Brownian Karhunen–Loève Modes
Compress a covariance operator into orthogonal modes and account for omitted variance.
Free theory articleCompress a covariance operator into orthogonal modes and account for omitted variance.
Concept & interpretation
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).
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.