Given the number of observations at a level, this function calculates bandwidth parameter of spherical basis function (SBF).
Usage
bandwidth(n)
Arguments
n
the number of observations at a level
Value
h
bandwidth parameter at a level
Details
This function is used for obtaining the bandwidth of the coarsest network level $L$, $h_L$.
From geometry, the surface area covered by surface mass distribution with variance $\sigma^2$ over
unit sphere $\Omega$ is $2 \pi (1 - \sqrt{1 - \sigma^2})$.
Since the total surface area of the unit sphere is $4 \pi$ and
the variance of SBF induced from Poisson kernel is
$\sigma^2=\left((1 - h^2)/(1 + h^2)\right)^2$,
the surface covered are is $2 \pi \left(1 - \sqrt{1 - ((1 - h^2)/(1 + h^2))^2}\right)$.
Under the assumption that the observations are
distributed equally over the sphere, it can be easily known how many observation are needed in order to cover
the whole sphere with fixed $h$, and how large the $h$ is needed to cover the sphere when the number
of observations are fixed as follows:
$$
\# \mbox{ of observations}=n=\frac{2}{1-\sqrt{1-(\frac{1-h^2}{1+h^2})^2}} \mbox{ and } h=\sqrt{\frac{1-a_n}{1+a_n}}
$$
where $a_n = \sqrt{1 - (1 - 2/n)^2}$.
References
Oh, H-S. (1999) Spherical wavelets and their statistical analysis with applications to meteorological data. Ph.D. Thesis,
Department of Statistics, Texas A\&M University, College Station.
