[R] How to find the parameter of a power function to fit simulation data to it for the tail?

[David Winsemius]

 From a search at Barons site with "fitting distribution truncated  
power"

http://finzi.psych.upenn.edu/R/library/brainwaver/html/fitting.html
http://finzi.psych.upenn.edu/R/library/gamlss/doc/gamlss-manual.pdf
http://finzi.psych.upenn.edu/R/library/bipartite/html/degreedistr.html
http://finzi.psych.upenn.edu/R/Rhelp02a/archive/90168.html

-- 
David.

On Aug 2, 2011, at 5:27 PM, Paul Menzel wrote:

> Dear R folks,
>
>
> having simulation data in a vector n2off, I know that they should be
> similar to a power function f [1], f(n) = n^(-1/r), r ∈ ℕ\{0},  
> and I
> want to find the value for r best fitting the simulation data.
> Furthermore I know that this is only true for big n, that means  
> n2off(n)
> ~ f(n) ⇔ n2off(n)/f(n) → 1 for n → ∞. (The vector n2off is  
> considered a
> function n2off(n).)
>
> I came up with the following example where I artificially munch(?) the
> values of a known function, n^(-½), and the fit should hopefully  
> return
> r = 2, that means n^(-½).
>
>> n <- 1:10 # Should be more data points, but not useful for  
>> including into an email.
>> n
>         [1]  1  2  3  4  5  6  7  8  9 10
>> n2 <- n**(-0.5)
>> n2
>         [1] 1.0000000 0.7071068 0.5773503 0.5000000 0.4472136  
> 0.4082483 0.3779645
>         [8] 0.3535534 0.3333333 0.3162278
>> set.seed(1); n2off <- n2 + runif(1)/100 # for greater n the divisor  
>> should also be increased I guess.
>> n2off
>         [1] 1.0026551 0.7097619 0.5800054 0.5026551 0.4498687  
> 0.4109034 0.3806196
>         [8] 0.3562085 0.3359884 0.3188829
>
> Weighting fits(?) larger n higher or only from certain n on, for  
> example
> n ≥ 100, is not considered in this example. And probably the data  
> points
> are too small in this function.
>
> I have to admit that I am new to this topic and I am just overwhelmed
> what I have found when searching for »gafit« in the r-help archive  
> [2]
> and »curve parameter fitting« in rseek.org [3].
>
> Reading ?nlm, ?nlminb, ?opitimze and ?optim there are just too many
> options there. Reading about gafit [4] it says that it is not
> maintained.
>
> Additionally I am not sure if this could be turned into a linear model
> using log(n^(-1/r)) = -1/r log(n). Somewhere it said that linear
> regression models have to fulfill certain assumptions.
>
> So if somebody of you experienced users could point me to the  
> “best”
> function or package to use here and some literature regarding this  
> issue
> (fit only for big n) that would be much appreciated.
>
>
> Thank you in advance,
>
> Paul
>
>
> PS: Is that question too long for sending to the list and should I be
> less elaborate for further problems?
>
>
> [1] https://secure.wikimedia.org/wikipedia/en/wiki/Power_function
> [2] http://tolstoy.newcastle.edu.au/~rking/R/
> [3] http://www.rseek.org/
> [4] http://cran.r-project.org/web/packages/gafit/gafit.pdf
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> and provide commented, minimal, self-contained, reproducible code.

David Winsemius, MD
West Hartford, CT