# Questions tagged [power-law]

A power-law is a function that increases proportionally to a power of its argument (ax^b). Often seen in fitted relationships or in densities (power-law distributions).

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### Understanding the Log-Sum-Exp (LSE) Operator in Chinchilla Replication Study, is e a set or exp?

I'm trying to understand the use of the Log-Sum-Exp (LSE) operator in the Chinchilla replication study, specifically the one referenced in the paper "Chinchilla Scaling: A replication attempt&...
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### Confidence and prediciton intervals for power law fit

I would like to determine confidence intervals and prediction intervals for a noisy dataset that follows a power law distribution. I have a dataset that (to my eye) follows power law behavior in the ...
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### Distribution of videos watched, given people drop out from course

An online training course involves 100 videos, which people watch one per session, starting with 1 and finishing at 100. People drop out of the course, having completed N sessions, and watched videos ...
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### Are these data points decaying exponentially or as a power law?

0 I have a set of data points. The first coordinate is time and the second coordinate is energy. I am trying to figure out how the energy is decaying over time. Particularly, I have to find if it is ...
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### How to include a survival function in non-linear mixed regression

I have a dataset consisting of time periods, at the end of each one the individual either develops a disease, or doesn't and is right-censored. I suspect that the rate of developing the disease is ...
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### Fitting a power law model with an additional linear term

A model of the form $y=a\cdot x^b$ can be linearly fitted by taking logs on both sides - giving $\ln(y)=\ln(a)+b\cdot\ln(x)$, where $\ln(y)$ is regressed against $\ln(x)$. This is a standard textbook ...
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### Is there a relationship between the alpha of fat tails and kurtosis

From Wikipedia I have the compliment of the CDF parameterized for fat-tails distributions. $$\Pr[X>x] \sim x^{- \alpha}\text{ as }x \to \infty,\qquad \alpha > 0.\$$ In forecasting literature a ...
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### Difference between Power law distribution and Exponential distribution?

What is the difference between Power law distribution and Exponential distribution? They both look similar!
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### Cumulative distribution function in powerlaw distribution

I am trying to fit the power law distribution to continuous sample (x) which has a sample size n = 360 I'm using the methodology proposed by Aaron Clauset in the python package power-law: A Python ...
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### Ranking log-linear distributions and the Lucas numbers

The wikipedia page on rank-size distributions claims: "When any log-linear factor is ranked, the ranks follow the Lucas numbers, which consist of the sequentially additive numbers 1, 3, 4, 7, 11,...
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### Convergence of Percentile in Power Law

I have a probability distribution, that in its tail follows a power law. I've noticed, while I was simulating samples, and determining parameters experimentally, that as I increase the value of a ...
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### Maximum likelihood estimator for power law with negative exponent

Background I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration): Now I ...
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### Correct GLM or NLS to model exponential model with response variable with positive and negative values

I have been struggling to find the right way to model this dataset, this is a Data Frame with the dataset: ...
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### What are the theoretical assumptions of a power curve versus an exponential or quadratic one?

I have a set of bivariate biological data that has a clearly non-linear distribution. I have found that a power curve best fits the data under several metrics (e.g., AIC, residuals versus fits plot, ...
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### Do stock price changes follow Pareto Distribution? [closed]

I calculated the distribution of the stock price daily ratios. The ratios are multiplicative, $d_t=p_{t} / p_{t-1}$. As far as I know the distribution should look like Power law distribution (Pareto ...
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### How do I calculate a proper sample size to get a suitable power of Kolmogorov-Smirnov test with an underlying Zipf distribution?

I have been trying to develop to calculate the sample size to maximize the power of KS test (±0.8) on an underlying Zipf distribution. I have tried estimating the power by performing simulations: <...
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### Using AIC vs Likelihood Ratio test for comparing Lognormal and Powerlaw distributions

I am interested in comparing whether a lognormal or a power law are a better fit for a given set of data. Both distributions have been fit using MLE, with $x_{min}$ determined using KS-minimization a ...
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### Reading on "Economic nowcasting" for a probabilist

I am about to attend an interview for a postdoc in "economic nowcasting". I am work mostly in probability, particularly complex networks and random graphs. I am looking for some good reading,...
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### Why is a Fréchet distribution slowly varying, and what is the intuition behind it?

