Brownian Karhunen–Loève Modes

Compress a covariance operator into orthogonal modes and account for omitted variance.

Brownian Karhunen–Loève Modes

The Karhunen–Loève (KL) theorem states that a stochastic process can be represented as an infinite linear combination of orthogonal functions. For the standard Brownian motion on the interval [0, 1], its covariance operator is given by min(s, t).

Worked Example

Let's calculate the approximation of the covariance for two retained modes (M = 2) at s = 0.5 and t = 1.

The eigenvalues are given by \lambda_k = 1/(((k - 0.5)π)²).
For k = 1: \lambda_1 = 1/((0.5π)²) ≈ 0.4052847345693511
For k = 2: \lambda_2 = 1/((1.5π)²) ≈ 0.04503163717437235
The eigenfunctions are \phi_k(t) = √2 sin((k - 0.5)π t).
For k = 1: \phi_1(0.5) = √2 sin(0.25π) = 1.0, \phi_1(1) = √2 sin(0.5π) = √2 ≈ 1.4142135623730951
For k = 2: \phi_2(0.5) = √2 sin(0.75π) = 1.0, \phi_2(1) = √2 sin(1.5π) = -√2 ≈ -1.4142135623730951
The truncated covariance is K_M(s, t) = \sum_(k=1)^M \lambda_k \phi_k(s) \phi_k(t).
Calculate the sum:
K_2(0.5, 1) = (0.4052847345693511 × 1.0 × 1.4142135623730951) + (0.04503163717437235 × 1.0 × -1.4142135623730951) ≈ 0.5731591682507563 - 0.0636843520278618 = 0.5094748162228945
The exact Brownian covariance is min(0.5, 1) = 0.5, showing our 2-mode approximation is already quite close.

Experiment

You can interact with the Karhunen–Loève expansion using the controls provided in the Brownian Karhunen–Loève Modes module.

Change the Retained modes to observe how the truncated covariance converges to the exact Brownian covariance. Notice the decrease in the integrated omitted variance.
Modify the Mode probe to inspect the specific shape of different eigenfunctions \phi_k(t).
Adjust Probe s and Probe t to see the exact and truncated covariance at specific points in the [0, 1] interval.
Set Retained modes to 2, Probe s to 0.5, and Probe t to 1, and verify that the Truncated covariance is approximately 0.5094748162228945.

Limitations

The Karhunen–Loève expansion here is for standard Brownian motion, providing an infinite series of orthogonal deterministic functions with uncorrelated random coefficients. While truncating this series yields a mean-square optimal approximation of the process, a finite sum produces sample paths that are smooth and differentiable. True Brownian motion paths are nowhere differentiable. Thus, the truncated expansion does not fully capture the local rough behavior of the process, which is why fractional Brownian models or other techniques are often used when simulating realistic sample paths.

References

Mascagni, M. Brownian Motion and the Karhunen-Loève Expansion.
https://www.cs.fsu.edu/~mascagni/Brownianbeamer.pdf

Brownian Karhunen–Loève Modes