Particle Filters
Particle filters represent any probability distribution with weighted samples, handling nonlinearity and multiple competing hypotheses.
Beyond Gaussian assumptions
Kalman-family filters assume the state distribution is a single bell curve. When it is not, for example when a system might be in one of several distinct regimes, a particle filter is needed. It represents the distribution by a cloud of weighted samples, called particles, each a complete possible state. The cloud can take any shape, including several separate peaks.
The algorithm
- Predict: advance every particle through the model, adding process noise
- Weight: score each particle by how well it explains the new measurement
- Resample: draw a new equally weighted set, favoring high-weight particles, discarding poor ones
The weighted average of the particles is the state estimate, and their spread is the uncertainty. Because each particle is a full nonlinear model run, no linearization and no Gaussian assumption is required.
The cost and the fix
Particle filters are expensive: reliable coverage of a high-dimensional state can need enormous numbers of particles, a difficulty known as the curse of dimensionality. They also suffer degeneracy, where one particle takes almost all the weight; resampling and improved proposal distributions mitigate this. In practice particle filters are used on low- to moderate-dimensional states, or on a few critical variables, rather than on full fields, where the ensemble filter is preferred.
Where multi-modality matters
In a fusion twin, competing hypotheses are real. A diagnostic pattern might be consistent with either a benign fluctuation or the onset of an instability; a fault signature might match two different failing components. A particle filter can carry both explanations at once, with their probabilities, until data resolves which is true. That honesty about ambiguity is exactly what a twin owes an operator. See anomaly detection and disruption prediction.