Calculate nodes and weights for Gaussian quadrature in terms of probability distributions.
gauss.quad.prob(n,dist="uniform",l=0,u=1,mu=0,sigma=1,alpha=1,beta=1)number of nodes and weights
distribution that Gaussian quadrature is based on, one of "uniform", "normal", "beta" or "gamma"
lower limit of uniform distribution
upper limit of uniform distribution
mean of normal distribution
standard deviation of normal distribution
positive shape parameter for gamma distribution or first shape parameter for beta distribution
positive scale parameter for gamma distribution or second shape parameter for beta distribution
A list containing the components
vector of values at which to evaluate the function
vector of weights to give the function values
This is a rewriting and simplification of gauss.quad in terms of probability distributions.
The probability interpretation is explained by Smyth (1998).
For details on the underlying quadrature rules, see gauss.quad.
The expected value of f(X) is approximated by sum(w*f(x)) where x is the vector of nodes and w is the vector of weights. The approximation is exact if f(x) is a polynomial of order no more than 2n-1.
The possible choices for the distribution of X are as follows:
Uniform on (l,u).
Normal with mean mu and standard deviation sigma.
Beta with density x^(alpha-1)*(1-x)^(beta-1)/B(alpha,beta) on (0,1).
Gamma with density x^(alpha-1)*exp(-x/beta)/beta^alpha/gamma(alpha).
Smyth, G. K. (1998). Polynomial approximation. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3425-3429. http://www.statsci.org/smyth/pubs/PolyApprox-Preprint.pdf
# NOT RUN {
# the 4th moment of the standard normal is 3
out <- gauss.quad.prob(10,"normal")
sum(out$weights * out$nodes^4)
# the expected value of log(X) where X is gamma is digamma(alpha)
out <- gauss.quad.prob(32,"gamma",alpha=5)
sum(out$weights * log(out$nodes))
# }
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