The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0 & & x\leq 0,\\[6pt]e^{-x^{-\alpha}} & & x>0,\end{cases}$$ is regularly varying as stated here (page 19): It is not ...
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### How can this CDF be decreasing? (power-law)

I am reading the first vignette of the powerRlaw R package. It uses the moby dataset, which is an array of word frequency. Each element is a word, and the value if ...
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### Difference Between SciPy.Stats.powerlaw and powerlaw package

I am trying to simulate random variables that are power law distributed based on my understanding of the definition in this Wikipedia article and several other resources where the consensus is that a &...
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### Sample size determination for power law

I am looking for a method to estimate a mean when the underlying samples are drawn from a skewed distribution. Concretely, I have a population $P$ whose pdf largely follows a power law, where only a ...
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### What is the mathematical meaning of when two variables retain a non-linear relationship even after log transformation?

I am working with a set of biological data drawn from a sample of species, specifically a skeletal measurement and body size, with the intent of calculating a regression line to try to predict the ...
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### Is Prices law a consequence of Benford's law?

Background Price's law says this: In an company, half of all value is created by the square root of the people. I'm sure Dr. Price looked at more than the median, but I suspect this was how the ...
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### Fit a function whose asymptotics is known

Consider the experimental blue curve in the figure that follows It turns out that I know the expression for the tails (-they are power laws- red and green curves) and I want to fit the curve using ...
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### Help with Power Curve MCMC

I'm trying to analyze some $D$ non-logistic cumulative data in a time series, bounded below by 0 and unbounded above. Splitting data into $W$ time windows of $d$ days, I know each window can be ...
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### Sampling distribution of the mean of the discrete-power law distribution

For a certain problem I wish to generate random integers $k$ so that their distribution follows $p_k \sim k^{-\alpha}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$. I am following the ...
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### In the poweRlaw package, is the location parameter estimated for a lognormal distribution the $median=\exp(\mu)=\theta$ or $\mu$?

I have a set of graphs (each with the same nodes but with edges' weights defined by different research subjects) for which I would like to report statistics. One of these statistics is the betweenness ...
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### Why do poweRlaw and fitdistrplus differ in fitted lognorm parameters

I am trying to evaluate whether a power-law fit is appropriate for some data of lake areas that we have, and whether the theoretically supported alternative of the log-normal, at least at the tails, ...
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### Appropriate Data Analysis when Criterion has Heavy-Tailed Distribution

I have a data set where my independent variables (i.e., personality assessment scores) are continuous and follow a normal distribution. The criterion, sales performance, is heavy-tailed and follows a ...
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### Is there a named distribution with the property $P(X>10^k) = p^k$?

If I'm doing my math correctly, the exponential distribution has the property $P(X >k) = p ^ k$ (with $p$ conventionally written as $e ^ {-\lambda}$). I'm wondering if there is a different ...
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### Why Are Bell Curves Used for Performance Appraisals?

Why do organizations that stack-rank employees force-fit employee rankings to a bell curve? Is there any theoretical or emperical evidence to support a normal or near-normal distribution of ...
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### Cut off long tail of distribution (no matter how long is the tail) [closed]

One of the algorithms that we are working on is designed to report the time it takes for people to finish the onboarding process in an app. We need an algorithm to eliminate the long tail of a ...
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### How to perform the Kolmogorov-Smirnov test on the truncated power-law distributed data?

Recently, I read papers that perform power-law fitting on their empirical data (estimate the alpha), some of them report corresponding p-value for the Kolmogorov-Smirnov test, but many of them do not. ...
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### Powerlaw Distribution and old nodes [closed]

I am working on the size of discussions (number of replies, replies to the replies) on Twitter data, and I am observing Power-law distribution for small and medium-size discussion but it is not valid ...
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### Rank-size log-log plot inflection: drooping tail of power law

Most of my data seems nicely to fit a power law but with a "drooping tail", which I believe is quite common, although in this case the drop-off is quite steep. I have two related questions if I may ...
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### How to Select the Largest Y Values for {X,Y} Pairs, for a Pareto-Distributed Dataset, for a Meaningful Fit?

First, apologies for the inelegant question. Second, on to the question: Background Information: I study impact craters, and the size-frequency distribution (number vs diameter) of impact craters ...
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### Random Samples from Spliced Distribution

I am studying Clauset, Shalizi, and Newman, Power Law Distributions in Empirical Data (preprint available here) in R. Packages used: ...
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### How can I calculate the exponent (parameter or tail index) of the Pareto distribution associated to the following paperclips experiment?

The experiment is the following: take a pool of paperclips for which in each round we follow these steps: 1) choose two paperclips randomly 2) if they are not linked with each other then link them ...
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I want to fit a multiple power law equation of the form $y = {x_1}^{\alpha_1} {x_2}^{\alpha_2}$ where I have many examples of $y, x_1, x_2$. (Note there is no intercept.) Is it possible for me to ...