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

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November 7

Goodness of a logarithmic fit

Hello. I have a set of data (5 points or so) and I need to test how well they fit my model, which is a log model. Unfortunately by some stupid and arbitrary stipulation I may only use "technology" (including calculators, computers, etc.) for "routine calculations" (doing stats is specifically excluded). What is a good way to calculate the goodness of the fit by hand (using a calculator for things like adding, taking logs, etc. of course)? I skimmed your article but didn't see anything I thought might work. thank you. 24.92.85.35 (talk) 03:04, 7 November 2011 (UTC)

You can test the fit by plotting the results on logarithmic graph paper (if available). This would not require any calculation at all, and the quality of fit can be seen by drawing the "best straight line". In the absence of special graph paper, the same can be achieved on ordinary graph paper, but calculating (or looking up) the logarithms of one set of data is necessary. Fitting data to models was done regularly long before calculators were available. Dbfirs 08:17, 7 November 2011 (UTC)

Just take the logs and apply a standard linear Lack-of-fit sum of squares. Except you won't get anything much useful with just five points! If you start of with more than one model that decreases the degrees of freedom you should use, otherwise you're like the commercials where they keep picking ten people until they get nine out of them saying xyz is best.. Dmcq (talk) 08:36, 7 November 2011 (UTC)

Binomial theorem

Ok this is a primary school question but I'm missing something... If n is prime, how do I show ${\tbinom {n}{k}}$ is a multiple of n when k is not 0 or n? Money is tight (talk) 11:25, 7 November 2011 (UTC)

Write ${\tbinom {n}{k}}$ in terms of a fraction of factorials, and argue that the n in the numerator (on the front of n!) can't cancel in the fraction because it's prime. Staecker (talk) 11:49, 7 November 2011 (UTC)

I still don't get it. If n doesn't cancel out, how does that make ${\frac {1\cdot 2\cdot ...\cdot (n-1)}{k!(n-k)!}}$ an integer? Money is tight (talk) 02:48, 8 November 2011 (UTC)

You know that ${\frac {1\cdot 2\cdot ...\cdot (n-1)\cdot n}{k!(n-k)!}}$ is an integer, which means that all factors in the denominator cancel with something from the numerator. But none of this cancellation can occur from n in the numerator since it's prime. So it must be that all the factors in the denominator cancel with the numerator factors other than n, which means that ${\frac {1\cdot 2\cdot ...\cdot (n-1)}{k!(n-k)!}}$ is an integer. Staecker (talk) 02:53, 8 November 2011 (UTC)

OMG I can't believe I missed that. Thanks for your help though. Money is tight (talk) 05:02, 8 November 2011 (UTC)

likelihood value

Assume that I have three models. I am given a "likelihood value" for each one: -880, -874, and -856. Is it possible to make a statement about comparative accuracy between the models using those values? For example, can I state something like "the third model is 12% more accurate than the first model."? -- kainaw™ 19:48, 7 November 2011 (UTC)

these look more like log-likelihoods to me. take the exponential of the numbers to give a likelihood function. hth 222.153.61.63 (talk) 07:21, 8 November 2011 (UTC)

I don't think there is a general relationship between the likelihood of a model and its accuracy, but there may be for specific cases. -- Meni Rosenfeld (talk) 08:06, 8 November 2011 (UTC)

Homework interpreting - "normalised absolute values'

Hello, I'm trying to do the following homework problem: Let L/K be finite extensions of the p-adic rationals $\mathbb {Q} _{p}$. Let $|\cdot |_{K},\,|\cdot |_{L}$ be the normalised absolute values, and put $n=[L:K]$. Show that $|x|_{L}=|x|_{K}^{n}$ for all $x\in K$.

