Re: [R] Incomplete Gamma Functions and GammaDistribution Doc errata.
On Thu, 18 Mar 2004, Adrian Custer wrote: In the course of trying to implement the CDF of an InverseGammaDistribution, I have run across the need for an igamma() function. Several others have needed this function but the answers I have found so far are not totally clear to me. I'm writing for three reasons: 1) to present a small error in the docs 2) to clarify the approach we are expected to take 3) to request,for the ease of use of myself and others, that the R developers implement igamma(), document the approach they want users to take or do both. We would rather you defer to someone who leaps to fewer false assumptions. 1) An error in the docs of the gamma distribution: - The GammaDistribution(param1,param2) is defined alternatively with two versions of the second parameter which are inverses of one another. 'Rate' arises from one derivation such as that presented in Mathworld: http://mathworld.wolfram.com/GammaDistribution.html (see just below equation (2) ) define $\theta \equiv 1 / \lambda$ to be the time between changes Do please read what your reference says. It calls \lambda the `rate of change', and that is the only occurrence of the word `rate' on the page. which makes \theta in equation (3) a rate parameter. In the R That seems an impossible interpretation of what is actually said. documentation (i.e. ?pgamma) this same equation is presented but the parameter \theta is now called s and is termed the scale. This goes against general usage of mathworld, wikipaedia, the COLT library, and others who call the s term a rate. `mathworld' plainly does not, and you have given us no others to verify. Please either change the equation or replace the name with rate and the label with r or nu or theta. Please look `rate' up in your dictionary, and look the gamma density up in a score of statistical references. Your reference calls what R calls `rate' a `rate of change', and you insist it means that 1/rate is `a rate parameter'! 2) Using existing functions to build the igamma(\alpha,\beta): - The incomplete gamma has been mentioned in two previous emails that I could find: http://www.r-project.org/nocvs/mail/r-help/2000/1006.html and http://www.r-project.org/nocvs/mail/r-help/2000/3618.html both of which suggest that the upper incomplete gamma can be obtained using the pgamma() function. Neither mention the `upper incomplete gamma'. Is Prof. Ripley's message (the first above) saying that upperIncompletegamma(x,\rate) = pgamma(x,\rate) * gamma(\rate) ? Let me quote it to you, since by this point I have problems with your veracity. Has anybody implemented the _incomplete_ gamma function in R? What do you think the cumulative distribution function of a gamma distribution is? (Now, there is a little room for disagreement here, but I think pgamma(x, nu)*gamma(nu) might be what you are looking for.) So - `Upper incomplete gamma' is not mentioned. - In your own reference, the parameters of an incomplete gamma do not contain a rate, and so what makes you think nu is \rate? The second argument of pgamma is the *scale*. I believe the `incomplete Gamma' of people like Pearson is what your reference calls the `lower incomplete Gamma', but note I did say there is a little room for disagreement here and might be what you are looking for. Let's look at Abramowitz and Stegun, http://jove.prohosting.com/~skripty/page_260.htm They have P(a, x) as the integral from 0 to x, whereas in Mathematica notation \gamma(a, x) is the integral from 0 to a with shape parameter x. (That is pretty strange, since it is a that is normally varied for fixed x.) So Abramowitz and Stegun's incomplete Gamma is the lower incomplete Gamma of Mathematica with a more sensible set of parameter names. Then in your reference the gamma cdf for \lambda=1 is D(x) = pgamma(x, shape = h) = P(h, x)/P(h, Inf) and hence P(nu, x) = pgamma(x, shape = h) * P(h, Inf) = pgamma(x, shape = h) * gamma(nu) just as I suggested. I have not yet been able to see why this would be true. I keep tripping myself up on the differing notation of R, mathworld, wikipaedia and Abramowitz and Stegun's Handbook of mathematical functions. Would someone please confirm that this is mathematically correct? It is pretty discourteous to ask other people to confirm that an expert posting is `mathematically correct', especially given the phrases you selectively omitted. 3) Adding an igamma() function or documentation for how to generate it. Since I am at least the third person not clever enough to see the way to derive the igamma directly, because the lack of an igamma() function has cost me a chunk of my day,
Re: [R] Incomplete Gamma Functions and GammaDistribution Doc errata.
