dTpdWou1D | R Documentation |
Computation of the transition probability density (tpd) for a WN diffusion.
dTpdWou1D(x, x0, t, alpha, mu, sigma, maxK = 2L, expTrc = 30,
vmApprox = 0L, kt = 0, logConstKt = 0)
x |
a vector of length |
x0 |
a vector of length |
t |
a scalar containing the times separating |
alpha |
drift parameter. |
mu |
mean parameter. Must be in |
sigma |
diffusion coefficient. |
maxK |
maximum absolute value of the windings considered in the computation of the WN. |
expTrc |
truncation for exponential: |
vmApprox |
whether to use the von Mises approximation to a wrapped normal ( |
kt |
concentration for the von Mises, a suitable output from |
logConstKt |
the logarithm of the von Mises normalizing constant associated to the concentration |
See Section 3.3 in García-Portugués et al. (2019) for details. See dTpdWou
for the general case (less efficient for 2D).
A vector of size n
containing the tpd evaluated at x
.
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11222-017-9790-2")}
t <- 0.5
alpha <- 1
mu <- 0
sigma <- 1
x0 <- 0.1
dTpdWou1D(x = seq(-pi, pi, l = 10), x0 = rep(x0, 10), t = t, alpha = alpha,
mu = mu, sigma = sigma, vmApprox = 0)
# von Mises approximation
kt <- scoreMatchWnVm(sigma2 = sigma^2 * (1 - exp(-2 * alpha * t)) / (2 * alpha))
dTpdWou1D(x = seq(-pi, pi, l = 10), x0 = rep(x0, 10), t = t, alpha = alpha,
mu = mu, sigma = sigma, vmApprox = 1, kt = kt,
logConstKt = -log(2 * pi * besselI(x = kt, nu = 0,
expon.scaled = TRUE)))
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