Wikipedia:Reference desk/Archives/Mathematics/2007 October 27

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October 27

Rounding of in financial mathematics

Sir, in financial mathematics, the answer for some questions should be "corrected to the nearest .1%, in some questions- corrected to the nearest 0.5%, in some rounded down to 0.1%. please explain.Thanks. Mohana. —Preceding unsigned comment added by 59.178.115.204 (talk) 03:46, 27 October 2007 (UTC)

In all cases, the inention is to have a result that is similar to the "true" result, but for easier reading, rounded to a multiple of some fixed quantity. For example, if the number is 34.791524863%, you will not write the entire number but round it. I think the intention in the three methods you mentioned are: "Corrected to the nearest 0.1%" will be 34.8%; this is the multiple of 0.1% which is closest to the number. "Corrected to the nearest 0.5%" will be 35.0%, the multiple of 0.5% closest to the number. And "rounded down to 0.1%" will be 34.7%, the highest multiple of 0.1% lower than the number. -- Meni Rosenfeld (talk) 12:41, 27 October 2007 (UTC)

See also Rounding.  --Lambiam 16:08, 27 October 2007 (UTC)

"in-order" scoring systems

.....please note: I have added this in an unusual way because I keep trying to add a question and the server keeps stuffing up somehow and won't let me add it to the ref. desk. It just adds to "Ref Desk/ Math (comment)" and "Math" turns out to be different to "Mathematics". I suspect it's coz I'm the first person for the day, and not logged in, so it treats it as a new page creation. Or whatever.

[explanation of cryptic comment: I added this by simply editing the section directly, not using the "add a question" link at the top of the page, and I thought it might come out garbled on the page. Actually, to my surprise, it looks normal, apart from the large size heading.]

I'm looking for a good scoring system for problems where a student has to put a list of events in some kind of order (say chronological). My basic (time consuming) one is to simply add up the absolute value of the position errors, so putting the 7th event incorrectly in 9th place would get 2 points (the perfect score being obviously zero). Is this system a good one, or is there a flaw in the same way all voting systems are flawed? Also, are there any better/ simpler ones? Thanks in advance to all you bright folks out there :) 203.221.126.254 10:51, 27 October 2007 (UTC)

Apparently, in this edit, User:Froth replaced the header with a transclusion to his userspace header, which happened to have a broken link. I have no idea why he did that, but for the time being I have fixed his link, so it should work now.

If I understand your problem correctly, you are in general looking for Non-parametric statistics; in particular, Spearman's rank correlation coefficient is sometimes used to determine the similarity between two orderings. It is not computationally faster than your suggestion, but it can be more informative, depending on the application.

I assume that by "all voting systems are flawed" you mean Arrow's impossibility theorem; This is a common interpretation of it (even more common is that the only flawless voting system is dictatorship), but wrong in my opinion. The "correct" interpretation is that flawless voting systems are not indenpendent of irrelevant alternatives. -- Meni Rosenfeld (talk) 12:33, 27 October 2007 (UTC)

I don't grok why "independence of irrelevant alternatives" bothers some people, but anyway — the condition that I'd discard first is that the procedure must give a complete ranking of all possible outcomes, since we usually only care about the first one (or few). —Tamfang 00:34, 30 October 2007 (UTC)

The system I use is a point for each correct answer and a point if the answer comes after the previous correct one in the "list". The first correct one is always in sequence. Examples using "List the 5 vowels in order: aeiou gets 5+5; abcde gets 2+2 (a&e are correct, a is after 'null' and e is after 'a'); eaozu 4+3 (eaou are correct, e is first, nothing for a as it isn't after e, 1 for o which is after a, 1 for u which is after o). -- SGBailey 22:30, 27 October 2007 (UTC)

The main problem with your original scoring system (126.254's system I mean) is that when a student gets all items ordered correctly, except for the first, which she puts at the end, none of the items have the correct rank, and yet she almost got a perfect ordering (she just didn't recognize one of the items, perhaps). I would use something like minimum edit distance here. The number of basic operations (delete, insert, transpose and substitute) to get from one string to another could be used to find the distance between the correct ordering and the student's ordering. In this case, our student would only be two operations away from the correct answer, and get a good grade. You'd have to figure out how to translate the edit distance to a grade by how students perform, but that shouldn't be to difficult. This is also a method of grading that is relatively easy to explain to the students (they don't have to know the algorithm). Other definitions of string distance (like dynamic time warping) may also be useful. risk 23:17, 27 October 2007 (UTC)

The first method that comes to mind for a pure mathematician is: perfect score is zero, and add a point for every pair of events that is out of order. For example, if 123 is the correct order, then 312 would get 2 points (the pair 1-3 is out of order, as is 2-3, but 1-2 is in the correct order), and 321 would get 3 points (each of the three pairs 1-2, 1-3, and 2-3 are out of order). This method is related to the word metric on the symmetric group --- i.e. it's a standard way to measure the distance between two orderings.

