How to fit a nonparametric distribution to a sample of known percentile values

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Xavier 2024-8-13

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评论： Jeff Miller 2024-8-14

Hello everyone

I have a sample of percentile values that describe the distribution of possible earthquake acceleration levels that lead to the failure of a building component. I would like to fit a nonparametric model to these data. I know that, for a random sample of these earthquake acceleration levels, I coiuld fit a nonparametric density using the the ksdensity function but is there a way to do a similar fit for the cumulative distribution function of this function?

Many thanks

Example data:

percentiles  = [3    11    27    33    52    66    75    87    92];
acc = [0.3339    0.3595    0.4209    0.4283    0.4645    0.5010    0.5080    0.5713    0.6025];

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Torsten 2024-8-14

@Jeff Miller

And the method doesn't allow to approximate only between acc(1) and acc(9) where 89 % of the mass is cumulated ?

Jeff Miller 2024-8-14

@Torsten Not completely. The smoothing would spill over at the edges, so for example the pdf at prctile 91 would depend a bit on what you assumed about the top 8%.

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回答（2 个）

Star Strider 2024-8-13

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在 MATLAB Online 中打开

The empirical cumulative distribution function ecdf would likely bea appropriate here. (There is also ecdf however it seems less applicable to me.) There are a number of associated functions as well, lilnked to in that documentation page.

percentiles = [3 11 27 33 52 66 75 87 92];

acc = [0.3339 0.3595 0.4209 0.4283 0.4645 0.5010 0.5080 0.5713 0.6025];

figure

ecdf(acc, 'Frequency',percentiles)

grid

axis('padded')

[f,x,flo,fup] = ecdf(acc, 'Frequency',percentiles)

f = 10x1

0 0.0067 0.0314 0.0919 0.1659 0.2825 0.4305 0.5987 0.7937 1.0000

x = 10x1

0.3339 0.3339 0.3595 0.4209 0.4283 0.4645 0.5010 0.5080 0.5713 0.6025

flo = 10x1

NaN 0 0.0152 0.0651 0.1314 0.2407 0.3845 0.5532 0.7562 NaN

fup = 10x1

NaN 0.0143 0.0476 0.1187 0.2004 0.3243 0.4764 0.6441 0.8313 NaN

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Torsten 2024-8-14

编辑：Torsten 2024-8-14

在 MATLAB Online 中打开

So you want a smooth version of

percentiles = [3 11 27 33 52 66 75 87 92];

acc = [0.3339 0.3595 0.4209 0.4283 0.4645 0.5010 0.5080 0.5713 0.6025];

plot(acc,percentiles/100)

to get an approximate cdf ? Maybe fit a sigmoid function ?

Star Strider 2024-8-14

在 MATLAB Online 中打开

The pdf plots might look something like this —

percentiles = [3 11 27 33 52 66 75 87 92];

acc = [0.3339 0.3595 0.4209 0.4283 0.4645 0.5010 0.5080 0.5713 0.6025];

[f,x,flo,fup] = ecdf(acc, 'Frequency',percentiles);

dfdx = gradient(f, x);

dpda = gradient(percentiles/100, acc);

figure

stairs(x, dfdx, 'DisplayName','From ‘ecdf’ Results')

hold on

stairs(acc, dpda, 'DisplayName','From Posted Vectors')

hold off

grid

xlabel('$x$', 'Interpreter','LaTeX')

ylabel('$\frac{dF(x)}{dx}$', 'Interpreter','LaTeX', 'FontSize',14)

legend('Location','best')

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Image Analyst 2024-8-14

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spline_demo.m

You could fit a spline through them. The spline doesn't take any parameters, it just fits a cubic equation between each pair of points. See attached demo.