Bayesian filtering

Hi, everyone!!! this time I am going to talk about optimal filtering and Bayesian filtering

The term optimal filtering traditionally refers to a class of methods that can be used for estimating the state of a time-varying system which is indirectly observed through noisy measurements. The term optimal in this context refers to statistical optimality. Bayesian filtering refers to the Bayesian way of formulating optimal filtering.

The Kalman filter (Kalman, 1960b) is the closed form solution to the Bayesian filtering equations for the filtering model, where the dynamic and measurement models are linear Gaussian:

x_k=A_k-1x_k-1+q_k-1,
y_k=H_kx_k+r_k,

where x_k in Rⁿ is the state, y_k in R^m is the measurement, q_k-1 =N(0;Q_k-1) is the process noise, rk=N(0;R_k) is the measurement noise, and the prior distribution is Gaussian x0=N(m₀; P₀). The matrix A_k-1 is the transition matrix of the dynamic model and H_k is the measurement model matrix. In probabilistic terms the model is

p( x_k | x_k-1)= N( x_k | A_k-1 x_k-1; Q_k-1);
p( y_k | x_k)= N( y_k | H_k x_k; R_k);

I have implemented using Matlab, filtering using the Kalman Filter of three state space models known as:

1. Gaussian random walk
2. Car Tracking model
3. Resonator model

If you want to know more I recommend reading the following book: https://users.aalto.fi/~ssarkka/pub/cup_book_online_20131111.pdf

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