If one picks two real numbers from the interval (independent, uniformly distributed), their sum has the triangular distribution.
The sum of three such numbers has a differentiable probability density function:
And the density of is smoother still: the p.d.f. has two
As the number of summands increases, these distributions converge to normal if they are translated and scaled properly. But I am not going to do that. Let’s keep the number of summands to four at most.
The p.d.f. of is a piecewise polynomial of degree . Indeed, for the density is piecewise constant, and the formula
provides the inductive step.
For each , the translated copies of function form a partition of unity:
The integral recurrence relation gives an easy proof of this:
And here is the picture for the quadratic case:
A partition of unity can be used to approximate functions by piecewise polynomials: just multiply each partition element by the value of the function at the center of the corresponding interval, and add the results.
Doing this with amounts to piecewise linear interpolation: the original function is in blue, the weighted sum of hat functions in red.
With we get a smooth curve.
Unlike interpolating splines, this curve does not attempt to pass through the given points exactly. However, it has several advantages over interpolating splines:
- Is easier to calculate; no linear system to solve;
- Yields positive function for positive data;
- Yields monotone function for monotone data