Adrian == Adrian Custer [EMAIL PROTECTED]
on Thu, 18 Mar 2004 18:47:54 -0800 writes:
Adrian Hello all,
Adrian In the course of trying to implement the CDF of an
Adrian InverseGammaDistribution, I have run across the need for an igamma()
Adrian function. Several others have needed this function but the answers I
Adrian have found so far are not totally clear to me. I'm writing for three
Adrian reasons:
Adrian 1) to present a small error in the docs
I don't agree.
Did you look at the postscript or PDF version of the help file?
{you can get with
help(pgamma, offline=TRUE)
when you have a working latex installation}
If you compare its formula with Abramowitz Stegun 6.5.1
its pretty pretty very very obvious
that P(a,x) == pgamma(x,a)
{so forget about 'rate' or 'scale' which are both 1 by default!)
Why should this be hard to see? Shuldn't you first do your
homework before starting to insult the R developers?
Regards,
Martin Maechler, Seminar fuer Statistik, ETH Zurich Switzerland
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Re: [R] Incomplete Gamma Functions and GammaDistribution Doc errata.
Note corrections to two typos, since Master Custer seems to get confused by tiny details. On Fri, 19 Mar 2004, Prof Brian Ripley wrote: On Thu, 18 Mar 2004, Adrian Custer wrote: In the course of trying to implement the CDF of an InverseGammaDistribution, I have run across the need for an igamma() function. Several others have needed this function but the answers I have found so far are not totally clear to me. I'm writing for three reasons: 1) to present a small error in the docs 2) to clarify the approach we are expected to take 3) to request,for the ease of use of myself and others, that the R developers implement igamma(), document the approach they want users to take or do both. We would rather you defer to someone who leaps to fewer false assumptions. 1) An error in the docs of the gamma distribution: - The GammaDistribution(param1,param2) is defined alternatively with two versions of the second parameter which are inverses of one another. 'Rate' arises from one derivation such as that presented in Mathworld: http://mathworld.wolfram.com/GammaDistribution.html (see just below equation (2) ) define $\theta \equiv 1 / \lambda$ to be the time between changes Do please read what your reference says. It calls \lambda the `rate of change', and that is the only occurrence of the word `rate' on the page. which makes \theta in equation (3) a rate parameter. In the R That seems an impossible interpretation of what is actually said. documentation (i.e. ?pgamma) this same equation is presented but the parameter \theta is now called s and is termed the scale. This goes against general usage of mathworld, wikipaedia, the COLT library, and others who call the s term a rate. `mathworld' plainly does not, and you have given us no others to verify. Please either change the equation or replace the name with rate and the label with r or nu or theta. Please look `rate' up in your dictionary, and look the gamma density up in a score of statistical references. Your reference calls what R calls `rate' a `rate of change', and you insist it means that 1/rate is `a rate parameter'! 2) Using existing functions to build the igamma(\alpha,\beta): - The incomplete gamma has been mentioned in two previous emails that I could find: http://www.r-project.org/nocvs/mail/r-help/2000/1006.html and http://www.r-project.org/nocvs/mail/r-help/2000/3618.html both of which suggest that the upper incomplete gamma can be obtained using the pgamma() function. Neither mention the `upper incomplete gamma'. Is Prof. Ripley's message (the first above) saying that upperIncompletegamma(x,\rate) = pgamma(x,\rate) * gamma(\rate) ? Let me quote it to you, since by this point I have problems with your veracity. Has anybody implemented the _incomplete_ gamma function in R? What do you think the cumulative distribution function of a gamma distribution is? (Now, there is a little room for disagreement here, but I think pgamma(x, nu)*gamma(nu) might be what you are looking for.) So - `Upper incomplete gamma' is not mentioned. - In your own reference, the parameters of an incomplete gamma do not contain a rate, and so what makes you think nu is \rate? The second argument of pgamma is the *scale*. the *shape*. I believe the `incomplete Gamma' of people like Pearson is what your reference calls the `lower incomplete Gamma', but note I did say there is a little room for disagreement here and might be what you are looking for. Let's look at Abramowitz and Stegun, http://jove.prohosting.com/~skripty/page_260.htm They have P(a, x) as the integral from 0 to x, whereas in Mathematica notation \gamma(a, x) is the integral from 0 to a with shape parameter x. (That is pretty strange, since it is a that is normally varied for fixed x.) So Abramowitz and Stegun's incomplete Gamma is the lower incomplete Gamma of Mathematica with a more sensible set of parameter names. Then in your reference the gamma cdf for \lambda=1 is D(x) = pgamma(x, shape = h) = P(h, x)/P(h, Inf) and hence P(nu, x) = pgamma(x, shape = h) * P(h, Inf) = pgamma(x, shape = h) * gamma(nu) P(nu, x) = pgamma(x, shape = nu) * P(nu, Inf) = pgamma(x, shape = nu) * gamma(nu) just as I suggested. I have not yet been able to see why this would be true. I keep tripping myself up on the differing notation of R, mathworld, wikipaedia and Abramowitz and Stegun's Handbook of mathematical functions. Would someone please confirm that this is mathematically correct? It is pretty discourteous to ask other people to confirm that an expert posting is `mathematically correct', especially given the
Re: [R] Incomplete Gamma Functions and GammaDistribution Doc errata.
