bandwidth             package:SpherWave             R Documentation

_C_a_l_c_u_l_a_t_i_o_n _o_f _B_a_n_d_w_i_d_t_h _P_a_r_a_m_e_t_e_r

_D_e_s_c_r_i_p_t_i_o_n:

     Given the number of observations at a level, this function
     calculates bandwidth parameter of spherical basis function (SBF).

_U_s_a_g_e:

     bandwidth(n) 

_A_r_g_u_m_e_n_t_s:

       n: the number of observations at a level

_D_e_t_a_i_l_s:

     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=((1 - h^2)/(1 + h^2))^2,  the
     surface covered are is 2 pi (1 - sqrt{1 - ((1 - h^2)/(1 +
     h^2))^2}).  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:

 # of observations = n = 2/(1 - sqrt{1 - ((1 - h^2)/(1 + h^2))^2}) and h = sqrt{(1 - a_n)/(1 + a_n)}

     where a_n = sqrt{1 - (1 - 2/n)^2}.

_V_a_l_u_e:

       h: bandwidth parameter at a level

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

     'eta.comp'

_E_x_a_m_p_l_e_s:

     bandwidth(20)

