General question, I've seen two descriptions of "logarithmic distribution". One is related to the frequency of digits called Benford's law (digit 1 occurs more frequently than 2, 2 than 3, etc) whose explanation is that it is the result of a mixture of distributions. The other description is a 2-page paragraph The logarithmic distribution in Kendall and Stuart (1977, The Advanced theory of statistics, Vol 1, 4th edition, pp 139-140), attributing the derivation to Fisher (1943). Are these concepts of logarithmic distribution the same or not?
Second question I would like to ask: Kendall and Stuart give an example of a distribution of the logarithmic type from Fisher (1943), "distribution of butterflies in Malaya, with theoretical frequencies given by the logarithmic distribution" No. of species Theoretical frequency Observed frequency 1 135.05 118 2 67.33 74 3 44.75 44 4 33.46 24 5 26.69 29 6 22.17 22 7 18.95 20 etc ... From what I've understood, the Theoretical frequency was generated by - ( q^r ) / ( r * ln(1-q) ) in which r is the No. of species, q is the probability of the presence of an attribute. How was, how can the fit be realized?
isn't the lognormal distribution the same as logarithmic? Just guessing. Else maybe you could look in the direction of exponential distributions. I am just guessing though... :) good luck! Edzo
"Vincent Vinh-Hung" <conrv...@az.vub.ac.be> wrote in message
> General question, > I've seen two descriptions of "logarithmic distribution". > One is related to the frequency of digits called Benford's law (digit 1 > occurs more frequently than 2, 2 than 3, etc) whose explanation is that > it is the result of a mixture of distributions. > The other description is a 2-page paragraph The logarithmic distribution > in Kendall and Stuart (1977, The Advanced theory of statistics, Vol 1, > 4th edition, pp 139-140), attributing the derivation to Fisher (1943). > Are these concepts of logarithmic distribution the same or not?
> Second question I would like to ask: Kendall and Stuart give an > example of a distribution of the logarithmic type from Fisher (1943), > "distribution of butterflies in Malaya, with theoretical frequencies > given by the logarithmic distribution" > No. of species Theoretical frequency Observed frequency > 1 135.05 118 > 2 67.33 74 > 3 44.75 44 > 4 33.46 24 > 5 26.69 29 > 6 22.17 22 > 7 18.95 20 > etc ... > From what I've understood, the Theoretical frequency was generated > by > - ( q^r ) / ( r * ln(1-q) ) > in which r is the No. of species, q is the probability of the presence > of an attribute. > How was, how can the fit be realized?
Lognormal I believe most often is used to describe a normal distribution after logarithm transform, while logarithmic distribution in the sense of Kendall-Stuart is else (I didn't really grasp KS' formalism).
BTW, I queried how the fit was done because I can't find the same figures as the Fisher 1943 example, assigning q=0.97293 I come with 135.05 (ok), 65.7 (instead of the published 67.33), 42.6 (instead of 44.75), 31.1 (instead of 33.46), etc.
> isn't the lognormal distribution the same as logarithmic? Just guessing. > Else maybe you could look in the direction of exponential distributions. > I am just guessing though... :) > good luck! > Edzo
> "Vincent Vinh-Hung" <conrv...@az.vub.ac.be> wrote in message > news:3909DA49.F17CA7AE@az.vub.ac.be... > > General question, > > I've seen two descriptions of "logarithmic distribution". > > One is related to the frequency of digits called Benford's law (digit 1
Vincent Vinh-Hung wrote: > General question, > I've seen two descriptions of "logarithmic distribution". > One is related to the frequency of digits called Benford's law (digit 1 > occurs more frequently than 2, 2 than 3, etc) whose explanation is that > it is the result of a mixture of distributions. > The other description is a 2-page paragraph The logarithmic distribution > in Kendall and Stuart (1977, The Advanced theory of statistics, Vol 1, > 4th edition, pp 139-140), attributing the derivation to Fisher (1943). > Are these concepts of logarithmic distribution the same or not?
> Second question I would like to ask: Kendall and Stuart give an > example of a distribution of the logarithmic type from Fisher (1943), > "distribution of butterflies in Malaya, with theoretical frequencies > given by the logarithmic distribution" > No. of species Theoretical frequency Observed frequency > 1 135.05 118 > 2 67.33 74 > 3 44.75 44 > 4 33.46 24 > 5 26.69 29 > 6 22.17 22 > 7 18.95 20 > etc ... > From what I've understood, the theoretical frequency was generated > by > - ( q^r ) / ( r * ln(1-q) ) > in which r is the No. of species, q is the probability of the presence > of an attribute. > How was, how can the fit be realized?
You will need a value of q first. This will either be estimated from the raw data or assumed by some hypothesis. Once you have this just plug in the value of r you want and multiply the resulting probability by the sum of the observed frequencies.
You might also be able to use the theorectical mean q/((q - 1 )*Log[1 - q]) to estimate q by equating it to the sample mean and solving for q.
> With thanks in advance, > Vincent Vinh-Hung
-- Dr Graeme Byrne La Trobe University, Bendigo PO Box 199, Bendigo, 3552 Phone: 61 3 5444 7263 Fax: 61 3 5444 7998 g.by...@bendigo.latrobe.edu.au
In answer to your second question, if N is the total number of butterflies and p(x) is the logarithmic probablity of x species, then the fitted frequency of x species is N*p(x).
Many thanks to Dr Byrne, the explicit expression of the mean hinted at the correct direction, the error I made was confusing natural logarithm and base 10 logarithm!
I apologize that I didn't post the complete example data, which follows:
> You will need a value of q first. This will either be estimated from the raw > data or assumed by some hypothesis. Once you have this just plug in the > value of r you want and multiply the resulting probability by the sum of > the observed frequencies.
> You might also be able to use the theorectical mean q/((q - 1 )*Log[1 - q]) > to estimate q by equating it to the sample mean and solving for q.
Thank you very much for the comment, I made the error of confusing neperian log and base 10 log when applying p(x), therefore couldn't find anything approaching the example results. Vincent Vinh-Hung Oncologisch Centrum, AZ-VUB B-1090 Jette
> In answer to your second question, if N is the total number of > butterflies and p(x) is the logarithmic probablity of x species, then > the fitted frequency of x species is N*p(x).
