Numerical validation

We now compare the methods of Sections 16.7.1 and 16.7.2 for a special case where $ p({\bf z})$ can be computed exactly. Consider the trivial case where $ {\bf A}=[1, \; 1, \ldots 1]^\prime$. In this case $ {\bf z}={\bf A}^\prime {\bf x}$ has the Irwin-Hall distribution

$\displaystyle p(z) = \frac{1}{2(N-1)!} \sum_{k=0}^N(-1)^k
\binom Nk (z-k)^{N-1} {\rm sign}(z-k),$

where $ {\rm sign}(0)=0.$ In Figure 16.1, we show the $ \log p({\bf z})$ error (compared with exact Irwin-Hall distribution) on the Y-axis for both methods for $ N=10$ and $ N=40$. The error is very small, less that .02 at $ N=10$ and less than .004 at $ N=40$. This error for a 40-dimensional PDF is very small and for all practical purposes, can be ignored. Notice that the error becomes smaller with increasing $ N$. For even larger $ N$, however, the round-off error in computing Irwin-Hall eventually dominates. Although almost always the same, the ML approach had slightly more error and the SPA was faster. Therefore, SPA is our method of choice.
Figure: Numerical comparison of SPA (dots) and ML (circles) with Irwin-Hall distribution (X-axis) for $ N=10$ and $ N=40$.
\includegraphics[width=5.4in]{irwinhall.eps}

Baggenstoss 2017-05-19