sigmaDiff: High-frequency estimate of the diffusion matrix

In egarpor/sdetorus: Statistical Tools for Toroidal Diffusions

High-frequency estimate of the diffusion matrix

Description

Estimation of the \Sigma in the multivariate diffusion

dX_t=b(X_t)dt+\Sigma dW_t

by the high-frequency estimate

\hat\Sigma = \frac{1}{N\Delta}\sum_{i=1}^N(X_i-X_{i-1})(X_i-X_{i-1})^T

Usage

sigmaDiff(data, delta, circular = TRUE, diagonal = FALSE,
  isotropic = FALSE)

Arguments

data	vector or matrix of size c(N, p) containing the discretized process.
delta	discretization step.
circular	whether the process is circular or not.
diagonal, isotropic	enforce different constraints for the diffusion matrix.

Details

See Section 3.1 in García-Portugués et al. (2019) for details.

Value

The estimated diffusion matrix of size c(p, p).

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11222-017-9790-2")}

Examples

# 1D
x <- drop(euler1D(x0 = 0, alpha = 1, mu = 0, sigma = 1, N = 1000,
                  delta = 0.01))
sigmaDiff(x, delta = 0.01)

# 2D
x <- t(euler2D(x0 = rbind(c(pi, pi)), A = rbind(c(2, 1), c(1, 2)),
               mu = c(pi, pi), sigma = c(1, 1), N = 1000,
               delta = 0.01)[1, , ])
sigmaDiff(x, delta = 0.01)
sigmaDiff(x, delta = 0.01, circular = FALSE)
sigmaDiff(x, delta = 0.01, diagonal = TRUE)
sigmaDiff(x, delta = 0.01, isotropic = TRUE)

egarpor/sdetorus documentation built on March 4, 2024, 1:23 a.m.