone makes intuitive sense to me (the more cheese the more energy needed). But seeing that the height factor is predictive suggests some sort of asymmetry. Perhaps the microwave is effectively aiming down, or it’s just that the cheese is placed at the bottom?

It’s fun to think about.

Without a physical explanation and very few data points, fitting an arbitrary curve with high R^2 values is pretty meaningless.

I can see volume being an important parameter, linearly related to time (total energy needed to melt that much cheese).

I can also see “thickness” as a important parameter (as energy is preferentially absorbed nearer the surface).

I have trouble with the y-intercept, though. Why should melting microscopic amounts of cheese take 13 seconds?

13.071507 + 0.18126714*(volume)*(height)

has R^2 of 0.977, not bad. Found using Eureqa.

Simon: What software did you use to combine all the videos together?

I used Adobe AfterEffects. There has to be a cheaper, simpler solution for the same effect, though.

Christopher: Did you try different orientations of your cheese? The melting time of your cheese ought to vary greatly depending on how it stands. And what is your cheese budget, anyway? Can you afford several dozen more trials so we can examine variation in your data?

Different orientations of the same block of cheese is inspired. My wife has put some serious constraints on my cheese budget, though, in light of my smelling up our place, so the reshoot of this activity, while inevitable, will have to wait.

(PS. Do they only sell cheese in my part of the world? Doesn’t someone else have sufficient curiosity to test this out.)

Karen: What’s the *math* goal you’re after?

Seriously, I haven’t thought about it. I’m not sure where the idea comes from that this is in any way a lesson. I was curious. I followed up on my curiosity. I got confused. I came here to share my curiosity and confusion with you folks. That’s about all I have right now.

On why it works, it seems the exposed surface area is how the microwaves get into the cheese. I imagine heat loss to be a minimal concern over the time scales involved here, and in comparison to the bombardment of microwaves.

And the exposed surface area idea is lovely. Kids often wonder about orientation of prisms. Abstractly, orientation doesn’t matter-a triangular prism is still a triangular prism no matter what face it is resting on (or whether it is resting on a face at all). But in the applied problem, this matters very much. Did you try different orientations of your cheese? The melting time of your cheese ought to vary greatly depending on how it stands. And what is your cheese budget, anyway? Can you afford several dozen more trials so we can examine variation in your data?

More on cubeyness:
http://wp.me/pAG7Q-3D

More on middle schoolers and prisms:
http://wp.me/pAG7Q-55