This is similar to your method, but not exactly the same (note that your method gives the same values on 312 and 321, which doesn't feel quite right). Really, though, any reasonable method could be ok --- just as long as you don't use the method I recall a grade-school teacher of mine once using in which I gave the answer 23451 and he gave it no credit because "none of them were in the correct place." An easier method to calculate might be the "longest correct string" method (looks like part of what SGBailey is talking about) --- find the longest string of events that are in the correct order (not caring whether there are other events placed in between). For example, 412635 would get 4 points (out of a possible 6 for the correct answer 123456) because 1-2-3-5 is in the correct order. Kfgauss 23:20, 27 October 2007 (UTC)

Hi all. Thanks for those highly interesting answers. Yes it was Arrow's impossibility theorem as mentioned above. Thanks for fixing the formatting etc., Meni. As for Kfgauss, interesting idea; that trick you are talking about is called a diff, which is a crucial Wikipedia utility, which helps in editing, and tracking changes to these pages. I think I'll stick with my way. 312 and 321 both give 4, and I agree, 312 is better, because only one of them is out. Even so, 312 is simply first among equals, since 321 at least has 2 in the correct place, and the teacher cannot really assume that the student got the right answer for the wrong reasons. If they are historical events, the student might just get 1 and 3 confused, but have a clear idea of 2. It seems like it's still a good system. So 321 is only slightly worse than 312. My system is not easy to compute, but easier than doing a diff by hand. That was a tricky suggestion though, and possibly the best in theory. Cheers, 203.221.127.33 16:49, 2 November 2007 (UTC)

Fractal probability distribution

Is there a family of naturally occuring phenomena (ie. datasets based on nature) that show a 'fractal distribution'? By this I mean, a multivariate distribution that can only finitely be described as a 'fractal' (by a recursive process, basically). A probability distribution that doesn't get less complex, no matter how much you zoom in to a specific sub-domain. It would be trivial to invent a probability distribution that works like this, but I wonder if any natural processes make this kind of 'picture' when two or more attributes are scatter-plotted against each other. There are plenty of fractal structures everywhere in the world, surely there are 'hidden variables' that paint the same kinds of pictures? I've been looking for research into this area, but I'm not coming up with much. I'm thinking that maybe I've got the wrong keywords in mind. Anybody have any ideas? Feel free to post anything that feels remotely relevant. risk 23:02, 27 October 2007 (UTC)

I'd say that the noise in the variables, however tiny, would destroy any sort of fractal structure in the distribution. -- Meni Rosenfeld (talk) 23:49, 27 October 2007 (UTC)

True, the infinite complexity of the fractal will at some point be cut off by noise, but I'm hoping that before that point the recursive structure of the distribution becomes apparent (and may be induced perfectly). risk 00:13, 28 October 2007 (UTC)

(after edit conflict) No, but inasmuch as this is a question about the real world, the Science section of the Wikipedia:Reference desk is a more appropriate spot to ask the question. But why multivariate? The Cantor function is a univariate fractal distribution, and the "Devil's staircase" article mentions some physical systems, but I don't know if they can be used to get a fractal distribution.  --Lambiam 23:54, 27 October 2007 (UTC)

I did wonder where to post this, but figured that I was mainly looking for statisticians and chaoticians. I figured I'd have better luck here. The reason I'm looking for multivariate distributions, is that I'm investigating the possibility of a thesis subject on machine learning in this general area, which means multivariate distributions. Even so, the devil's staircase is a starting point, thanks. risk 00:13, 28 October 2007 (UTC)

I imagine that many features of the natural world would show fractal-like behaviour over a range of scales if you sampled then with sufficient accuracy and frequency. For example, if you had a minute-by-minute record of wind speeds over the sourse of several years then you could look for similarities between the distribution of individual gusts, storms and seasonal variations. Or you could look for a fractal distribution of terrain roughness by sampling sizes of sand grains, pebbles, boulders and mountains over a wide territory.

However, if you are looking for an actual example of such a dataset, I think your problem will be that natural data is not usually collected over a sufficiently wide range of scales to suit your purpose. Also, your mention of "hiden variables" confuses me - I would have thought that fractal models would be most appropriate when describing epiphenomena, and they are unlikely to provide any insights into underlying physical laws.

Anyway, the only example that I can come up with of real datasets covering a wide range of scales is price data in financial markets. Not sure if that meets your definition of "naturally occuring phenomena", but one source of information (which you may already know of) is Mandelbrot's book The (Mis)behaviour of Markets, which is all about deriving fractal models of financial markets. Gandalf61 07:03, 28 October 2007 (UTC)

I see your point about scale. Most datasets are indeed collected specifically a at a single scale. I may need to look at time series and similar data. That way, if I get enough data at a small resolutions, I'll also have all the scales above it. Perhaps hidden variables is not the right term. I simply meant that we see fractals everyday, I expect them to be present 'under the surface' as well, in things that aren't readily observable. These datasets would still be epiphenomena, but not directly observable ones. I wasn't aware of Mandelbrot's book. I'll definitely try to get a copy. My definition of naturally occuring is very broad (I'm basically looking for anything not explicitly constructed for this purpose, to show tjat the whole idea has real-world relevance), and financial data should be easy to obtain, so I'll look into that. risk 15:02, 29 October 2007 (UTC)

The statistical aspect of the question, may in some sense smooth out the fractal nature. For example take trees which can be well modelled by a fractal structure. If you take a gross measure of these, say it height, and build the distribution of heights you will find that you will get a smooth distribution. The central limit theorem may play a part in this. --Salix alba (talk) 08:53, 28 October 2007 (UTC)

Yes that's true. Anything averaged over many fractal instances will end up a representation of 'fractal space'. Which would usually be very smooth, and non-recursive, so nobody's going to invent a fractal distribution, even if the underlying process is fractal-like, per instance. Maybe looking at the distribution is not the right angle. Thank you. risk 15:02, 29 October 2007 (UTC)

Coastlines. 68.231.151.161 02:54, 30 October 2007 (UTC)