rotations (version 1.5)

Maxwell: The modified Maxwell-Boltzmann distribution

Description

Density, distribution function and random generation for the Maxwell-Boltzmann distribution with concentration kappa $\kappa$ restricted to the range $[-\pi,\pi)$.

Usage

dmaxwell(r, kappa = 1, nu = NULL, Haar = TRUE)
pmaxwell(q, kappa = 1, nu = NULL, lower.tail = TRUE)
rmaxwell(n, kappa = 1, nu = NULL)

Arguments

r,q

vector of quantiles.

kappa

concentration parameter.

nu

circular variance, can be used in place of kappa.

Haar

logical; if TRUE density is evaluated with respect to the Haar measure.

lower.tail

logical; if TRUE (default) probabilities are $P(X\le x)$ otherwise, $P(X>x)$.

n

number of observations. If length(n)>1, the length is taken to be the number required.

Value

dmaxwell

gives the density

pmaxwell

gives the distribution function

rmaxwell

generates a vector of random deviates

Details

The Maxwell-Boltzmann distribution with concentration $\kappa$ has density $$C_\mathrm{{M}}(r|\kappa)=2\kappa\sqrt{\frac{\kappa}{\pi}}r^2e^{-\kappa r^2}$$ with respect to Lebesgue measure. The usual expression for the Maxwell-Boltzmann distribution can be recovered by setting $a=(2\kappa)^0.5$.

References

Bingham MA, Nordman DJ and Vardeman SB (2010). "Finite-sample investigation of likelihood and Bayes inference for the symmetric von Mises-Fisher distribution." Computational Statistics & Data Analysis, 54(5), pp. 1317-1327.

See Also

Angular-distributions for other distributions in the rotations package. @cite bingham2010

Examples

r <- seq(-pi, pi, length = 500)

#Visualize the Maxwell-Boltzmann density fucntion with respect to the Haar measure
plot(r, dmaxwell(r, kappa = 10), type = "l", ylab = "f(r)")

#Visualize the Maxwell-Boltzmann density fucntion with respect to the Lebesgue measure
plot(r, dmaxwell(r, kappa = 10, Haar = FALSE), type = "l", ylab = "f(r)")

#Plot the Maxwell-Boltzmann CDF
plot(r,pmaxwell(r,kappa = 10), type = "l", ylab = "F(r)")

#Generate random observations from Maxwell-Boltzmann distribution
rs <- rmaxwell(20, kappa = 1)
hist(rs, breaks = 10)