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## Introduction to the Particle Filter Computer Practical

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**Introduction to the Particle Filter Computer Practical**Peter Jan van Leeuwen and Phil Browne**The Particle filter**Use ensemble with the weights.**What are these weights?**• For Gaussian distributed variables is is given by: • One can just calculate this value • That is all !!!**How to make particle filters useful?**• Introduce localisation to reduce the number of observations. • Use proposal-density freedom. • Several ad-hoc combinations of Particle Filters and Ensemble Kalman Filters**Bayes Theorem and the proposal density**Bayes Theorem now becomes: We have a set of particles at time n-1 so we can write and use this in the equation above to perform the integral:**The standard particle filter**Performing the integral over the sum of delta functions gives: The posterior is now given as a sum of transition densities. This is the standard particle filter, also called Sequential Importance Resampling, or SIR. This scheme is degenerate when the number of observations Is large.**The proposal transition density**Multiply numerator and denominator with a proposal density q: Note that 1) the proposal depends on the future observation, and 2) the proposal depends on all previous particles, not just one.**Between observations: relaxation**We add a relaxation term to the model equation: How strong should the relaxation be?**How are the weights affected?**Draw samples from the proposal transition density q, to find: with weights Likelihood weight Proposal weight**The equivalent-weights Particle Filter**At the last time step towards the observations make sure all weights are approximately equal: • Set a target weight • Move each particle such that it has the target weight, using: • Give each particle small random perturbation.**The weights as function of the position in state space**1 4 3 Target weight 2 5 xin What should the target weight be?**Practical: the barotropic vorticity equation**• Stochastic barotropic vorticity equation: • 256 by 256 grid - 65,536 variables • Double periodic boundary conditions • Semi-Langrangian time stepping scheme • Identical twin experiments over 1200 time steps • Observations every 50 time steps – decorrelation time of 42 • 24-48 particles**¼ Observations over half of state**Truth Mean of particle filter ensemble**Experiment**• Study influence of relaxation strength, or nudging strength (nudgefac) • Study influence of target weight, so how many particles can reach that weight, so percentage of particles to keep (keep). Looking at: • Trajectories, fields, RMSE • Histograms, pdfs**Rank Histograms**SST (observed) Meridional wind high up in Atmosphere (unobserved)