# 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

 ```1 2 3 4 5 6``` ```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 colums.

## Author(s)

Philip Rinn

`timeseries1D`