dTpdPde1D {sdetorus}

Transition probability density in 1D by PDE solving

Description

Computation of the transition probability density (tpd) of the Wrapped Normal (WN) or von Mises (vM) diffusion, by solving its associated Fokker–Planck Partial Differential Equation (PDE) in 1D.

Usage

dTpdPde1D(Mx = 500, x0, t, alpha, mu, sigma, type = "WN",
  Mt = ceiling(100 * t), sdInitial = 0.1, ...)

Arguments

Mx

size of the equispaced spatial grid in [-\pi,\pi).

x0

point giving the mean of the initial circular density, a WN with standard deviation equal to sdInitial.

t

time separating x0 and the evaluation of the tpd.

alpha

drift parameter.

mu

mean parameter. Must be in [\pi,\pi).

sigma

diffusion coefficient.

type

either "WN" or "vM".

Mt

size of the time grid in [0, t].

sdInitial

the standard deviation of the concentrated WN giving the initial condition.

...

Further parameters passed to crankNicolson1D.

Details

A combination of small sdInitial and coarse space-time discretization (small Mx and Mt) is prone to create numerical instabilities. See Sections 3.4.1, 2.2.1 and 2.2.2 in García-Portugués et al. (2019) for details.

Value

A vector of length Mx with the tpd evaluated at seq(-pi, pi, l = Mx + 1)[-(Mx + 1)].

References

García-Portugués, E., Sørensen, M., Mardia, K. V. and Hamelryck, T. (2019) Langevin diffusions on the torus: estimation and applications. Statistics and Computing, 29(2):1–22. doi:10.1007/s11222-017-9790-2

Examples

Mx <- 100
x <- seq(-pi, pi, l = Mx + 1)[-c(Mx + 1)]
x0 <- pi
t <- 0.5
alpha <- 1
mu <- 0
sigma <- 1
isRStudio <- identical(.Platform$GUI, "RStudio")
if (isRStudio) {
  manipulate::manipulate({
  plot(x, dTpdPde1D(Mx = Mx, x0 = x0, t = t, alpha = alpha, mu = 0,
                    sigma = sigma), type = "l", ylab = "Density",
       xlab = "", ylim = c(0, 0.75))
  lines(x, dTpdWou1D(x = x, x0 = rep(x0, Mx), t = t, alpha = alpha, mu = 0,
                      sigma = sigma), col = 2)
  }, x0 = manipulate::slider(-pi, pi, step = 0.01, initial = 0),
  alpha = manipulate::slider(0.01, 5, step = 0.01, initial = 1),
  sigma = manipulate::slider(0.01, 5, step = 0.01, initial = 1),
  t = manipulate::slider(0.01, 5, step = 0.01, initial = 1))
}