Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

This example shows how to use spline commands from Curve Fitting Toolbox™ to smooth a histogram.

Here is a histogram of some random values that might represent data that were collected on some measurement.

y = randn(1,5001); hist(y);

We would like to derive from this histogram a smoother approximation to the underlying distribution. We do this by constructing a spline function `f`

whose average value over each bar interval equals the height of that bar.

If `h`

is the height of one of these bars, and its left and right edges are at `L`

and `R`

, then we want the spline `f`

to satisfy

` integral {f(x) : L < x < R}/(R - L) = h,`

or, with `F`

the indefinite integral of `f`

, i.e., `DF = f`

,

` F(R) - F(L) = h*(R - L).`

[heights,centers] = hist(y); hold on ax = gca; ax.XTickLabel = []; n = length(centers); w = centers(2)-centers(1); t = linspace(centers(1)-w/2,centers(end)+w/2,n+1); p = fix(n/2); fill(t([p p p+1 p+1]),[0 heights([p p]),0],'w') plot(centers([p p]),[0 heights(p)],'r:') h = text(centers(p)-.2,heights(p)/2,' h'); dep = -70; tL = text(t(p),dep,'L'); tR = text(t(p+1),dep,'R'); hold off

So, with `n`

the number of bars, `t(i)`

the left edge of the `i`

-th bar, `dt(i)`

its width, and `h(i)`

its height, we want

` F(t(i+1)) - F(t(i)) = h(i) * dt(i), for i = 1:n,`

or, setting arbitrarily `F(t(1))`

= 0,

` F(t(i)) = sum {h(j)*dt(j) : j=1:i-1}, for i = 1:n+1.`

dt = diff(t); Fvals = cumsum([0,heights.*dt]);

Add to this the two end conditions `DF(t(1)) = 0 = DF(t(n+1))`

, and we have all the data we need to get `F`

as a complete cubic spline interpolant.

F = spline(t, [0, Fvals, 0]);

The two extra zero values in the second argument indicate the zero endslope conditions.

Finally, the derivative, `f = DF`

, of the spline `F`

is the smoothed version of the histogram.

DF = fnder(F); % computes its first derivative h.String = 'h(i)'; tL.String = 't(i)'; tR.String = 't(i+1)'; hold on fnplt(DF, 'r', 2) hold off ylims = ylim; ylim([0,ylims(2)]);

Was this topic helpful?