# Estimating the parameter of a power law distribution using maximum likelihood estimation (MLE)

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Kashif Naukhez il 3 Lug 2023
Commentato: Kashif Naukhez il 3 Lug 2023
I have a set of data points. I want to fit a power law distibution to these data points using the relation:
P(E) = CE^-m
where,
E = Set of data points
P(E) is the probability density function of E
C = a constant
m = power law exponent
I need to determine "C" and "m" using MLE and overlap the power law curve over P(E) versus E plot.
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### Risposte (2)

Animesh il 3 Lug 2023
To fit a power law distribution to your data points using Maximum Likelihood Estimation (MLE) in MATLAB, you can follow these steps:
1.Define the power law function powerLaw with parameters C and m:
powerLaw = @(C, m, E) C * E.^(-m);
2.Create a probability density function (PDF) of your data points E using the power law function:
pdf = @(C, m) powerLaw(C, m, E);
3.Define the negative log-likelihood function negLogLikelihood to be minimized:
negLogLikelihood = @(params) -sum(log(pdf(params(1), params(2))));
4.Use the fminsearch function to find the parameters C and m that minimize the negative log-likelihood:
initialGuess = [1, 1]; % Initial guess for C and m
params = fminsearch(negLogLikelihood, initialGuess);
C = params(1);
m = params(2);
5.Plot the data points E and the fitted power law curve:
scatter(E, P, 'b', 'filled');
hold on;
x = linspace(min(E), max(E), 100);
plot(x, powerLaw(C, m, x), 'r', 'LineWidth', 2);
xlabel('E');
ylabel('P(E)');
legend('Data', 'Power Law Fit');
Make sure to replace E and P with your actual data points and their corresponding probabilities.
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Kashif Naukhez il 3 Lug 2023
I am having trouble creating the functions as I am new to MATLAB and getting some errors while running the code. Can u send the full code, if possible

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Shantanu Dixit il 3 Lug 2023
Modificato: Shantanu Dixit il 3 Lug 2023
Hi Kashif,
You can use fminsearch - MATLAB and obtain the C and m for the power law fit. See the below snippet for minimizing the negative log likelihood using fminsearch.
negLogLikelihood = @(params) -sum(log(params(1) * data.^(-params(2))));
initialGuess = [0.5,0.5]; % initial guess for c and m
options = optimset('MaxFunEvals', 1000);
params = fminsearch(negLogLikelihood, initialGuess, options);
C = params(1);
m = params(2);
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