# How do I generate a pdf from some known percentile values

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Christopher Stokes on 12 Oct 2021
Commented: Star Strider on 13 Oct 2021
Hi Matlab community,
I have percentile values that describe a distribution of possible sea level rise magnitudes. I would like to be able to generate a probability density function that closely approximates the actual distribution from which the percentile values were generated. Can anyone suggest how to achieve this please?
Example data:
prctiles = [5 10 30 33 50 67 70 90 95];
SLR = [3.2760 3.5265 4.1286 4.2013 4.5566 4.9151 4.9836 5.6045 5.9105];

Star Strider on 12 Oct 2021
A slightly different approach —
prctiles = [5 10 30 33 50 67 70 90 95];
SLR = [3.2760 3.5265 4.1286 4.2013 4.5566 4.9151 4.9836 5.6045 5.9105];
B = fminsearch(@(b) norm(prctiles/100 - cdf('Normal',SLR,b(1),b(2))), [SLR(prctiles==50);rand])
B = 2×1
4.5589 0.8095
SLRv = linspace(min(SLR), max(SLR));
yfit = cdf('Normal', SLRv, B(1), B(2));
figure
plot(SLR, prctiles/100, 'p')
hold on
plot(SLRv, yfit, '-r')
hold off
grid
title(sprintf('\$p = N(%.2f, %.3f)\$',B), 'Interpreter','latex')
legend('SLR','Fitted Noprmal Distribution', 'Location','NW') Experiment to get different results.
.
Star Strider on 13 Oct 2021
As always, my pleasure!
It was an education for me as well!
.

Image Analyst on 12 Oct 2021
Have you seen fitdist() in the Stats toolbox?
Of course you'd be better off with much more data.
Image Analyst on 12 Oct 2021
Here's an example:
clc; % Clear the command window.
fprintf('Beginning to run %s.m ...\n', mfilename);
close all; % Close all figures (except those of imtool.)
clear; % Erase all existing variables. Or clearvars if you want.
workspace; % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 17;
SLR = [3.2760 3.5265 4.1286 4.2013 4.5566 4.9151 4.9836 5.6045 5.9105];
% Plot data.
subplot(2, 1, 1);
bar(SLR)
grid on;
title('Original SLR Data', 'FontSize', fontSize);
xlabel('Index', 'FontSize', fontSize);
ylabel('SLR Value', 'FontSize', fontSize);
% Get distribution.
d = fitdist(SLR(:), 'Normal')
% Make curve, plot distribution.
% https://en.wikipedia.org/wiki/Normal_distribution
x = linspace(min(SLR), max(SLR), 1000);
amp = 1 / (d.sigma * sqrt(2*pi));
y = amp * exp(-(1/2) * ((x - d.mu) / d.sigma) .^ 2)
subplot(2, 1, 2);
plot(x, y, 'b-', 'LineWidth', 2);
grid on;
title('Estimated Distribution of SLR', 'FontSize', fontSize);
xlabel('SLR', 'FontSize', fontSize);
ylabel('PDF', 'FontSize', fontSize); Pick the distribution that fits the theory of what the distribution should actually be. Hopefully you know this in advance. Actually you need to if you're going to model it. Otherwise just normalize your histogram and that is the actual PDF.

Christopher Stokes on 12 Oct 2021
Hi, I have been using that function quite a bit recently and the excellent wrapper function allfitdist() from the file exchange. My problem is that they rely on relatively large data samples from which to build a PDF when what I have is a small number of summary statistics that describe the distribution (i.e. not data samples). I imagine there is a way to use fitdist to do what I need to do, but I can't envisage how this would work.

R2021b

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