My appologies to everyone. Prof. Ripley, please realize that I did not intend to suggest that your mathematics were wrong. My question about mathematical correctness refered to my presention of your formula as an equation. I have great admiration for your work and its influences on my field and, therefore, start with the presumption that all errors must be my own. Mr. Maechler, please understand that I had no intent to insult anyone. I wanted to communicate three things: 1) to point out what I understood to be an error in the documentation (which you both point out to be my own misunderstanding of the terminology). 2) to ask for confirmation that I was interpreting past emails correctly. 3) to argue for an R implementation of an incomplete gamma mathematical function. To the R community, please accept my appologies for a message which you may have taken to be insulting. I am deeply thankful for your work, for your code and for this email list. --adrian __ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
[R] Incomplete Gamma Functions and GammaDistribution Doc errata.
Hello all, In the course of trying to implement the CDF of an InverseGammaDistribution, I have run across the need for an igamma() function. Several others have needed this function but the answers I have found so far are not totally clear to me. I'm writing for three reasons: 1) to present a small error in the docs 2) to clarify the approach we are expected to take 3) to request,for the ease of use of myself and others, that the R developers implement igamma(), document the approach they want users to take or do both. 1) An error in the docs of the gamma distribution: - The GammaDistribution(param1,param2) is defined alternatively with two versions of the second parameter which are inverses of one another. 'Rate' arises from one derivation such as that presented in Mathworld: http://mathworld.wolfram.com/GammaDistribution.html (see just below equation (2) ) define $\theta \equiv 1 / \lambda$ to be the time between changes which makes \theta in equation (3) a rate parameter. In the R documentation (i.e. ?pgamma) this same equation is presented but the parameter \theta is now called s and is termed the scale. This goes against general usage of mathworld, wikipaedia, the COLT library, and others who call the s term a rate. Please either change the equation or replace the name with rate and the label with r or nu or theta. 2) Using existing functions to build the igamma(\alpha,\beta): - The incomplete gamma has been mentioned in two previous emails that I could find: http://www.r-project.org/nocvs/mail/r-help/2000/1006.html and http://www.r-project.org/nocvs/mail/r-help/2000/3618.html both of which suggest that the upper incomplete gamma can be obtained using the pgamma() function. Is Prof. Ripley's message (the first above) saying that upperIncompletegamma(x,\rate) = pgamma(x,\rate) * gamma(\rate) ? I have not yet been able to see why this would be true. I keep tripping myself up on the differing notation of R, mathworld, wikipaedia and Abramowitz and Stegun's Handbook of mathematical functions. Would someone please confirm that this is mathematically correct? 3) Adding an igamma() function or documentation for how to generate it. Since I am at least the third person not clever enough to see the way to derive the igamma directly, because the lack of an igamma() function has cost me a chunk of my day, because using functions related to distributions to derive pure mathematical functions is counter intuitive, and because two implementations already exist, I hope the R developers would be willing to incorporate an igamma() function directly into R. The implementation could either use the code posted to the list: http://www.r-project.org/nocvs/mail/r-help/1998/0295.html or be a convinience function around the pgamma and gamma solution suggested in the emails presented above. Alternatively, if this approach doesn't suit you, would someone please add an explanation to the documentation for the gamma function explaining how to generate an igamma(\alpha, \beta) using the current functions? Thanks to all, adrian __ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
