14
votes
Accepted
How can this CDF be decreasing? (power-law)
The vignette by Gillespie (2020) cites Clauset et al (2009) as another source of the figure. Reading that latter paper, it looks like they are actually plotting the complementary CDF (better known as ...
12
votes
Accepted
What is the discrete equivalent of the powerlaw distribution?
If you want a discrete distribution where $P(X=x)\propto x^{-\alpha}$ for $x=1,2,3,...$ then you would have a zeta distribution.
If you want a discrete distribution where $P(X=x)\propto x^{-\alpha}$ ...
10
votes
Accepted
"Law of large numbers" for distribution with infinite variance?
Not only the (weak and strong) laws of large numbers hold for averages
of iid random variables $X_i$ with no assumption on the existence of a
variance, but the weak [and only the weak] law of ...
8
votes
Accepted
Is KS test really appropriate when validating a power law/estimating power law parameters?
The issue raised in the Geller paper is not about the ranking of the data, but rather that the Kolmogorov-Smirnov test has different critical values when the parameters of the distribution you are ...
7
votes
Random Sample from Power Law Distribution
This is an answer to the question 3: how to sample from a power-law distribution. The answer is based on the article pointed by @Sycorax: Power-Law Distributions in Empirical Data by Clauset et al. ...
6
votes
Regression for power law
A paper by Lin and Tegmark nicely summarizes the reasons why lognormal and/or markov process distributions fail to fit data displaying critical, power law behaviors... https://ai2-s2-pdfs.s3.amazonaws....
6
votes
Regression for power law
If you want equal error-variance on every observation in the untransformed scale, you can use nonlinear least squares.
(This will often not be suitable; errors over many orders of magnitude are ...
6
votes
Characterizing/Fitting Word Count Data into Zipf / Power Law / LogNormal
The distribution of word frequencies is often characterized by Zipf's law, which states that it has Pareto distribution $p(k) \sim k^{-s}$, so-called power law.
This power law can be well seen as a ...
5
votes
Accepted
Extracting power of a power law from data
As in my first comment on the question I see this as being entirely about power laws for bivariate data. (The inclination to read it otherwise is puzzling.)
Based on the posted data, I did local ...
5
votes
Accepted
Besides the Pareto and Zipfian distributions, which distributions obey the power-law?
That's basically the complete list in your question, (the Pareto and the zeta/Zipf).
A power law is one where the pdf/pmf is proportional to $x^{-p}\,$ ($1$).
People use power laws for either ...
5
votes
Fitting a Pareto distribution to two dimensional data in R
You need to expand you data frame to generate the raw data (silly I know, on my list of things to fix)
...
5
votes
Accepted
Student's t as a power law distribution
$$\dfrac{(1 + x^2 / n)^{-(n+1)/2}}{x^{-(n+1)}} = \left(\dfrac{1}{x^2} +\dfrac1n \right)^{-(n+1)/2} \to \left(\dfrac1n \right)^{-(n+1)/2}$$ as $x\to \infty$ while $n$ remains fixed, i.e. the limit of ...
4
votes
Accepted
Understanding power laws and log plots (Newman paper)
I can explain some of the differences.
His first plot is a histogram density-estimate, not a count-histogram, so the y-axis will be scaled so that the total area under the curve is 1
For later plots ...
4
votes
Extracting power of a power law from data
Edit: The question is actually about data in which $x$ and $y$ are allowed to vary. This answer is not relevant to that case, but only to the case when $x$ is a quantity and you want to fit a ...
4
votes
Is KS test really appropriate when validating a power law/estimating power law parameters?
TL;DR
However, in an old paper on the Whitworth distribution by Nancy Geller, she mentions that once observations are ranked, they are no longer independently and identically distributed and ...
4
votes
Accepted
Discrete Pareto Distribution vs Zipf Distribution and Power Law vs Zipf Law
[In relation to the relationship between the Zipf and the zeta distributions, the Wikipedia definitions absolutely address your main question. It's possible that you didn't understand what was there.]
