# sigmaDiff: High-frequency estimate of the diffusion matrix In sdetorus: Statistical Tools for Toroidal Diffusions

## Description

Estimation of the Σ in the multivariate diffusion

dX_t=b(X_t)dt+Σ dW_t

by the high-frequency estimate

\hatΣ = \frac{1}{NΔ}∑_{i=1}^N(X_i-X_{i-1})(X_i-X_{i-1})^T.

## Usage

 1 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. doi: 10.1007/s11222-017-9790-2

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 # 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) 

sdetorus documentation built on Aug. 19, 2021, 9:06 a.m.