Now the problem is, I don't know what a 'normalised' absolute value is, and it hasn't been defined in the lecture notes for the course I'm taking. I know what a 'normalised valuation' is, i.e. one for which the image of the valuation on the nonzero elements of the field, $v(K^{*})$, is the integers. However, I've never seen a definition for a normalised absolute value: is it just one for which the corresponding valuation is normalised? Presumably if you know what the question is asking you will know what it means and hopefully be able to help me! Thanks in advance. Simba31415 (talk) 20:23, 7 November 2011 (UTC)

Eigenvector question

I am reading a paper and there is a statement that I believe is false, but I'm no expert on Eigenvalues/vectors. It claims: "Given a set of vectors of n-degree in an n-degree space, a normal vector exists that is the Eigenvector of the set." My understanding may be completely off. I understand that every vector has multiple Eigenvectors - vectors that are parallel to the original vector. Given a set of vectors that are not necessarily parallel, it is not possible to claim that there is a single Eigenvector for the whole bunch. But, what I think is meant here, there is an Eigenvector for an "average" vector for the set - assuming you pick a function that takes a bunch of vectors and averages them into a single vector. So, is the statement in the paper valid? Is my understanding valid? Are we both completely off? (I know Meni can dumb this down for me) -- kainaw™ 21:24, 7 November 2011 (UTC)

Unless there is another meaning of "eigenvector" which I am not aware of, there is no such thing as an eigenvector of a vector (or of a set of vectors). A matrix has eigenvectors, and a linear transformation from a space to itself has eigenvectors.

Maybe the paper is trying to say something about eigenvectors of a martix whose columns (or rows) are the given vectors. -- Meni Rosenfeld (talk) 08:05, 8 November 2011 (UTC)

Yes. There is a 4x8 matrix which is created by eight 4-dimension vectors. Then, it makes the claim quoted above. Then, it uses this "normal vector" to be a sort of average of the set of eight vectors. My gut feeling is to explain to the author that reducing a set of vectors to one vector is never going to be a one-to-one relationship. So, even if he claims it is a magical Eigenvector, his idea of reducing it to one vector isn't going to be valid. -- kainaw™ 13:36, 8 November 2011 (UTC)

Only square matrices have eigenvectors. It seems likely that the author of the paper does not understand what an eigenvector is. Looie496 (talk) 15:42, 8 November 2011 (UTC)

I agree, but stating so is uncomfortable because I cannot describe what an Eigenvector is. I can only describe what it does (changes magnitude, but not direction - allowing for negative magnitude - when multiplied by the square matrix). I also cannot calculate an Eigenvector for any n-square matrix by hand. Examples online tend to only be for 2x2 matrices. So, I was afraid that any note I wrote would do more to show my ignorance than note a real problem with the statement in the paper. -- kainaw™ 19:36, 8 November 2011 (UTC)

"Eigenvector" is defined as any vector that does not change its direction when you act on it with a square matrix (or with a linear transformation that can be described by a square matrix). It is not defined for a 4x8 matrix or for a set of eight 4-dimension vectors. That's all you need to say.

It is possible that the author uses the word "eigenvector" to mean "the average of a set of vectors", which is not what the term is supposed to mean. --Itinerant1 (talk) 20:16, 8 November 2011 (UTC)

Finding an eigenvector for a matrix isn't hard. Find $\det(\lambda I-A)$, find a root $\lambda$, solve the linear system $(A-\lambda I)v=0$. -- Meni Rosenfeld (talk) 20:15, 9 November 2011 (UTC)

Hi kainaw. Our curiosity is piqued, so can you give us a link to the paper? It's been emotional (talk) 15:39, 10 November 2011 (UTC)

It is a submission for publication. It hasn't been published (and I seriously doubt it will be without some major revisions). I get a couple papers every month to look over and make comments on. For some ungodly reason, someone thinks I know a few things one or two computery topics. -- kainaw™

Also, this is my note, based on the context, which I hope is correct: "This is incorrect. I believe it should that that with n-degree space, an nxn matrix will have an eigenvector that is perpendicular to the principal plane of each degree." -- kainaw™ 16:42, 10 November 2011 (UTC)