...
4
votes
Accepted
Definition of heavy-tailed distribution
For a fixed $\lambda>0$, we can rewrite the product as a fraction, so we can see the complementary cdf (right tail) of $Y_\lambda\sim Exp(\lambda)$:
$$ P(Y_\lambda >x)=e^{-\lambda x}. $$
We ...
4
votes
Accepted
Piecewise continuous power law distribution sampling
You can see this distribution as a mixture of two power law distributions,
$$f(x)=\alpha f_1(x) + (1-\alpha) f_2(x)$$
with
$$f_1(x)=\frac{𝑥^{−a_1}\mathbb I_{x_1\le x\le \tilde x}}
{\underbrace{\int_{...
3
votes
Accepted
Fitting a power law to the relationship between two variables
Here's the unflipped version of your plot:
No choices of $a$ and $b$ in $y=ax^{-b}$ (where $y$ is adult density and $x$ is algae concentration) will fit that shape of relationship; they don't ...
3
votes
"Law of large numbers" for distribution with infinite variance?
The strong law of large numbers requires only that the random variables have finite mean $\mu$ for the sample average to converge almost surely to $\mu$. There is no requirement that the variance be ...
3
votes
Estimate power law exponent for node degree distribution in scale free networks
Not sure if it is still of interest, however, you are estimating the exponent of the CDF of the degree which is equal to $-\alpha+1$, where $\alpha$ is 3 for the Barabasi Albert model, therefore, you ...
3
votes
Accepted
What can we infer about the probability distribution $P(X \ge x) = {x^{- E(x)}}$
Consider the expression
$$P(X \ge k) = {k^{- \beta}},\;\; \beta>0,\;\; k \in \{1,2,3,...\}$$
We have
$$P(X < k) = 1- k^{- \beta}$$
Then
$$P(X=1) = P(X<2) = 1- 2^{- \beta}$$
$$P(X=2) = P(...
3
votes
Sampling distribution of the mean of the discrete-power law distribution
Your distribution $p_k \sim k^{-\alpha-1}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$ is a truncated zeta distribution.
The distribution has no finite variance for $\alpha<2$ and the ...
3
votes
Accepted
Sampling distribution of the mean of the discrete-power law distribution
The distribution you are dealing with is a truncated zeta distribution, with mass function given by:
$$p_K(k) = \frac{k^{-\alpha}}{\zeta (\alpha,k_\min)}
\quad \quad \quad \text{for all integers } k \...
3
votes
Fit a function whose asymptotics is known
A rather good fit is obtained with the function below. The numerical values of parameters are shown on the figure where the red curve represents the fitted function.
This leads to asymptotics : $\...
3
votes
What is the mathematical meaning of when two variables retain a non-linear relationship even after log transformation?
You might have a case with a sum of components
$$y = a_1x^{n_1} + a_2x^{n_2} + ...$$
See below how it looks like in a log-log plot. It is a curve that bends at some point and goes from one straight ...
2
votes
Statistical test for power law samples
I do not believe there is a simple answer to your question without more details about the distribution. Power law distributions can have infinite variance, in which cases the large sample guarantees ...
2
votes
Interpreting the difference between lognormal and power law distribution (network degree distribution)
Check out this 2019 article: https://www.nature.com/articles/s41467-019-08746-5
Contrary to the claims of much of network science, applying robust statistical tools to nearly 1000 social, biological, ...
2
votes
Intuition behind power law distribution
One interesting property of the power-law distribution comes from looking at it on a log-scale. If we have $X \sim \text{Power}(x_\min, \alpha)$ then the logarithmic transformation $Y = \ln(x/x_\min) ...
2
votes
Accepted
Why is the slope not back transformed in a regression equation for allometric relationships
You have probably found estimates for $p$ and $q$ of the following model:
$$\log_{10}Y = p +q\log_{10}X$$
This is equivalent to
$$10^{\log_{10}Y} = 10^{p +q\log_{10}X} =10^{p +\log_{10}X^q} $$
$$\...
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