# Determining the type of data distribution

I am looking on expanding my introductory knowledge in stat, and I have come up with this challenge. I have an application that is designed to analyze and index news articles. The goal of the application is to count the number of occurrences of a certain word, and to show how many times this count occurs in a dataset. In a following bar chart, I have made one such analysis.

As I said, from the chart it is visible that there are 942 examples of a words that repeat only once. However, there is only 1 example of a word that is repeated 160 times.

So, my question is how can I, with a certain degree of probability, confirm that my word distribution is following a theoretically defined distribution? In order to answer to this question, I have used R's

fitdistrplus library

to plot a Cullen - Frey graph that could point me in the right direction of choosing a probable distribution. The code in R, to plot my y-axis values looks like this:

library(fitdistrplus)
library(logspline)

x <- c(942, 125, 54, 28, 10, 6, 11, 5, 9, 1, 5, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1,1)
descdist(x, discrete = TRUE, boot = 500)


This produces a following chart:

From the chart it can be deduced that the data is suggesting to be in the negative binomial distribution area. However, I am not sure how to confirm my claim with greater level of confidence, and I would like an opinion on the process done so far?

Revised data

After a comment by a fellow colleague, I have revised my data that is being import into R's descdist() function. The newly formulated data now accounts for the zero values that are obviously part of my count-data set.

The revised R code looks like this:

library(fitdistrplus)

xx <- 0:160
y <- c(0,942,125,54,28,10,6,11,5,0,9,1,5,3,0,0,1,2,0,0,1,0,0,0,2,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)
obs <- rep(xx,y)

# plot the histogram
hist(obs, freq=TRUE, breaks=1000)
# Describe with Cullen & Frey Graph
descdist(obs, discrete = TRUE, boot = 500)


It produces the following CF graph:

What I think is critical, and I have changed it in this revised addition is that now I am modeling with 2D data set, in which both of my X and Y axises are present.

What I would still like to find out, how to confirm my suspicion that the distribution is Negative Binomial? And if my R code valid now?

P.S. I have even fitted the distributions NB and Poisson distribution to get AIC score, which is the following:

fit.nbinom <- fitdist(obs, "nbinom")
fit.pois <- fitdist(obs, "pois")

plot(fit.nbinom)
summary(fit.nbinom)
fit.nbinom$$aic plot(fit.pois) summary(fit.pois) fit.pois$$aic


1 4707.648 (AIC for NB)

2 7378.281 (AIC for Poisson)

• Your data are not consistent with any unimodal distribution whatsoever. This can be proven by minimizing a chi-squared statistic over the set of all such distributions. Since it is extremely rare for "theoretically defined distributions" to be multimodal--I cannot think of any--this looks like a fruitless endeavor. Please note that the R code is invalid because it does not include all the many zero counts that are not shown in your plot!
– whuber
Nov 9, 2018 at 21:12
• Kudos for a nicely framed, instructive question. Nov 9, 2018 at 21:17
• Thank you for the kind comment. @whuber I see your point, but could it be that I have just inputed wrong data into R, because my Java analysis app needed it in such format to perform calculations and plotting. The original data and revised Cullen - Frey graph is now added in the post. Thanks again.
– H.G.
Nov 9, 2018 at 22:09
• @Tackler529, it may be possible to model this with a mixture such as a Hyperexponential. We can use moment matching. It won't be perfect but if it works for your application we could try (example matching 2 moments) Nov 10, 2018 at 4:22
• @SecretAgentMan thanks on the comment, but correct me if I am wrong, any kind of exponential distribution, including hyperexponential dist., presumes to have continuous variables in the dataset. Since I am counting words, in the finite space of one article, shouldn't that indicate that my variable is discrete? My logic comes from the conclusion that even if all the word were the same the final count of occurrences can not be greater than the total count of words in the article.
– H.G.
Nov 10, 2018 at 10:32