splaplace: Spherical Laplace Distribution

In Riemann: Learning with Data on Riemannian Manifolds

Spherical Laplace Distribution

Description

This is a collection of tools for learning with spherical Laplace (SL) distribution on a (p-1)-dimensional sphere in \mathbf{R}^p including sampling, density evaluation, and maximum likelihood estimation of the parameters. The SL distribution is characterized by the following density function,

f_{SL}(x; μ, σ) = \frac{1}{C(σ)} \exp ≤ft( -\frac{d(x,μ)}{σ} \right)

for location and scale parameters μ and σ respectively.

Usage

dsplaplace(data, mu, sigma, log = FALSE)

rsplaplace(n, mu, sigma)

mle.splaplace(data, method = c("DE", "Optimize", "Newton"), ...)

Arguments

data	data vectors in form of either an (n\times p) matrix or a length-n list. See wrap.sphere for descriptions on supported input types.
mu	a length-p unit-norm vector of location.
sigma	a scale parameter that is positive.
log	a logical; TRUE to return log-density, FALSE for densities without logarithm applied.
n	the number of samples to be generated.
method	an algorithm name for concentration parameter estimation. It should be one of "Newton", "Optimize", and "DE" (case-sensitive).
...	extra parameters for computations, including maxiter maximum number of iterations to be run (default:50). eps tolerance level for stopping criterion (default: 1e-6). use.exact a logical to use exact (TRUE) or approximate (FALSE) updating rules (default: FALSE).

Value

dsplaplace gives a vector of evaluated densities given samples. rsplaplace generates unit-norm vectors in \mathbf{R}^p wrapped in a list. mle.splaplace computes MLEs and returns a list containing estimates of location (mu) and scale (sigma) parameters.

Examples

# -------------------------------------------------------------------
#          Example with Spherical Laplace Distribution
#
# Given a fixed set of parameters, generate samples and acquire MLEs.
# Especially, we will see the evolution of estimation accuracy.
# -------------------------------------------------------------------
## DEFAULT PARAMETERS
true.mu  = c(1,0,0,0,0)
true.sig = 1

## GENERATE A RANDOM SAMPLE OF SIZE N=1000
big.data = rsplaplace(1000, true.mu, true.sig)

## ITERATE FROM 50 TO 1000 by 10
idseq = seq(from=50, to=1000, by=10)
nseq  = length(idseq)

hist.mu  = rep(0, nseq)
hist.sig = rep(0, nseq)

for (i in 1:nseq){
  small.data = big.data[1:idseq[i]]             # data subsetting
  small.MLE  = mle.splaplace(small.data)        # compute MLE
  
  hist.mu[i]  = acos(sum(small.MLE$mu*true.mu)) # difference in mu
  hist.sig[i] = small.MLE$sigma
}

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(idseq, hist.mu,  "b", pch=19, cex=0.5, 
     main="difference in location", xlab="sample size")
plot(idseq, hist.sig, "b", pch=19, cex=0.5, 
     main="scale parameter", xlab="sample size")
abline(h=true.sig, lwd=2, col="red")
par(opar)

Riemann documentation built on March 18, 2022, 7:55 p.m.