Wikipedia:Reference desk/Archives/Mathematics/2013 February 4

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

February 4

Testing Support Vector Machine

Hi. Should the performance of a soft-margin support vector machine on unseen data be assessed in terms of its misclassification error (i.e. simply whether each data point is correctly labelled as 1 or 0), or as the sum of the SVM slack error variables (i.e. the extent of the error is taken into account, with correctly classified points scored as zero)? --82.13.141.22 (talk) 15:49, 4 February 2013 (UTC)

The only canonical way to assess performance is in terms of the objective function that the method minimizes. The Soft margin section of the article gives the objective function that applies in this case. Of course there are an infinite number of other ways you could assess performance, but there is no canonical way to choose amongst them. Looie496 (talk) 03:38, 5 February 2013 (UTC)

OK, so seeing as the weight vector and C value will be the same for all evaluated points, then this implies I should use the sum of the SVM slack error variables. I'm confused though because I nearly always see misclassification rate used as the method of evaluation. --81.101.105.36 (talk) 09:45, 5 February 2013 (UTC)

void space in crushed stone

It seems to me off the top of my head there ought to be an optimal answer to this question, but I bogged down trying to figure it. I'm trying to give runoff water from my yard a place to go, i.e. an underground space full of crushed stone (with layer of fabric to keep dirt from clogging the spaces), and I want to maximize the void space with respect to size of the stone. So, if I assume cubical stone, and the average void space between two contacting stones would be some function of the angle at which they contact, which sould be some function of pi, maybe? It seems that a couple of huge stones would not have much total void space, and conversely filling the space with tiny stones (i.e. sand) wouldn't either, so somewhere in between would be optimal; maybe the important variable would be average stone dimension divided by length/diameter of the space to be filled (assume cubical? spherical?) I'd be grateful for any calculation that might get me near the right order of magnitude, even. Thanks. Gzuckier (talk) 19:55, 4 February 2013 (UTC)

The question falls under the general genre of "packing problems". There are many models for packings, most of which concern optimal (or nearly optimal) packings that minimize the amount of void space for a certain grain size and shape (this is known as a close packing). Our articles on sphere packing and random close pack are also relevant. For spheres of equal size, the optimal density (ratio of filled to total space) is always $\pi /3{\sqrt {2}}$ (a result of Thomas Callister Hales), and is achieved by the face-centered cubic packing. For non-uniform packings (with grains of variable size) it's possible to achieve any higher density by filling in all the holes with smaller and smaller grains. Sławomir Biały (talk) 20:08, 4 February 2013 (UTC)

Percolation theory is also relevant to the specific problem of channeling runoff, Sławomir Biały (talk) 12:14, 5 February 2013 (UTC)

A critical consideration is whether the stones are placed at random or placed, and held, in the position of your choosing, where presumably you can achieve larger voids. Also note that too much void space could leave it vulnerable to collapse, say if one stone holding the rest up cracks in two. BTW, with your design, what keeps the covering fabric from itself being clogged ? StuRat (talk) 04:00, 5 February 2013 (UTC)

A better solution might be to leave the space void and put a grid over it, but, if it has to be filled, then I think the stones must be as near as you can get to spherical, and all around the same size (as mentioned above), with random arrangement. The actual size of the stones makes hardly any difference. Perhaps someone can provide a mathematical proof that this configuration is optimal (or disprove my claim if they think I'm wrong). There are much better solutions if you are allowed to use flagstones in a triangular or rectangular framework, but I assume you want stone that can be poured in and dug out easily. As mentioned above, the system will eventually stop working because of silt, so it will need regular cleaning. An exit channel for the water will reduce the maintenance required. Dbfirs 17:55, 6 February 2013 (UTC)

I believe it's true that the scale of the stones won't matter for packing efficiency, but only if you neglect boundary conditions. If you included those, then smaller stones are better. Picture a single stone, with no gaps at all, for the reductio ad absurdum case. However, too small of stones are too easily silted up, or wash away themselves, so there's a limit on that approach. StuRat (talk) 21:18, 6 February 2013 (UTC)

A single spherical stone in a cubical void of equal diameter would allow more empty space (47.64%) than smaller spherical stones packed tightly (face-centered cubic packing allows only 25.95% in the limit) [and exactly the same 47.64% with loosest regular packing if the diameter of the stones is a factor of the diameter of the hole]. I agree that boundary conditions will make a (variable) difference depending on the shape of the hole, but loose packing will have a greater effect. The difficulty with small stones is that they more quickly tend towards tight packing (maybe 30% free space) under gravity and tiny vibrations. Dbfirs 09:28, 7 February 2013 (UTC)