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

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];

[Jeff Miller]

• What do you want to do with this nonparametric density once you have it? For example, would it be enough just to determine its moments, or do you actually want to visualize the whole density?
• Can you identify lower and upper bounds on acc so that there are some constraints on the lower 3% and upper 8% of the distribution (i.e., outside your observed percentiles)? It doesn't seem like there can be a unique solution without such bounds.

[Star Strider]

@Jeff Miller — Geophysics is not an area of my expertise, however I would assume there is some background ‘noise’ that is impossible to completely remove, that being the lower limit, although that could arbitrarily be defined as zero. However as I understand it, the Richter scale is open-ended, so while there is a theoretical upper limmt (the highest acceleration the seismometer can record, at least before it is destroyed), there may be no actual upper limit.

[Jeff Miller]

@Star Strider Fair enough. I was asking about limits because I have a possible method to suggest, but it would only be workable if those limits were known (or stipulated).

[Torsten]

@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]

@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%.

Answer 1

[Star Strider]

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

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

x = 10x1

0.3339 0.3339 0.3595 0.4209 0.4283 0.4645 0.5010 0.5080 0.5713 0.6025

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

flo = 10x1

NaN 0 0.0152 0.0651 0.1314 0.2407 0.3845 0.5532 0.7562 NaN

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

fup = 10x1

NaN 0.0143 0.0476 0.1187 0.2004 0.3243 0.4764 0.6441 0.8313 NaN

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

.

[Xavier]

Thanks Star Strider, but I was hoping for an approach that would be able to yield a smooth CDF instead of a more stepwise function.

[Torsten]

Smooth would mean: you assume a certain known pdf and estimate its parameters or you cheat between the values.

[Star Strider]

@Xavier — My pleasure! To get anything smooth, you would need to fit a parametric distribution to your data.

@Torsten — Exactly!

[Xavier]

I get what you are saying but if I fit a nonparametric density to a data sample using ksdensity, I also get a smooth fit. What I was asking is for the equivalent of that to fit a sample of percentile data instead of a random sample.

Thanks

[Star Strider]

I cannot comment on your ksdensity call because you did not post your code.

The ksdensity function uses a kernel smoothing function, so it will produce a smooth curve.

Also, the default is to fit 'Function','pdf' so you would have to specify 'Function','cdf' for your data, as wellk as 'Support','positive'. I also do not see any provision to specifically use the percentile values. The pts argument can apparently be anything, so long as it is the same size as the first argument.

I am not certain that the ksdensity function will do what you want it to do, using the data you have. That is the reason that I did not use it in my answer.

[Xavier]

Indeed, the ksdensity function does not allow fitting the data in the way I want, as you say. That's why I was looking for help on this. Maybe someone is aware of an alternative approach.

[Torsten]

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]

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')

.

Answer 2

[Image Analyst]

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.

[Xavier]

Thanks for this idea, but fitting a spline does not ensure that the fitted function will comply with the necessary conditions for being a CDF.