Key Formulas in Particle Filters Explained
Here are the key formulas for particle filters, explained in plain text:
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Bayes' Rule: This rule forms the foundation of particle filters, enabling the update of beliefs about a system's state based on new observations.
The formula for Bayes' Rule is: Posterior Probability = (Likelihood * Prior Probability) / Evidence
where:
- Posterior Probability is the probability of the state given the observation
- Likelihood is the probability of the observation given the state
- Prior Probability is the initial probability of the state
- Evidence is the probability of the observation
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Likelihood Calculation: This calculation assesses how well the predicted measurements of a particle align with the actual measurements. Particles exhibiting higher likelihoods are considered more likely to represent the system's true state.
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Weight Update: The weight assigned to each particle reflects its probability of representing the true state. Weights are updated based on the likelihood that a particle's predicted measurements match the actual measurements.
The weight update formula is: New Weight = Current Weight * Likelihood
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Normalization: Following the weight update, normalization is performed to ensure that the weights sum up to 1, creating a valid probability distribution.
The normalization formula is: Normalized Weight = Weight / Sum of all Weights
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Resampling: This step is essential to prevent particle degeneracy. Particles with low weights are discarded and replaced with copies of particles possessing higher weights. This process aims to concentrate computational resources on the most promising regions of the state space.
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State Estimation: The estimated state of the system can be derived by calculating the weighted average of the particles.
The state estimation formula is: Estimated State = Sum(Weight * Particle State) for all particles
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Motion Update: Particles are moved according to the system's motion model. This model describes how the system's state evolves over time, typically incorporating some noise to account for motion uncertainties.