Finding an initial estimate of the intrinsic volumes

init_ivols find an initial estimate of the intrinsic volumes via moment-fitting.

init_ivols(d, init_mode = 0, delta = d/2, var = d/4)

Arguments

d

the dimension of the bivariate chi-bar squared distribution.
init_mode

specifies the way through which the initial estimate is found:

  • init_mode==0: uniform distribution

  • init_mode==1: discretized normal distribution

  • init_mode==2: circular cone fitting statdim

  • init_mode==3: circular cone fitting variance

  • init_mode==4: circular cone fitting statdim and variance (geometric mean)
delta

an estimate of the statistical dimension of the cone.
var

an estimate of the variane of the intrinsic volumes.

Value

The output of init_ivols is a (d+1)-column vector.

See also

rbichibarsq, circ_rbichibarsq, rbichibarsq_polyh, loglike_ivols

Package: conivol

Examples

D <- c(5,5)
d <- sum(D)
alpha <- c(pi/3,pi/4)
v_exact <- circ_ivols(D, alpha, product=TRUE)
m_samp <- rbichibarsq(10^6,v_exact)
est <- estim_statdim_var(d, m_samp)

list( v_exact = v_exact , v_init0 = init_ivols( d ) ,
      v_init1 = init_ivols( d, 1, delta=est$delta, var=est$var ) ,
      v_init2 = init_ivols( d, 2, delta=est$delta ) ,
      v_init3 = init_ivols( d, 3, delta=est$delta, var=est$var ) ,
      v_init4 = init_ivols( d, 4, delta=est$delta, var=est$var ) )
#> $v_exact
#>  [1] 0.0007466705 0.0059021152 0.0238862411 0.0650168700 0.1304771451
#>  [6] 0.1959118384 0.2206475000 0.1866928441 0.1151708399 0.0464763322
#> [11] 0.0090716034
#> 
#> $v_init0
#>  [1] 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909
#>  [7] 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909
#> 
#> $v_init1
#>  [1] 0.001027626 0.005615565 0.022273421 0.064145431 0.134169050 0.203861994
#>  [7] 0.225045217 0.180495106 0.105170839 0.044513592 0.013682161
#> 
#> $v_init2
#>  [1] 0.003195552 0.015418752 0.042267282 0.085501810 0.136724769 0.177800371
#>  [7] 0.189545420 0.164326460 0.112616650 0.056952606 0.015650327
#> 
#> $v_init3
#>  [1] 0.0003151408 0.0021400851 0.0084990689 0.0249073521 0.0577011768
#>  [6] 0.1087064603 0.1678885992 0.2108631213 0.2093538516 0.1533829812
#> [11] 0.0562421628
#> 
#> $v_init4
#>  [1] 0.001066540 0.006105089 0.020143723 0.049045970 0.094399095 0.147756265
#>  [7] 0.189591624 0.197836291 0.163190050 0.099333854 0.031531499
#>