Calculate the Drift and Diffusion of one-dimensional stochastic processes

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Description

Langevin1D calculates the Drift and Diffusion vectors (with errors) for a given time series.

Usage

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Langevin1D(data, bins, steps, sf = ifelse(is.ts(data), frequency(data), 1),
  bin_min = 100, reqThreads = -1)

Arguments

data

a vector containing the time series or a time-series object.

bins

a scalar denoting the number of bins to calculate the conditional moments on.

steps

a vector giving the τ steps to calculate the conditional moments (in samples (=τ * sf)).

sf

a scalar denoting the sampling frequency (optional if data is a time-series object).

bin_min

a scalar denoting the minimal number of events per bin. Defaults to 100.

reqThreads

a scalar denoting how many threads to use. Defaults to -1 which means all available cores.

Value

Langevin1D returns a list with thirteen components:

D1

a vector of the Drift coefficient for each bin.

eD1

a vector of the error of the Drift coefficient for each bin.

D2

a vector of the Diffusion coefficient for each bin.

eD2

a vector of the error of the Driffusion coefficient for each bin.

D4

a vector of the fourth Kramers-Moyal coefficient for each bin.

mean_bin

a vector of the mean value per bin.

density

a vector of the number of events per bin.

M1

a matrix of the first conditional moment for each τ. Rows corespond to bin, columns to τ.

eM1

a matrix of the error of the first conditional moment for each τ. Rows corespond to bin, columns to τ.

M2

a matrix of the second conditional moment for each τ. Rows corespond to bin, columns to τ.

eM2

a matrix of the error of the second conditional moment for each τ. Rows corespond to bin, columns to τ.

M4

a matrix of the fourth conditional moment for each τ. Rows corespond to bin, columns to τ.

U

a vector of the bin borders.

Author(s)

Philip Rinn

See Also

Langevin2D

Examples

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# Set number of bins, steps and the sampling frequency
bins <- 20;
steps <- c(1:5);
sf <- 1000;

#### Linear drift, constant diffusion

# Generate a time series with linear D^1 = -x and constant D^2 = 1
x <- timeseries1D(N=1e6, d11=-1, d20=1, sf=sf);
# Do the analysis
est <- Langevin1D(x, bins, steps, sf, reqThreads=2);
# Plot the result and add the theoretical expectation as red line
plot(est$mean_bin, est$D1);
lines(est$mean_bin, -est$mean_bin, col='red');
plot(est$mean_bin, est$D2);
abline(h=1, col='red');

#### Cubic drift, constant diffusion

# Generate a time series with cubic D^1 = x - x^3 and constant D^2 = 1
x <- timeseries1D(N=1e6, d13=-1, d11=1, d20=1, sf=sf);
# Do the analysis
est <- Langevin1D(x, bins, steps, sf, reqThreads=2);
# Plot the result and add the theoretical expectation as red line
plot(est$mean_bin, est$D1);
lines(est$mean_bin, est$mean_bin - est$mean_bin^3, col='red');
plot(est$mean_bin, est$D2);
abline(h=1, col='red');