Kernel function interpretation problem

Dear MatLab comunity,

I have a distribution that I fitted with the Kernel function. See below.

A1 = load('trajrmsd_I_1_1_c.txt');
A1_RMSD = A1(:,2);
figure
[f,xi] = ksdensity(A1_RMSD);
plot(xi,f,'color','r','linewidth',2);

The point is that in the y axis i get values higher than 1. Is that normal? How is it rationalized?

All the best,

Alessandro

Answer 1

[Aditya Patil]

As ksdensity returns the probability density, it can be higher than one. The integral of this function, which is the total probability, will be 1.

See the following code for example,

A1 = load('trajrmsd_I_1_1_c.txt');
A1_RMSD = A1(:,2);
[f,xi] = ksdensity(A1_RMSD);
plot(xi,f);

fitobj = fit(xi', f', 'linearinterp');
integralfun = @(x)(fun(x, fitobj));
integral(integralfun, 0, 2.5) % this gives integral over entire range

function y = fun(x, fitobj)
    y = feval(fitobj, x)';
    y = max(0, y);
end

To consider an analogous example, if you consider a rectangle with unit area, it's height can be made arbitrarily large by making the width smaller than 1 for e.g., height = 5, and width = 0.2. The area will however remain 1.