timeseries2D: Generate a 2D Langevin process

In Langevin: Langevin Analysis in One and Two Dimensions

Description

timeseries2D generates a two-dimensional Langevin process using a simple Euler integration. The drift function is a cubic polynomial, the diffusion function a quadratic.

Usage

timeseries2D(
  N,
  startpointx = 0,
  startpointy = 0,
  D1_1 = matrix(c(0, -1, rep(0, 14)), nrow = 4),
  D1_2 = matrix(c(0, 0, 0, 0, -1, rep(0, 11)), nrow = 4),
  g_11 = matrix(c(1, 0, 0, 0, 0, 0, 0, 0, 0), nrow = 3),
  g_12 = matrix(c(0, 0, 0, 0, 0, 0, 0, 0, 0), nrow = 3),
  g_21 = matrix(c(0, 0, 0, 0, 0, 0, 0, 0, 0), nrow = 3),
  g_22 = matrix(c(1, 0, 0, 0, 0, 0, 0, 0, 0), nrow = 3),
  sf = 1000,
  dt = 0
)

Arguments

N	a scalar denoting the length of the time-series to generate.
startpointx	a scalar denoting the starting point of the time series x.
startpointy	a scalar denoting the starting point of the time series y.
D1_1	a 4x4 matrix denoting the coefficients of D1 for x.
D1_2	a 4x4 matrix denoting the coefficients of D1 for y.
g_11	a 3x3 matrix denoting the coefficients of g11 for x.
g_12	a 3x3 matrix denoting the coefficients of g12 for x.
g_21	a 3x3 matrix denoting the coefficients of g21 for y.
g_22	a 3x3 matrix denoting the coefficients of g22 for y.
sf	a scalar denoting the sampling frequency.
dt	a scalar denoting the maximal time step of integration. Default dt=0 yields dt=1/sf.

Details

The elements a_{ij} of the matrices are defined by the corresponding equations for the drift and diffusion terms:

D^1_{1,2} = ∑_{i,j=1}^4 a_{ij} x_1^{(i-1)}x_2^{(j-1)}

with a_{ij} = 0 for i + j > 5.

g_{11,12,21,22} = ∑_{i,j=1}^3 a_{ij} x_1^{(i-1)}x_2^{(j-1)}

with a_{ij} = 0 for i + j > 4

Value

timeseries2D returns a time-series object with the generated time-series as columns.

Author(s)

Philip Rinn

See Also

timeseries1D

Langevin documentation built on Oct. 19, 2021, 5:06 p.m.