Wikipedia:Reference desk/Archives/Mathematics/2011 May 7

= May 7 =

Neural nets for time series forecasting
Are there any up to date introductory textbooks on this subject that people would recommend please? Any online primers or tutorials? It is neural nets in relation to time series that I am interested in, not just neural nets.

People using neural nets claim that they are superior to time series statistical methods, but textbooks about time series forecasting do not cover them. Thanks 92.15.21.162 (talk) 09:41, 7 May 2011 (UTC)


 * You can find some details in this paper and in the references. Count Iblis (talk) 16:00, 7 May 2011 (UTC)

Sorry, I cannot see any significant relevant details in that PDF which is about Global Warming. The textbook mentioned in the references is from 1997 - not what I would call up to date. Thanks for trying anyway. 2.97.208.37 (talk) 12:40, 8 May 2011 (UTC)

Continued fractions and transcendentality
Hello everyone,

I'm working with continued fractions, and I've been asked to give an example of a series of quotients $$a_n$$ for a transcendental number $$\alpha $$ and prove it's transcendental. Theorems I have to play with are Liouville's theorem (if $$x$$ is algebraic of degree n then there exists some constant c such that $$|x-\frac{p}{q}|>\frac{c}{q^n} $$ dependent on the coefficients of the minimal polynomial) and of course that for any x with convergents $$p_n ,\, q_n$$ we have $$|x-\frac{p_n}{q_n}|<\frac{1}{q_n q_{n+1}}$$: if I recall correctly this holds for all x, not just algebraic irrationals. Lastly, I might be able to get away with using the fact that a continued fraction expansion terminated iff the value we're expanding is rational.

The example I've always seen for transcendality using continued fractions is the sum $$\sum_n \frac{1}{10^{n!}} $$: however, we're asked explicitly to give the quotients of the transcendental number (or rather, to give a series of quotients which give a transcendental number and then show it's transcendental), and I've never actually seen the continued fraction expansion of $$\sum_n \frac{1}{10^{n!}} $$: it looks like it would be messy. Now I suppose I could give a well-known transcendental number and figure out the coefficients but I don't think that's what's intended. Does anyone have any idea of a good, quick answer I could use? I presume the intention is to show it doesn't satisfy Liouville's theorem but can't be rational. I seem to recall reading that the coefficients have to be unbounded if it's transcendental, but I'm not sure if that's true or not: maybe an application of the pigeonhole principle. Could anyone suggest anything to help? Thank you! :) Tasterpapier (talk) 13:09, 7 May 2011 (UTC)


 * I'm not sure what the question is. It doesn't say you you're required to give an answer as a continued fraction. Then why can't you use the partial sums of $$\sum_n \frac{1}{10^{n!}} $$? There is a well known sequence of quotients 3/1, 19/7, 193/71, ... (see OEIS A053518 and A053519) which converges to e if that helps.--RDBury (talk) 18:55, 7 May 2011 (UTC)


 * I'm sorry, I looked it up and I think I was using a different terminology to the wikipedia standard: by the 'quotients' I meant the numbers a_0, a_1, etc. such that our number is expanded in continued fraction form as [a_0, a_1, a_2, ...]. The question is to choose a series of these continued fraction quotients (a_i) which correspond to a transcendental number, and show that the number to which they correspond is transcendental. I am saying that while my above example is certainly transcendental, I have no idea what the quotients are so that example isn't any use to me. I presume there is a nice choice of quotients I could use which correspond to some transcendental number which fails to obey Liouville's theorem, hence isn't algebraic: I could certainly use a well-known transcendental number and find the quotients, but I don't think that's what's intended of me. I was wondering if there were any good sensible choices I should use? For example, something simple like a_i = i: though it isn't clear how I'd show that didn't obey Liouville's theorem. This is for an exam question past paper, so ideally I need a nice sequence I can work with easily, rather than one on the online sequence database I would have to look up. Tasterpapier (talk) 21:48, 7 May 2011 (UTC)


 * Choose a_n huge --- $$a_n = 2^{n!}$$ should do it. Then $$q_n q_{n+1} > q_n^n$$, so $$|x-\frac{p_n}{q_n}|<\frac{1}{q_n q_{n+1}} < \frac{1}{q_n^n}$$, and Liouville's theorem applies.--203.97.79.114 (talk) 22:11, 7 May 2011 (UTC)


 * That's perfect, thank you! Tasterpapier (talk) 23:07, 7 May 2011 (UTC)

Actuary
does an actuarial calculate both risk to the insured and the risk to the insurer? —Preceding unsigned comment added by 76.118.147.89 (talk) 14:15, 7 May 2011 (UTC)
 * I'm an actuary (in pensions, rather than insurance, but it's the same idea) and I calculate whatever I'm paid to calculate. Usually, it would be the insurer that hires an actuary rather than the insured, so it's the risk to the insurer that the actuary would calculate. Actuarial calculations also aren't much use when dealing with individuals. If you want to know the chance of a particular individual dying in the next year, you ask a doctor, not an actuary. The insurer will have thousands of policies, though, so the actuary can average everything out and get useful answers. --Tango (talk) 20:05, 7 May 2011 (UTC)