Description Usage Arguments Details Value Note Author(s) References See Also Examples
Transform from a simplex (either regular or right-angled) to a hypercube.
1 2 3 |
e |
Vector with all elements between 0 and 1 (hypercube) |
p |
Vector of positive elements whose sum is either equal to, or less than, one (simplex) |
Function e_to_p()
takes one from e
-space to
p
-space.
Function p_to_e()
takes one from p
-space to
e
-space. This is useful when integrating over a simplex; use
Jacobian()
to evaluate the Jacobian of the transform.
Forward transformation:
e_1=∑_{i=1}^d p_i
e_i=\frac{p_{i-1}}{∑_{j=i-1}^d p_j}, 2 <= i <= d
Backward transformation:
p_1=e_1 e_2
p_i=e_1 e_{i+1} ∏_{i=2}^{i}(1-e_i), 2 <= i <= d
Jacobian:
J=∏_{i=2}^{d-1}(1-e_i)^{d-i}
The functions documented here return a scalar.
To do a regular simplex, use the “di” of the right-angled simplex; see the examples.
Robin K. S. Hankin
M. Evans and T. Swartz 2000. Approximating Integrals via Monte Carlo and Deterministic Methods, Oxford University Press; page 28
Robin K. S. Hankin (2010). “A Generalization of the Dirichlet Distribution”, Journal of Statistical Software, 33(11), 1-18, http://www.jstatsoft.org/v33/i11/
1 2 3 4 5 6 7 8 9 10 11 | ## Not run:
# First, try to calculate the volume of a regular 4-simplex:
adapt(5,rep(0,5),rep(1,5),functn=function(x){Jacobian(c(1,x))})
# Should be close to 1/5! = 1/120 ~= 0.008333
# (that was the 'di trick')
# Now, try to calculate the volume of a triangular-based pyramid:
adapt(3,rep(0,3),rep(1,3),functn=Jacobian)
# Should be close to 1/8=0.125
## End(Not run)
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