Langevin1D: Calculate the Drift and Diffusion of one-dimensional...
In Langevin: Langevin Analysis in One and Two Dimensions

Description Usage Arguments Value Author(s) See Also Examples

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

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

`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*)). Only used if `kernel` is `FALSE`.
`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. Only used if `kernel` is `FALSE`.
`kernel`	a logical denoting if the kernel based Nadaraya-Watson estimator should be used to calculate drift and diffusion vectors.
`h`	a scalar denoting the bandwidth of the data. Defaults to Scott's variation of Silverman's rule of thumb. Only used if `kernel` is `TRUE`.

Langevin1D returns a list with thirteen (six if kernel is TRUE) 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 Diffusion 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`. If `kernel` is `FALSE`.
`M1`	a matrix of the first conditional moment for each τ. Rows correspond to `bin`, columns to τ. If `kernel` is `FALSE`.
`eM1`	a matrix of the error of the first conditional moment for each τ. Rows correspond to `bin`, columns to τ. If `kernel` is `FALSE`.
`M2`	a matrix of the second conditional moment for each τ. Rows correspond to `bin`, columns to τ. If `kernel` is `FALSE`.
`eM2`	a matrix of the error of the second conditional moment for each τ. Rows correspond to `bin`, columns to τ. If `kernel` is `FALSE`.
`M4`	a matrix of the fourth conditional moment for each τ. Rows correspond to `bin`, columns to τ. If `kernel` is `FALSE`.
`U`	a vector of the `bin` borders. If `kernel` is `FALSE`.

Philip Rinn

Langevin2D

# 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(data = x, bins = bins, steps = steps, sf = sf)
# 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(data = x, bins = bins, steps = steps, sf = sf)
# 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")