fdars (version 0.3.3)

kernel.brownian: Brownian Motion Covariance Function

Description

Computes the Brownian motion (Wiener process) covariance function: $$k(s, t) = \sigma^2 \min(s, t)$$

Usage

kernel.brownian(variance = 1)

Value

A covariance function object of class 'kernel_brownian'.

Arguments

variance

Variance parameter \(\sigma^2\) (default 1).

Details

The Brownian motion covariance produces sample paths that start at 0 and have independent increments. The covariance between two points equals the variance times the minimum of their positions.

This covariance is only defined for 1D domains starting at 0.

See Also

kernel.gaussian, make.gaussian.process

Examples

# Generate Brownian motion paths
cov_func <- kernel.brownian(variance = 1)
t <- seq(0, 1, length.out = 100)
fd <- make.gaussian.process(n = 10, t = t, cov = cov_func)
plot(fd)